Information Flux Theory: A Reinterpretation of the Standard Model with a Single Fermion and the Origin of Gravity

1. Introduction

1.1. Status of the Standard Model and Open Questions

1.1.1. Achievements

The Standard Model (SM), established in the 1970s, is built on the gauge symmetry $\mathrm{SU}{\left(3\right)}_{\mathrm{C}}\otimes \mathrm{SU}{\left(2\right)}_{\mathrm{L}}\otimes \mathrm{U}{\left(1\right)}_{\mathrm{Y}}$ and spontaneous symmetry breaking via the Higgs mechanism. Through (1) precision tests of electroweak interactions at LEP/SLC, (2) the consistent running of parameters such as ${\alpha }_s\left(M_Z\right)$ and ${sin}^2{\theta }_W$, and (3) the complete observation of the particle spectrum—including the discovery of the Higgs boson in 2012— it has almost entirely covered the phenomenology in the 100 GeV– 10 TeV range[1]. Theoretically, it functions as a well-defined perturbative quantum field theory thanks to (i) a strictly fixed interaction structure enforced by local gauge symmetry, (ii) a commutative operator algebra on four-dimensional commutative spacetime, and (iii) the fulfillment of anomaly-cancellation conditions. Consequently, it enjoys exceptionally high experimental credibility, as demonstrated by the ${10}^{-10}$ precision of quantum electrodynamics and the unitarity tests of the CKM matrix in flavour physics.

1.1.2. Outstanding Problems

From the viewpoints of parameter minimality and an origin-based explanation, the SM leaves the following fundamental issues unresolved:

• Origin of fermion masses and mixings The Yukawa matrices $Y_f$ contain 13 mass parameters and 10 mixing parameters; their hierarchical structure (e.g., $m_t/m_u\sim {10}^5$) and the texture of the CKM matrix are not fixed intrinsically but must be supplied externally.
• Neutrino masses and CP phases The SM predicts strictly massless neutrinos, yet oscillation experiments show $\Delta m_{ij}^2\ne 0$. Whether neutrinos are Majorana or Dirac particles and the origin of lepton CP violation remain open questions[2].
• Stability and naturalness of the scalar sector The Higgs mass is quadratically sensitive to radiative corrections (the hierarchy problem); stabilisation up to ${\Lambda }_{\mathrm{Pl}}$ demands a dedicated mechanism.
• The strong-CP problem The experimental requirement ${\theta }_{\mathrm{QCD}}<{10}^{-10}$ is not naturally accommodated within the SM.
• Consistency with gravitational and cosmological phenomena Cosmological observables such as dark matter, dark energy, and inflation are inadequately explained by SM+GR alone, calling for unification at the quantum-gravity scale.
• Multiplicity of free parameters and aesthetic concerns The $\mathcal{O}\left(20\right)$ free parameters of the SM violate the principle of theoretical minimality, and the search for a more fundamental reduction principle is ongoing.

1.1.3. Position of the Present Work

The Information Flux Theory (IFT) proposed here aims to resolve these outstanding issues by

• simultaneously describing all fermion families with a single fermion operator, automatically generating the Yukawa matrices via an exponential rule and operator contraction;
• reproducing masses, mixings, and the Higgs sector without additional parameters while explicitly preserving the gauge group $\mathrm{SU}{\left(3\right)}_{\mathrm{C}}\otimes \mathrm{SU}{\left(2\right)}_{\mathrm{L}}\otimes \mathrm{U}{\left(1\right)}_{\mathrm{Y}}$;
• introducing a Unified Evolution Equation as the foundational equation, naturally extendable to gravitational and cosmological terms.

In this way, IFT seeks to preserve the successes of the SM while simultaneously resolving the fundamental problems (i)–(vi) in one stroke. This section organises the achievements and limitations of the SM, and the construction of IFT is developed in the following sections.

1.2. Conceptual Basis of Information Flux Theory

1.2.1. Core Idea—A Single Fermion and Self-Information Flux

All observable quantities in the universe can be reduced to the conserved 4-vector

$$J^{\mu }\left(x\right):=\overline{\Psi }\left(x\right){\gamma }^{\mu }\Psi \left(x\right),{\partial }_{\mu }{J}^{\mu }=0,$$

namely the self-information flux of a single fermion Ψ. Here $\Psi$ is the unique field in the fundamental representation of $\mathrm{SU}{\left(3\right)}_C\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$. “Generations’’ are replaced by a series of projectors ${\Psi }_n={\Pi }_n\Psi$ with ${\Pi }_n^2={\Pi }_n$ and ${\Pi }_m{\Pi }_n=0(m\ne n)$, while the mass hierarchy is fixed by an exponential rule $m_n\propto {\epsilon }^n$ ($\epsilon$: information-dissipation rate). The Yukawa matrices are not inputs but outcomes, drastically reducing the free constants of the Standard Model.

1.2.2. Unified Evolution Equation (UEE)

The time evolution of the information flux obeys the Lindblad (GKLS) equation

$$\dot{\rho }=-i[\mathcal{H},\rho ]+\sum _{\alpha }\left(L_{\alpha }\rho L_{\alpha }^{†}-\frac{1}{2}\{L_{\alpha }^{†}{L}_{\alpha },\rho \}\right),$$

(1)

such that in the IR limit $\mathcal{H}\to {\mathcal{H}}_{\mathrm{GR}}$ it coincides with the Einstein–Hilbert action, while in the UV limit $\mathcal{H}\to {\mathcal{H}}_{\mathrm{SM}}$, thereby linking quantum theory and gravity through a single principle.

1.2.3. Masses and Mixings from Minimal Degrees of Freedom

With dissipators chosen as $L_{\alpha }\simeq \sqrt{\gamma }{\Pi }_n\Psi {\Pi }_m$ ($\gamma$: dissipation coefficient), mass generation and mixing are induced automatically through the contractions of ${\Pi }_n$. Because the construction employs only the gauge-covariant derivative $D_{\mu }={\partial }_{\mu }-i{g}_a{A}_{\mu }^a{T}^a$, symmetry is preserved.

1.2.4. Methodological Outline

The theory is developed through:

(i)

a rigorous derivation of the UEE and anomaly-cancellation conditions,

(ii)

deduction of exponential-rule Yukawa matrices from the projector series,

(iii)

comparison of the dissipation rate $\epsilon$ with experimental data

and consequently shown to reproduce the Standard Model in its entirety.

1.3. Unified Evolution Equation and Construction Method of the Single-Fermion Framework

1.3.1. Design Principle—Coexistence of Conservation and Dissipation

This theory is founded on the dual principle that “local gauge quantities are conserved, yet environmental dissipation organises the system.” The dynamics of the density operator $\rho \left(t\right)$ are given by

$$\begin{array}{|c|}\hline \dot{\rho }=-i[\mathcal{H},\rho ]+{\displaystyle \sum _{\alpha }}\left(L_{\alpha }\rho L_{\alpha }^{†}-\frac{1}{2}\{L_{\alpha }^{†}{L}_{\alpha },\rho \}\right)\\ \hline\end{array}$$

(UEE)

of GKLS type. The trace $\rho =1$ is strictly conserved, while the von Neumann entropy satisfies $\dot{S}\left(\rho \right)=\frac{1}{2}{\sum }_{\alpha }[L_{\alpha },L_{\alpha }^{†}]\rho \ge 0$, explicitly manifesting time irreversibility.

1.3.2. Minimal Building Blocks

Field operators are placed in ${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_{\mathrm{Mink}}\otimes {\mathcal{H}}_{\mathrm{int}}$, and only the gauge-covariant derivative $D_{\mu }={\partial }_{\mu }-i{g}_a{A}_{\mu }^a{T}^a$ is employed. The effective Hamiltonian is

$$\mathcal{H}=\int d^{3}x\overline{\Psi }\left(-i{\gamma }^0{\gamma }^j{D}_j\right)\Psi +{\mathcal{H}}_{\mathrm{gauge}},$$

with no mass term at the outset; masses are generated automatically by the projector contractions described below.

1.3.3. Single Fermion and Projector Series

The 12 SM fermions are unified into a single Dirac operator $\Psi$. “Generations’’ are represented by the projector series

$${\Psi }_n={\Pi }_n\Psi ,{\Pi }_n^2={\Pi }_n,{\Pi }_m{\Pi }_n=0(m\ne n).$$

Choosing the dissipators as $L_{\alpha }\propto \sqrt{\gamma }{\Pi }_n\Psi {\Pi }_m$, one induces the exponential rule $m_n=m_0{\epsilon }^n,\epsilon =\gamma /\Lambda ,$ so that the Yukawa matrices are determined as a consequence of $\epsilon$.

1.3.4. Construction Algorithm (Outline)

1)

Anomaly Cancellation: Impose ${\sum }_{\alpha }[T^a,L_{\alpha }]=0$ to fix the gauge representations identical to those of the SM.

2)

Projector Contraction: Use ${\Pi }_m\Psi {\Pi }_n={\delta }_{mn}{\Psi }_n$ to derive the exponential-rule Yukawa matrices.

3)

RG Consistency: Require ${\beta }_g=0,{\beta }_{\epsilon }=0$ to reproduce ${\alpha }_s\left(M_Z\right)$ and ${sin}^2{\theta }_W$ within experimental accuracy.

4)

Gravitational Limit: Add $L_{\mathrm{grav}}\sim \sqrt{{\gamma }_G}R\Psi$ and recover the Einstein equation in the IR.

The following chapters rigorously formalise each of these steps and perform detailed comparisons with experimental data.

1.4. Bridge to Chapter 2: Introduction of the Five-Operator Functionally Complete Set

1.4.1. Position and Purpose

We have already emphasised that the dynamics of the universe can be described solely with the single fermion $\Psi$. However, for clarity it is preferable to modularise the operator content so that physical functions become visible. Chapter 2 therefore adopts the set

$${\mathcal{S}}_5=\left\{D,{\Pi }_n,V_n,\Phi ,R\right\},$$

a five-operator functionally complete set. The aim is to establish the Functional-Completeness Proposition (5-Op)—that “five operators suffice to reconstruct the full functionality of $\Psi$”—rather than to assert minimality or uniqueness. This subsection organises (i) the roles of the five operators, (ii) the proof roadmap of Chapter 2, and (iii) the links to subsequent chapters, thereby bridging inter-chapter logic.

1.4.2. Five Operators and Their Roles

At the beginning of Chapter 2 an elimination experiment shows that omitting any element of ${\mathcal{S}}_5$ obscures specific functionalities. The correspondence is summarised in Table 1.

1.4.3. Claim of Functional Completeness

Although the theory closes when folded into the single $\Psi$, the introduction of ${\mathcal{S}}_5$ dramatically enhances functional separation, readability, and computational convenience. The conclusion of the elimination experiment is that ${\mathcal{S}}_5$ constitutes a usefully small basis, though not minimal, for decomposing the functions of $\Psi$ into information-theoretic, dissipative, and geometric sectors.

1.4.4. Structure and Roadmap of Chapter 2

• §2.1 Declaration Presents the Functional-Completeness Proposition (5-Op).
• §2.2 Foundations Defines $C^{*}$-algebras, CPTP maps, and fractal measures.
• §2.3–2.7 Constructs each operator and verifies its assigned role.
• §2.8 Proof of Functional Completeness Demonstrates algebraic closure and preservation of CPTP maps.
• §2.9 Bridge Specifies where these operators are used in later chapters.

1.4.5. Links to Subsequent Chapters

• Chapter 3 — With $\{D,{\Pi }_n,V_n\}$ proves the Three-Form Equivalence Theorem (operator, variational, and field-equation forms).
• Chapters 4–6 — Analyse information dissipation and measurement processes (thermalisation, quantum Zeno effect, etc.).
• Chapters 7–10 — Derive Yukawa matrices and the mass hierarchy from the exponential rule of ${\Pi }_n$ and $V_n$.
• Chapters 11–13 — Use $\Phi$ and R to coherently treat GR reduction, the BH information problem, and cosmological parameters.

1.4.6. Summary

The five-operator functionally complete set decomposes the full behaviour of $\Psi$ into the aspects of time evolution, projection, dissipation, geometry, and vacuum stability. Hereafter, this paper adopts ${\mathcal{S}}_5$ as the standard set for explanation and calculation, reintegrating it into $\Psi$ where necessary to streamline the discussion.

2. Five Operators and the Canonical Decomposition Theorem (Functional Completeness)

2.1. Statement of the Theorem and Proof Strategy

2.1.1. Introduction and Notational Conventions [3,4,5]

For the sake of visual clarity and computational convenience, the functions contained in the single fermion $\Psi$ are operationally partitioned into the following five-operator set

$$\left\{D,{\Pi }_n(n=1,\dots ,18),V_n=\sqrt{\gamma }{\Pi }_n,\Phi ,R\right\},(\gamma >0),$$

denoted by ${\mathcal{S}}_5$. Here D — reversible generator, ${\Pi }_n$ — mutually orthogonal projection operators, $V_n$ — GKLS-type dissipative jump operators, $\Phi$ — scalar field with normalised four-gradient (to be specified in Eq. (3)), R — zero-area resonance kernel with exponential area convergence.

This subsection declares:

• that ${\mathcal{S}}_5$ provides a canonical decomposition (functional completeness) whose elements satisfy all functional requirements without redundancy;
• the existence of a bijective map

  $$\mathcal{G}:\Phi ⟼(D,{\Pi }_n,V_n,R)$$

  between the scalar $\Phi$ and the remaining operators ($\Phi$ Generating Map Theorem);
• that omitting any element of ${\mathcal{S}}_5$ breaks one of the functional requirements, making it the minimal practical basis that preserves all functions without loss.

A roadmap for the proofs is also provided.

2.1.2. Theorem 2.1 — Canonical Decomposition Theorem and $\Phi$ Generating Map Theorem [6,7]

Theorem 1

(Canonical Decomposition Theorem (Functional Completeness) and $\Phi$ Generating Map Theorem).

(i) 

On a Hilbert space $\mathcal{H}$ there exists a set of operators $(D,{\Pi }_n,V_n,R)$ simultaneously satisfying the following conditions. Any two such sets are related by a unitary transformation ${\Pi }_n\to U{\Pi }_n{U}^{†}(U\in \mathrm{U}\left(\mathcal{H}\right))$ and a rescaling of γ:

(a)

Reversible unitary generator D — self-adjoint, $\left[D\right]=0$, locally Lorentz covariant.

(b)

Measurement basis $\left\{{\Pi }_n\right\}$ — ${\Pi }_n{\Pi }_m={\delta }_{nm}{\Pi }_n$, ${\sum }_{n=1}^{18}{\Pi }_n=I$.

(c)

Dissipative jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ — generate a CPTP semigroup.

(d)

GR-reduction scalar Φ — normalised four-gradient ${\nabla }_a\Phi {\nabla }^a\Phi =1$.

(e)

BH information-retention kernel R — zero-area kernel with area-exponential convergence $\parallel R\parallel \le A{e}^{-\lambda A}$ and information-preservation constraint $\left[R\rho \right]=0$.

(ii) 

If a scalar Φ satisfies

$${\nabla }_a\Phi {\nabla }^a\Phi =1,$$

(2)

then the map $\mathcal{G}:\Phi \mapsto (D,{\Pi }_n,V_n,R)$ is bijective. The inverse map ${\mathcal{G}}^{-1}$ is uniquely given by

$$\Phi \left(x\right)={\int }^x\sqrt{g_{ab}{J}^a{J}^b}ds,J^a:={{ϵ}^a}_{bcd}\left({\Pi }_n{\nabla }^b{\Pi }_n{\nabla }^c{\Pi }_n{\nabla }^d{\Pi }_n\right).$$

(iii) 

Removing any single element of ${\mathcal{S}}_5$ results in the loss of at least one functional requirement—reversible unitarity, CPTP dissipation, measurement basis, GR reduction, or BH information-retention/vacuum stability. Hence ${\mathcal{S}}_5$ is apractically irreducible basisthat preserves all functionality.

2.1.3. Overview of the Proof Strategy [8,9]

(S1) 

Uniqueness of $\Phi$ normalisation — Eq. (3) determines $\Phi$ up to an additive constant and an overall sign.

(S2) 

Construction of the generating map $\mathcal{G}$ — Starting from $\Phi$, sequentially define

$$D:=i\left[\cancel{\nabla },·\right],{\Pi }_n:={\chi }_{{\Omega }_n}(\Phi ),V_n:=\sqrt{\gamma }{\Pi }_n,R:=\underset{A\to 0}{lim}{A}^{-1}{K}_A[\Phi ],$$

and verify conditions (a)–(e) (§2.3–§2.7).

(S3) 

Elimination of redundant degrees of freedom — Show that conditions (a)–(e) fix all degrees of freedom except for unitary transformations and scale rescalings, which reduce to projector equivalence classes.

(S4) 

Construction of the inverse map ${\mathcal{G}}^{-1}$ — Prove that $(D,{\Pi }_n,V_n,R)$ uniquely reconstruct $\Phi$ via the $J^a$-current integral formula.

Conclusion

The five-operator set ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,\Phi ,R\}$ constitutes a functionally complete basis for the single-fermion UEE theory, jointly implementing the five principal functions— reversible unitarity, CPTP dissipation, measurement basis, GR reduction, and BH information-retention + vacuum stability— without mutual interference. A bijective map $\mathcal{G}$ exists between this set and the scalar field $\Phi$, enabling flexible transitions between operator-decomposed and $\Psi$-reintegrated representations. The subsequent sections provide detailed constructions of each operator and line-by-line proofs.

2.2. Mathematical Preliminaries: C*-Algebras, CPTP Semigroups, and Tetrad Normalization

In this subsection we arrange the mathematical foundations necessary to construct the five-operator set ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,\Phi ,R\}$ rigorously and to prove the Canonical Decomposition Theorem (Theorem 1). The topics covered are

• C*-algebras and GNS representations,
• Completely positive trace-preserving (CPTP) maps and the Kraus representation,
• Quantum dynamical semigroups generated by GKLS operators,
• Four-gradient–normalised scalars and tetrad construction.

2.2.1. Basics of C*-Algebras and GNS Representation [10,11,12]

Definition 1

(C*-Algebra). A norm-complete *-algebra $(\mathcal{A},\parallel ·\parallel ,*)$ that satisfies the spectral condition $\parallel A^{*}{A\parallel =\parallel A\parallel }^2$ is called aC*-algebra.

Lemma 1

(Uniqueness of the GNS Representation). For a positive linear functional $\omega :\mathcal{A}\to \mathbb{C}$, the GNS triple $\left({\pi }_{\omega },{\mathcal{H}}_{\omega },|{\Omega }_{\omega }\rangle \right)$ constructed from ω is unique up to unitary equivalence.

Proof. 

Let ${\mathcal{N}}_{\omega }:=\{A\in \mathcal{A}\mid \omega \left(A^{*}A\right)=0\}$. On the quotient $\mathcal{A}/{\mathcal{N}}_{\omega }$ introduce the inner product ${\langle \left[A\right],\left[B\right]\rangle }_{\omega }:=\omega \left(A^{*}B\right)$. Completing this space yields ${\mathcal{H}}_{\omega }$. The map ${\pi }_{\omega }\left(A\right)\left[B\right]:=\left[AB\right]$ is a *-homomorphism, and the standard argument gives the claimed uniqueness.    □

2.2.2. Completely Positive Trace-Preserving Maps and the Kraus Representation [13,14,15,16]

Definition 2

(CPTP Map). For the finite-dimensional C*-algebra $\mathcal{A}=B\left(\mathcal{H}\right)$, a linear map $\mathcal{E}:\mathcal{A}\to \mathcal{A}$ is calledcompletely positive and trace-preserving(CPTP) if, for every $n\in \mathbb{N}$, $\mathcal{E}\otimes {\mathrm{id}}_n$ is positive and $\left[\mathcal{E}\right(A\left)\right]=\left[A\right]$ holds.

Theorem 2

(Kraus Representation Theorem). A linear map $\mathcal{E}$ is CPTPiffthere exists a finite set $\left\{K_{\alpha }\right\}\subset \mathcal{A}$ such that

$$\mathcal{E}\left(A\right)=\sum _{\alpha }{K}_{\alpha }A{K}_{\alpha }^{†},\sum _{\alpha }{K}_{\alpha }^{†}{K}_{\alpha }=\mathbb{1}.$$

Proof. 

Diagonalise the Choi matrix $C_{\mathcal{E}}:={\sum }_{ij}|i\rangle \langle j|\otimes \mathcal{E}\left(\right|i\rangle \langle j\left|\right)$ as $C_{\mathcal{E}}={\sum }_{\alpha }|{\varphi }_{\alpha }\rangle \langle {\varphi }_{\alpha }|$. Then define $K_{\alpha }:=\langle \alpha |{\varphi }_{\alpha }\rangle$, which serve as Kraus operators. The converse follows from the Choi–Jamiołkowski isomorphism.    □

2.2.3. GKLS Generators and Quantum Dynamical Semigroups [17,18,19,20]

Theorem 3

(GKLS Generator). Let ${\left\{{\mathcal{T}}_t\right\}}_{t\ge 0}$ be a CPTP semigroup with continuous parameter $t\ge 0$. Its infinitesimal generator $L:={\displaystyle \frac{d}{dt}}{|}_{t=0}{\mathcal{T}}_t$ necessarily takes the form

$$L\left[\rho \right]=-i[H,\rho ]+\sum _{\alpha }\left(L_{\alpha }\rho L_{\alpha }^{†}-\frac{1}{2}\{L_{\alpha }^{†}{L}_{\alpha },\rho \}\right),$$

and conversely, any such $H=H^{†}$ and set $\left\{L_{\alpha }\right\}$ uniquely determine the semigroup.

Proof. 

Follow the standard proof combining Lindblad’s matrix-element calculation with the diagonalisation method of Gorini–Kossakowski–Sudarshan–Lindblad.    □

2.2.4. Four-Gradient–Normalised Scalars and Tetrad Construction

Definition 3

(Four-Gradient–Normalised Scalar). A scalar field Φ satisfying

$${\nabla }_a\Phi {\nabla }^a\Phi =1$$

is called afour-gradient–normalised scalar. Defining the unit timelike vector $u_a:={\nabla }_a\Phi$ and choosing an orthonormal spatial triad ${\left\{e_i^a\right\}}_{i=1}^3$ orthogonal to $u_a$, one obtains a uniquely determined tetrad $e_{\mu }^a=(u^a,e_i^a)$.

Lemma 2

(Uniqueness of the Tetrad). Under the above normalisation, $e_{\mu }^a$ is unique up to local $\mathrm{SO}\left(3\right)$ rotations.

Proof. 

Since $u^a$ fixes the timelike direction, the remaining freedom is precisely the three-dimensional rotation in the spatial subspace.    □

2.2.5. Conclusion and Bridge to Subsequent Sections

In this subsection we have systematically organised (i) C*-algebras and GNS representations, (ii) CPTP maps and the Kraus representation, (iii) quantum dynamical semigroups generated by GKLS operators, (iv) four-gradient–normalised scalars and tetrad construction. These tools prepare us to construct and canonicalise

$${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,\Phi ,R\}$$

from the scalar field $\Phi$ in the next sections and to prove functional completeness at the line-by-line level.

2.3. Normalization of the Master Scalar $\Phi$ and the Generating Map

2.3.1. Normalization Condition and Phase Degrees of Freedom [21,22]

The master scalar $\Phi :\mathcal{M}\to \mathbb{R}$, which lies at the heart of the single-fermion UEE, satisfies on the space–time manifold $(\mathcal{M},g_{ab})$

$$\begin{array}{|c|}\hline {\nabla }_a\Phi {\nabla }^a\Phi =1\\ \hline\end{array}$$

(3)

This condition guarantees that

• $\Phi$ is a Cauchy time function;
• its level sets possess a unit normal $u_a:={\nabla }_a\Phi$;
• $\Phi$ is unique up to the phase freedoms $\Phi \to \Phi +c$ and $\Phi \to -\Phi$.

Lemma 3

(Uniqueness of $\Phi$). A pure, integrable scalar field Φ satisfying (3) is unique except for a constant shift and an overall sign.

Proof. 

Set $u_a:={\nabla }_a\Phi$; then $u_a{u}^a=1$ and—by the Frobenius condition— $u_{[a}{\nabla }_b{u}_{c]}=0$. Hence $\Phi$ coincides with the proper time $\tau$ along $u^a$, leaving only the freedoms $\tau \mapsto \tau +c$ and $\tau \mapsto -\tau$.    □

2.3.2. Mapping from $\Phi$ to the Tetrad [23,24]

Definition 4

($\Phi$-Induced Tetrad). Define ${e^a}_0:=u^a={\nabla }^a\Phi$ and, with ${h^a}_b:={\delta }_b^a-u^a{u}_b$, set

$${e^a}_{\widehat{ı}}:={h^a}_b{\mathcal{L}}_u^{\widehat{ı}-1}{u}^b,\widehat{ı}=1,2,3.$$

Gram–Schmidt orthonormalisation then yields the tetrad ${\left\{{e^a}_{\mu }\right\}}_{\mu =0}^3$.

Lemma 4

($\Phi$–Tetrad Correspondence). Under condition (3), Φ and the tetrad ${e^a}_{\mu }$ are in one-to-one correspondence.

Proof. 

The relation ${e^a}_0=u^a={\nabla }^a\Phi$ follows immediately. The spatial triad ${e^a}_{\widehat{ı}}$ is uniquely fixed as an orthonormal basis of $h_{ab}$; conversely, line integration of $u_a$ reconstructs $\Phi \left(x\right)={\int }_{\gamma }{u}_a{\xi }^a$.    □

2.3.3. Construction of the $\Phi$ Generating Map [25,26]

From the master scalar $\Phi$ we define the generating map $\mathcal{G}$ that constructs the operator set ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,R\}$ (excluding $\Phi$ itself):

$$\begin{array}{cc}\hfill D& :=i{\gamma }^{\mu }{e_{\mu }}^a{\nabla }_a,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\Pi }_n& :=\frac{1}{2}\left[1+{\sigma }_n\left(u_a{\Gamma }^a-{\lambda }_n\right)\right],n=1,\dots ,18,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill V_n& :=\sqrt{\gamma }{\Pi }_n,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill R& :=\underset{A\to 0}{lim}{\displaystyle \frac{1}{A}}exp\left[-A{\mathcal{L}}_u\right]\hfill \end{array}$$

(7)

Here ${\gamma }^{\mu }$ are the Dirac matrices, ${\sigma }_n=\pm 1$ encode the exponential rule, and ${\lambda }_n$ are real constants uniquely fixed by the Yukawa hierarchy indices $\{0,1,3,5,\dots \}$.

2.3.4. Invertibility of the Generating Map [27]

Theorem 4

($\Phi$ Generating Map Theorem). The map $\mathcal{G}:\Phi \mapsto (D,{\Pi }_n,V_n,R)$ is bijective. Its inverse is uniquely given by

$$\Phi \left(x\right)={\int }_{x_0}^x\sqrt{g_{ab}{J}^a{J}^b}\xi ,J^a:={{ϵ}^a}_{bcd}\left({\Pi }_n{\nabla }^b{\Pi }_n{\nabla }^c{\Pi }_n{\nabla }^d{\Pi }_n\right)$$

(8)

Proof. 

Injectivity: If $\Phi \ne {\Phi }^{\prime }$ then $u_a\ne u_a^{\prime }$, hence the tetrads differ and at least D differs, so $\mathcal{G}(\Phi )\ne \mathcal{G}\left({\Phi }^{\prime }\right)$.

Surjectivity: Suppose a set $(D,{\Pi }_n,V_n,R)$ satisfies (4)–(7). Then $u_a:={e_a}^0$ is a closed one-form, so there exists $\Phi$ with $u_a={\nabla }_a\Phi$, uniquely determined by (8).    □

2.3.5. Conclusion

In this subsection we have proved at the line-by-line level (1) that under the normalization condition (3) the scalar $\Phi$ is unique up to phase freedom; (2) that the explicit formulas (4)–(7) construct ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,R\}$ from $\Phi$; and (3) that the mapping $\mathcal{G}$ is invertible. Hence the master scalar $\Phi$ is established as the absolute generator of the single-fermion UEE.

2.4. Canonical Form of the Reversible Generator $D={\mathcal{G}}_D[\Phi ]$

2.4.1. Definition and Assumptions [28]

Definition 5

($\Phi$-Induced Dirac Operator). For the tetrad $\left\{{e^a}_{\mu }\right\}$ induced by the four-gradient–normalised scalar Φ (see Lemma 2), define the reversible generator ($\Phi$-induced Dirac operator) by

$$\begin{array}{|c|}\hline D:=i{\gamma }^{\mu }{e_{\mu }}^a\left({\nabla }_a+\frac{1}{4}{{\omega }_a}^{bc}{\gamma }_{[b}{\gamma }_{c]}\right)\\ \hline\end{array}$$

(9)

In this subsection we show that (9) is the canonical form that simultaneously satisfies

• self-adjointness,
• local Lorentz covariance,
• the fixed point ${\beta }_D=0$.

2.4.2. General Candidate and the Self-Adjointness Condition [29]

A general first-order spinor operator can be written as

$$\tilde{D}=i{\gamma }^{\mu }{e_{\mu }}^a\left({\nabla }_a+\frac{1}{4}{{\omega }_a}^{bc}{\gamma }_{[b}{\gamma }_{c]}+A_a+i{B}_a{\gamma }_5\right)+M+i{M}_5{\gamma }_5,$$

(10)

where $A_a,B_a$ are vector fields and $M,M_5$ are scalar fields.

Lemma 5

(Self-Adjointness Criterion). The operator $\tilde{D}$ is self-adjoint with respect to the Dirac inner product $(\psi ,\phi ):=\int \overline{\psi }\phi \sqrt{-g}{\mathrm{d}}^{4}x$ (${\tilde{D}}^{†}=\tilde{D}$) iff

$$A_a=0,B_a=0,M=0,M_5=0.$$

Proof. 

Take the Hermitian adjoint using ${\left({\gamma }^{\mu }\right)}^{†}={\gamma }^0{\gamma }^{\mu }{\gamma }^0$. Comparing the coefficients of $\tilde{D}-{\tilde{D}}^{†}$, any of the four fields left non-zero would yield an anti-Hermitian contribution, which is forbidden.    □

2.4.3. Requirement of Local Lorentz Covariance [21]

Dirac spinors transform under the double-cover representation of $\mathrm{SL}(2,\mathbb{C})$. For $\tilde{D}$ to be covariant, the extra terms in (10)— $A_a,B_a,M,M_5$—must be Lorentz scalars; by Lemma 5 they are all zero, reducing the operator to (9).

Lemma 6

(Torsion-Free Spin Connection). The spin connection ${{\omega }_a}^{bc}$ of the tetrad induced by Φ coincides with the Levi-Civita connection and satisfies torsion-free condition ${T^a}_{\left[bc\right]}=0$.

Proof. 

From ${\nabla }_a\Phi {\nabla }^a\Phi =1$ and the Frobenius condition $u_{[a}{\nabla }_b{u}_{c]}=0$ with $u_a:={\nabla }_a\Phi$, the torsion three-form in Cartan’s structure equation vanishes.    □

2.4.4. ${\beta }_D=0$ Fixed Point [30,31]

For the reversible generator the effective action $S_D=\int \overline{\psi }D\psi \sqrt{-g}{\mathrm{d}}^{4}x$ has 1-loop $\beta$-function

$${\beta }_D={\displaystyle \frac{N_f-{\tilde{N}}_f}{24{\pi }^2}}{M}^3,$$

where $N_f$ is the number of fermionic degrees of freedom and ${\tilde{N}}_f:=16\mathrm{Tr}\left(B_a{B}^a\right)$. With $M=B_a=0$ from Lemma 5 we obtain

$$\begin{array}{|c|}\hline {\beta }_D=0\\ \hline\end{array}.$$

2.4.5. Canonical-Form Theorem

Theorem 5

(Canonical Form of the Reversible Generator). Given the tetrad induced by Φ, any first-order Dirac operator that simultaneously satisfies

1. 

self-adjointness,

2. 

local Lorentz covariance,

3. 

the fixed point ${\beta }_D=0$,

is equivalent to (9) up to unitary projector equivalence $D\mapsto UD{U}^{†}$ with $U\in \mathrm{U}\left(\mathcal{H}\right)$.

Proof. 

Starting from the general form (10) and applying Lemmas 5 and 6 in succession, all surplus parameters are removed except for a phase and projector equivalence. These do not affect the physics, leaving (9) as the unique canonical form.    □

2.4.6. Conclusion

The reversible generator D is fixed uniquely—up to projector equivalence—by the mapping ${\mathcal{G}}_D[\Phi ]$ from the normalised scalar $\Phi$. Its explicit form is

$$D=i{\gamma }^{\mu }{e_{\mu }}^a\left({\nabla }_a+\frac{1}{4}{{\omega }_a}^{bc}{\gamma }_{[b}{\gamma }_{c]}\right),$$

the only first-order Dirac operator that simultaneously fulfils self-adjointness, local Lorentz covariance, and the fixed-point condition ${\beta }_D=0$.

2.5. Pointer Projector Family ${\Pi }_n={\mathcal{G}}_{\Pi }[\Phi ]$ and Minimality

2.5.1. Definition of the Projector Family and the Internal Hilbert Space [32,33]

Definition 6

(Internal Hilbert Space). The internal degrees of freedom of Standard-Model fermions are the direct product of colour $(dim=3)$, weak isospin $(dim=2)$, and generation $(dim=3)$:

$${\mathcal{H}}_{\mathrm{int}}:={\mathbb{C}}_{\mathrm{color}}^3\otimes {\mathbb{C}}_{\mathrm{weak}}^2\otimes {\mathbb{C}}_{\mathrm{generation}}^3\simeq {\mathbb{C}}^{18}.$$

We choose an orthonormal basis $\left\{|c_i\rangle \otimes |w_j\rangle \otimes |g_k\rangle \right\}(i=1:3,j=1:2,k=1:3).$

Definition 7

(Pointer Projector Operators). For the triple index $n=(i,j,k)$ define

$$\begin{array}{|c|}\hline {\Pi }_{ijk}:=\left(|c_i{w}_j{g}_k\rangle \langle c_i{w}_j{g}_k|\right)\\ \hline\end{array},n\equiv (i,j,k),n=1,\dots ,18.$$

(11)

Collectively we denote the 18 projectors by ${\left\{{\Pi }_n\right\}}_{n=1}^{18}$.

2.5.2. Verification of Orthogonality and Completeness [34,35]

Lemma 7

(Orthogonality). For any $n\ne m$ one has ${\Pi }_n{\Pi }_m=0$, and ${\Pi }_n^2={\Pi }_n$.

Proof. 

Equation (11) defines one-dimensional projectors, so ${\Pi }_n^2={\Pi }_n$. Because the basis vectors are orthogonal, the product vanishes for $n\ne m$.    □

Lemma 8

(Completeness). ${\displaystyle \sum _{n=1}^{18}{\Pi }_n={\mathbb{1}}_{\mathrm{int}}.}$

Proof. 

The 18 basis vectors form an orthonormal system spanning ${\mathbb{C}}^{18}$; hence the projectors give a complete resolution of the identity.    □

2.5.3. Minimality Theorem [36]

Theorem 6

(Minimality of the Pointer Projector Family). Any projector family satisfying simultaneously

1. 

orthogonality: ${\Pi }_n{\Pi }_m={\delta }_{nm}{\Pi }_n$,

2. 

completeness: ${\sum }_n{\Pi }_n={\mathbb{1}}_{\mathrm{int}}$,

3. 

each image of ${\Pi }_n$ is one-dimensional,

requires at least 18 projectors. The set $\left\{{\Pi }_{ijk}\right\}$ defined in (11) is thereforeminimalin both number and structure.

Proof. 

Since $dim{\mathcal{H}}_{\mathrm{int}}=18$, a complete resolution by one-dimensional projectors necessitates at least 18 of them. Lemmas 7 and 8 show that (11) meets conditions (1) and (2); with fewer projectors completeness would be lost.    □

2.5.4. Generating Map ${\mathcal{G}}_{\Pi }$ from $\Phi$ [32]

On each level surface ${\Sigma }_{\tau }$ of the master scalar $\Phi$ we employ the reference tetrad ${e^a}_{\widehat{ı}}$ and define an index map ${\Xi }_{\mathrm{int}}:{\Sigma }_{\tau }\to \{1,\dots ,18\}$ (unique from the topological structure and group representations). We set

$$\begin{array}{|c|}\hline {\mathcal{G}}_{\Pi }[\Phi ]:{\Pi }_n\left(x\right)={\chi }_{\{{\Xi }_{\mathrm{int}}\left(x\right)=n\}}|c_i{w}_j{g}_k\rangle \langle c_i{w}_j{g}_k|\\ \hline\end{array},n=1,\dots ,18.$$

Thus the family $\left\{{\Pi }_n\right\}$ is generated from $\Phi$ bijectively.

2.5.5. Uniqueness up to Projector Equivalence

Lemma 9

(Uniqueness under Projector Equivalence). With Φ fixed, the projector family $\left\{{\Pi }_n\right\}$ is unique up to unitary conjugation $U{\Pi }_n{U}^{†}={\Pi }_n$ ($U\in \mathrm{U}\left({\mathcal{H}}_{\mathrm{int}}\right)$).

Proof. 

Unitary transformations preserving conditions (1)–(3) are restricted to diagonal unitaries that attach phases to each basis vector. Physical observables are phase-independent, so these families are considered equivalent.    □

2.5.6. Conclusion

The pointer projector family ${\left\{{\Pi }_{ijk}\right\}}_{i=1..3,j=1..2,k=1..3}$ is the minimal set of 18 projectors satisfying simultaneously (i) orthogonality, (ii) completeness, and (iii) one-dimensional images. It can be generated uniquely—up to projector equivalence—from the master scalar $\Phi$ via the map ${\mathcal{G}}_{\Pi }$. Hence the distinctions of fermion “generation, colour, and weak isospin’’ in the UEE appear as internal labels automatically endowed by the topological structure of $\Phi$.

2.6. Jump Operators $V_n=\sqrt{\gamma }{\Pi }_n$ and Canonical Dissipation

2.6.1. Definition of the Jump Operators [17,18]

Given the pointer projector family ${\left\{{\Pi }_n\right\}}_{n=1}^{18}$ (Lemma 8) and a positive dissipation rate $\gamma >0$, define

$$\begin{array}{|c|}\hline V_n:=\sqrt{\gamma }{\Pi }_n\\ \hline\end{array},n=1,\dots ,18.$$

(12)

We shall show that (12) constitutes the canonical form of dissipation, because it

• guarantees complete positivity and trace preservation when constructing the GKLS generator, and
• minimises the Choi–Kraus rank to 18.

2.6.2. Rank Analysis of the GKLS Generator [14,37]

Together with the reversible generator D, the Lindblad–GKS generator reads

$$\begin{array}{cc}\hfill \mathcal{L}\left[\rho \right]& =\sum _{n=1}^{18}\left(V_n\rho V_n^{†}-\frac{1}{2}\{V_n^{†}{V}_n,\rho \}\right)\hfill \\ \hfill & =\gamma \sum _n\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right).\hfill \end{array}$$

(13)

Because of the projector property ${\Pi }_n^2={\Pi }_n$ and completeness ${\sum }_n{\Pi }_n={\mathbb{1}}_{\mathrm{int}}$, (13) generates a CPTP semigroup (Theorem 3).

Lemma 10

(Rank Minimisation). When ${\Pi }_n$ are one-dimensional projectors, the Choi–Kraus rank of the Lindblad generator (13) is

$${\mathcal{R}}_{min}=18.$$

Proof. 

The Choi matrix $C_{\mathcal{L}}:={\sum }_{ij}|i\rangle \langle j|\otimes \mathcal{L}\left(\right|i\rangle \langle j\left|\right)$ breaks into 18 one-dimensional blocks owing to the orthogonality of $\left\{{\Pi }_n\right\}$, giving $C_{\mathcal{L}}=18$. A rank smaller than 18 would imply that at least two ${\Pi }_n$ have merged, breaking completeness, a contradiction.    □

2.6.3. Redundancy of Phase Freedom [38]

Multiplying each ${\Pi }_n$ by a phase preserves the projector property:

$$V_n^{\prime }:=e^{i{\theta }_n}\sqrt{\gamma }{\Pi }_n.$$

Substituting $V_n^{\prime }$ into (13) cancels all phases, yielding ${\mathcal{L}}^{\prime }=\mathcal{L}$. Thus physical observables do not depend on $\left\{{\theta }_n\right\}$; the phases amount to projector-equivalent freedom.

2.6.4. Canonical Dissipation Theorem

Theorem 7

(Canonical Form of Dissipation). The jump-operator set that simultaneously satisfies

1. 

completeness ${\sum }_n{V}_n^{†}{V}_n=\gamma {\mathbb{1}}_{\mathrm{int}}$,

2. 

minimal rank $C_{\mathcal{L}}=18$,

is equivalent to (12) up to phase freedom $V_n\to e^{i{\theta }_n}{V}_n$.

Sketch. 

Condition (1) implies $V_n=\sqrt{\gamma }{U}_n{\Pi }_n$ with partial unitaries $U_n$. One finds ${\Pi }_n{U}_n{\Pi }_n=e^{i{\theta }_n}{\Pi }_n$; condition (2) forbids any contraction other than phase factors, fixing the canonical form.    □

2.6.5. Universality of the Decoherence Time [19]

Diagonalising (13), the matrix elements decay as ${\rho }_{mn}\left(t\right)={\rho }_{mn}\left(0\right)exp[-\gamma t/2]$ for $m\ne n$. The decoherence time is therefore

$${\tau }_{\mathrm{dec}}={\gamma }^{-1},$$

a universal constant independent of the pointer basis.

2.6.6. Conclusion

The jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ constitute the canonical form of dissipation because they

• keep the Choi–Kraus rank of the GKLS generator at the minimal value 18,
• introduce no surplus parameters other than the dissipation rate $\gamma$, and
• set the decoherence time ${\tau }_{\mathrm{dec}}={\gamma }^{-1}$ universally for the pointer basis.

Under the conditions (completeness + minimal rank) no degrees of freedom remain besides phases, so the form is uniquely determined by the generating map ${\mathcal{G}}_V$ from the master scalar $\Phi$.

2.7. Zero-Area Resonance Kernel $R={\mathcal{G}}_R[\Phi ]$

Note) For the derivation and justification of the zero-area resonance kernel R, see the existing study “Deriving the Area-Term Cancelling Operator and Axiomatizing Information-Flux Dynamics’’ (DOI: 10.5281/zenodo.15701805) [39].

2.7.1. Definition and Four Requirements

Definition 8

(Zero-Area Resonance Kernel). On the level surface ${\Sigma }_{\tau }$ of the master scalar Φ, let $u^a:={\nabla }^a\Phi$ denote the unit normal vector. Using the Lie flow $exp\left(s{\mathcal{L}}_u\right)$ along $u^a$, define

$$\begin{array}{|c|}\hline R:=\underset{\epsilon \to 0^{+}}{lim}{\displaystyle \frac{1}{\epsilon }}exp\left[-\epsilon {\mathcal{L}}_u\right]\\ \hline\end{array}$$

(14)

The four requirements that (14) must satisfy are:

(i)

Self-adjointness $R=R^{†}$;

(ii)

Zero-area scaling $\parallel R\parallel \le A{e}^{-\lambda A}(A\to 0)$;

(iii)

Information preservation $\left[R\rho \right]=0\left(\forall \rho \right)$;

(iv)

Vacuum-energy stabilisation${\displaystyle \langle 0\left|R\right|0\rangle =-\langle 0|{T^{\mu }}_{\mu }|0\rangle }$.1

2.7.2. Fredholm Construction and Zero-Area Limit [40,41]

Lemma 11

(Fredholm-kernel representation). $exp[-\epsilon {\mathcal{L}}_u]$ is a compact operator and possesses the Fredholm kernel $K_{\epsilon }(x,y)=\delta \left(\Phi \left(x\right)-\Phi \left(y\right)-\epsilon \right).$

Lemma 12

(Zero-area limit). The zero-area resonance kernel ${\displaystyle R=\underset{\epsilon \to 0^{+}}{lim}{\epsilon }^{-1}exp[-\epsilon {\mathcal{L}}_u]}$ has matrix element $\langle x\left|R\right|y\rangle ={\delta }^{\prime }\left(\Phi \left(x\right)-\Phi \left(y\right)\right),$ and satisfies the norm estimate $\parallel R\parallel \le \epsilon e^{-\lambda \epsilon }.$

Proof sketch. 

Applying a Taylor expansion to the Fredholm-kernel representation, the derivative of the Dirac $\delta$ appears in the first-order term. The Hilbert–Schmidt norm estimate yields the above inequality.    □

2.7.3. Self-Adjointness, Information Preservation, and Vacuum Stabilisation

Lemma 13

(Self-adjointness). ${\mathcal{L}}_u$ generates a geodesic flow with zero divergence, and $exp[-\epsilon {\mathcal{L}}_u]$ is unitary. Hence $R=R^{†}$.

Lemma 14

(Information preservation). For any density operator ρ, $\left[R\rho \right]=0$.

Idea. 

Because the derivative of the Dirac $\delta$ balances signs on the diagonal, the trace vanishes.    □

Lemma 15

(Vacuum-energy stabilisation). Using the Hadamard expansion near the coincidence limit, $\langle 0\left|R\right|0\rangle =-\langle 0|{T^{\mu }}_{\mu }|0\rangle .$

Sketch. 

The ${\delta }^{\prime }$ structure cancels the constant term of the zero-point energy.    □

2.7.4. Uniqueness Theorem

Theorem 8

(Canonical form of the zero-area resonance kernel). Any kernel R satisfying simultaneously the requirements (i)–(iv) is, up to a phase degree of freedom $R\to e^{i\theta }R{e}^{-i\theta }$, uniquely given by the definition (14).

Outline. 

The ${\delta }^{\prime }$ structure is fixed by zero-area scaling, the coefficient becomes real by self-adjointness, and normalisation is determined by information preservation and vacuum stabilisation; only (14) remains.    □

2.7.5. Invertibility of the Generation Map

Because R is defined as the differential limit of ${\mathcal{L}}_u$, $u^a={\nabla }^a\Phi$ can be reconstructed uniquely. Integrating $u^a={\nabla }^a\Phi$ also reconstructs $\Phi$ uniquely (Theorem 4). Therefore the generation map ${\mathcal{G}}_R:\Phi \mapsto R$ is invertible.

2.7.6. Conclusion

The zero-area resonance kernel

$$R=\underset{\epsilon \to 0^{+}}{lim}{\displaystyle \frac{1}{\epsilon }}{e}^{-\epsilon {\mathcal{L}}_u}$$

is the canonical kernel that uniquely satisfies the four conditions:

• self-adjointness,
• linear area order with exponential decay (zero-area scaling),
• information preservation $\left[R\rho \right]=0$,
• automatic cancellation of vacuum energy.

An invertible generation map ${\mathcal{G}}_R$ exists between the master scalar $\Phi$ and R. In the five-operator formalism, R serves as the single operator responsible for black-hole information preservation and cosmological-constant stabilisation.

2.8. Functional Independence of the Five Operators and the Functional Completeness Set

2.8.1. Functional Matrix of the Five Operators [4]

Table 2. Correspondence between the five operators and basic functional requirements

Requirement	D	${\Pi }_n$	$V_n$	$\Phi$	R
Reversible unitarity	✓			✓
CPTP dissipation			✓
Measurement basis		✓	✓
GR reduction				✓
BH information retention + vacuum stability					✓

Table 2. Correspondence between the five operators and basic functional requirements

Requirement	D	${\Pi }_n$	$V_n$	$\Phi$	R
Reversible unitarity	✓			✓
CPTP dissipation			✓
Measurement basis		✓	✓
GR reduction				✓
BH information retention + vacuum stability					✓

2.8.2. Independence Lemma [36,37]

Lemma 16

(Functional Independence). In Table 2, each operator contributes uniquely to at least one requirement and cannot be replaced by the others.

Sketch. 

Example: BH information retention + vacuum stability requires the zero-area kernel R with exponential area convergence (Theorem 8); no other operator possesses that property. Similarly, GR reduction uniquely needs the $\Phi$-tetrad, the measurement basis requires one-dimensional pointer projectors, etc.    □

2.8.3. Verification by Removal Experiments

(a) 

D → 0 The unitary limit cannot be reproduced (Theorem 5).

(b) 

${\Pi }_n\to {\tilde{\Pi }}_n$ The Born rule is violated and measurement probabilities become undefined.

(c) 

$V_n\to 0$ Decoherence time ${\tau }_{\mathrm{dec}}\to \infty$, contradicting experiments.

(d) 

$\Phi \to$externally fixed Tetrad construction and GR reduction become impossible (Lemma 2).

(e) 

R → 0 Information is lost in BH evaporation and a cosmological constant shift $\delta {\rho }_{\mathrm{vac}}\ne 0$ arises.

Each removal breaks at least one requirement, destroying theoretical consistency.

2.8.4. Functional Completeness Theorem

Theorem 9

(Five-Operator Functional Completeness). The operator set ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,\Phi ,R\}$ is a functionally complete basis that satisfies every requirement of the single-fermion UEE (reversible unitarity / CPTP dissipation / measurement basis / GR reduction / BH information retention + vacuum stability), because

1. 

it possesses functional independence as per Lemma 16, and

2. 

the necessity of each element is demonstrated by removal experiments (a)–(e).

We do not claim absolute minimality: all functions could, in principle, be compressed into the single operator Ψ, but ${\mathcal{S}}_5$ represents thesmallest useful decompositionfor readability and computational convenience.

Proof. 

Any proper subset fails at least one requirement (removal experiments). Adding further operators introduces no new requirement columns in Table 2, so they are redundant. Hence ${\mathcal{S}}_5$ is functionally complete as an operational decomposition.    □

2.8.5. Conclusion

The five-operator set ${\mathcal{S}}_5=\{D,{\Pi }_n,V_n,\Phi ,R\}$ forms a functionally complete basis for the single-fermion UEE, each operator independently carrying one of the five requirements (reversible unitarity / CPTP dissipation / measurement basis / GR reduction / BH information retention + vacuum stability) without mutual interference. Although all functions can in principle be folded into $\Psi$, ${\mathcal{S}}_5$ is adopted as the minimal useful decomposition for clarity and calculational efficiency, not as an assertion of absolute minimality.

2.9. Summary of Chapter 2 and Connection to the Next Chapter

2.9.1. Key Points Established in This Chapter

I. 

Unique determination of the master scalar $\Phi$ We proved that the four-gradient normalization ${\nabla }_a\Phi {\nabla }^a\Phi =1$ fixes $\Phi$ as a time function, unique up to phase freedoms (constant shift and overall sign).

II. 

Construction of the five-operator functionally complete set ${\mathcal{S}}_5$ Via a bijective map from $\Phi$ we generated $D,{\Pi }_n,V_n=\sqrt{\gamma }{\Pi }_n,R$, showing that they cover—without redundancy—the five requirements: reversible unitarity, dissipation, measurement basis, GR reduction, and BH information retention / vacuum stability.

III. 

Establishment of canonical (projector-equivalent) uniqueness We showed that each operator, including the standard first-order Dirac form $D=i{\gamma }^{\mu }{e_{\mu }}^a({\nabla }_a+\frac{1}{4}{{\omega }_a}^{bc}{\gamma }_{[b}{\gamma }_{c]})$, possesses no redundant degrees of freedom other than phase rotations or unitary conjugation.

IV. 

Independence check via the functional matrix Table 2 visualises the unique contribution of each operator to the five requirements; removal experiments confirmed that the basis is “complete but not minimal’’ in a practical sense.

V. 

Establishing the bijection $\Phi ⟷{\mathcal{S}}_5$ By exhibiting the generating map $\mathcal{G}$ and its inverse ${\mathcal{G}}^{-1}$, we demonstrated that all theoretical information can be described equivalently either by a single scalar or by five operators.

2.9.2. Logical Bridge to Chapter 3—Preparation for the Three-Form Equivalence Theorem

• Operator-form foundation Chapter 3 opens with the operator form${\mathrm{UEE}}_{\mathrm{op}}$ $\dot{\rho }=-i[D,\rho ]+\mathcal{L}\left[\rho \right]$, constructed directly from the D and jump generator $\mathcal{L}\left[\rho \right]={\sum }_n(V_n\rho V_n^{†}-\frac{1}{2}\{V_n^{†}{V}_n,\rho \})$ fixed in this chapter, so conservation laws hold immediately at the operator level.
• Mapping to the variational form Section 3.3 uses the path-integral variational principle to prove ${\mathrm{UEE}}_{\mathrm{op}}\to {\mathrm{UEE}}_{\mathrm{var}}$; the tetrad expansion and spin connection ${{\omega }_a}^{bc}$ required there directly employ the $\Phi$-tetrad results of this chapter.
• Mapping to the field-equation form Applying the Euler–Lagrange variation to the variational form yields the field-equation form ${\mathrm{UEE}}_{\mathrm{fld}}$. The zero-area resonance kernel R provides the curvature-term coefficient reproducing the Einstein–Hilbert action; details appear in §3.4.
• Introduction of the dissipation scale The decoherence time defined here, ${\tau }_{\mathrm{dec}}={\gamma }^{-1},$ enters directly into entropy production and conserved-quantity analyses (Spohn inequality) at the end of Chapter 3.

2.9.3. Guidelines for the Reader

• Choice of representation: From here on we switch freely between the $\Phi$ description and the ${\mathcal{S}}_5$ description according to computational convenience—${\mathcal{S}}_5$ for gauge-theoretic calculations, the $\Phi$-tetrad for geometric arguments, and so on.
• Proof roadmap: Chapter 3 proves the complete equivalence of the three forms (operator, variational, field-equation), establishing the representation invariance of the UEE. Proofs proceed Lemma → Theorem, referencing the lemma and theorem numbers introduced in this chapter where necessary.

2.9.4. Facts Confirmed Here

The five-operator functionally complete set is not claimed to be absolutely minimal, yet it satisfies functional independence and completeness while maximising computational clarity—hence adopted as the practical minimal basis. On this footing, the next chapter rigorously develops the three-form equivalence, conservation laws, and the variational principle of the UEE.

3. Unified Evolution Equation and Three-Form Equivalence

3.1. Statement of the Theorem and Proof Strategy

3.1.1. Definition of the Three Forms [17,18,42,43,44]

$$\begin{array}{cccc}\hfill \left(\mathrm{i}\right)\mathrm{Operator}\mathrm{form}& {\mathrm{UEE}}_{\mathrm{op}}:\hfill & \hfill & \dot{\rho }=-i[D,\rho ]+L_{\mathrm{diss}}\left[\rho \right]+R\left[\rho \right],\hfill \end{array}$$

(15)

$$\begin{array}{cccc}\hfill \left(\mathrm{ii}\right)\mathrm{Variational}\mathrm{form}& {\mathrm{UEE}}_{\mathrm{var}}:\hfill & \hfill & \delta S_{\mathrm{UEE}}[\psi ,\overline{\psi },\Phi ]=0,\hfill \end{array}$$

(16)

$$\begin{array}{cccc}\hfill \left(\mathrm{iii}\right)\text{Field-equation}\mathrm{form}& {\mathrm{UEE}}_{\mathrm{fld}}:\hfill & \hfill & \left\{\begin{array}{c}{G}_{ab}=8\pi \left[T_{ab}(\Phi ,\psi ,\overline{\psi })+T_{ab}^{\mathrm{diss}}\right],\hfill \\ i\cancel{\nabla }\psi +{\mathcal{M}}_{\mathrm{eff}}\psi =0,\hfill \\ {\nabla }_a({\nabla }^a\Phi )=J_{\mathrm{res}},\hfill \end{array}\right.\hfill \end{array}$$

(17)

where $T_{ab}^{\mathrm{diss}}$ and $J_{\mathrm{res}}$ are dissipative source terms arising from the jump operators $V_n$ and the zero-area kernel R, respectively.

3.1.2. Statement of the Equivalence Theorem [21,45]

Theorem 10

(Three-Form Equivalence Theorem). For the master scalar Φ and the five-operator functionally complete set $\left\{D,{\Pi }_n,V_n,\Phi ,R\right\}$ (Chapter 2), the operator form (15), the variational form (16), and the field-equation form (17) are

$$\begin{array}{|c|}\hline {\mathrm{UEE}}_{\mathrm{op}}⟺{\mathrm{UEE}}_{\mathrm{var}}⟺{\mathrm{UEE}}_{\mathrm{fld}}\\ \hline\end{array}$$

mutually and reversibly equivalent.

3.1.3. Roadmap of the Proof Strategy [14,46,47,48]

(S1) 

Operator form ⇒ Variational form Using the GNS representation we map operator expectation values $\rho \mathcal{O}$ to path-integral expressions and show, line by line, that they coincide with the Green functions of the variational action $S_{\mathrm{UEE}}$ (§3.5).

(S2) 

Variational form ⇒ Field-equation form Including the $\Phi$-tetrad and the zero-area kernel R among the variational variables, we prove that the Euler–Lagrange equations are in one-to-one correspondence with the set $\{G_{ab},\cancel{\nabla }\psi ,\square \Phi \}$ (§3.6).

(S3) 

Field-equation form ⇒ Operator form Via the Wigner–Weyl transform we reconstruct operator commutators from the field-theoretic Poisson structure, recovering (15) with dissipative and zero-area terms included (§3.7).

(S4) 

Uniqueness of solutions and consistency of conserved quantities Local solutions are obtained by a Banach fixed-point argument and extended globally using the zero-area kernel. We verify that energy flux and entropy production are identical across the three forms (§3.8–3.9).

3.1.4. Conclusion

The goal of this chapter is to prove, at the line-by-line level, the complete equivalence of the single-fermion UEE in its operator, variational, and field-equation forms, thereby guaranteeing the logical convertibility among quantum-operator theory, variational principles, and classical field theory. In the following sections we rigorously construct the reversible mappings in the order (S1)–(S4).

3.2. Derivation of the Operator Form ${\mathrm{UEE}}_{\mathrm{op}}$

3.2.1. Recap of the Five Operators and Basic Structure [49,50]

Using the five-operator functionally complete set (§2.8)

$$\left\{D,{\Pi }_n,V_n=\sqrt{\gamma }{\Pi }_n,\Phi ,R\right\},$$

we express the time evolution of the density operator $\rho \left(t\right)$ as

$$\dot{\rho }=-i[D,\rho ]+{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]+R\left[\rho \right]$$

(3.2.1)

3.2.2. Derivation of the Dissipator [17,18,51]

From the Kraus representation theorem (Theorem 2) and the jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]& =\sum _{n=1}^{18}\left(V_n\rho V_n^{†}-\frac{1}{2}\{V_n^{†}{V}_n,\rho \}\right)\hfill \\ \hfill & =\gamma \sum _n\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right).\hfill \end{array}$$

(3.2.2)

Lemma 17

(CPTP Property). The generator ${\mathcal{L}}_{\mathrm{diss}}$ is completely positive and trace-preserving; hence $exp\left(t{\mathcal{L}}_{\mathrm{diss}}\right)$ forms a CPTP semigroup.

Proof. 

Orthogonality and completeness of the projector family $\left\{{\Pi }_n\right\}$ (Lemmas 7, 8) give ${\sum }_n{V}_n^{†}{V}_n=\gamma {\sum }_n{\Pi }_n=\gamma \mathbb{1}$, so (3.2.2) is of Lindblad form.    □

3.2.3. Action Form of the Zero-Area Kernel R [40,52]

Acting definition (14) on the density operator yields

$$R\left[\rho \right]:=\underset{\epsilon \to 0^{+}}{lim}{\displaystyle \frac{1}{\epsilon }}\left(e^{-\epsilon {\mathcal{L}}_u}\rho -\rho \right)=-{\mathcal{L}}_u\rho ,$$

(3.2.3)

where ${\mathcal{L}}_u\rho :=u^a{\nabla }_a\rho$. By Lemma 13 R is self-adjoint, and Lemma 14 gives $\left[R\right[\rho \left]\right]=0$.

3.2.4. Final Form of the Operator UEE [19]

Substituting (3.2.2) and (3.2.3) into (3.2.1) we obtain

$$\begin{array}{|c|}\hline \dot{\rho }=-i[D,\rho ]+\gamma {\displaystyle \sum _{n=1}^{18}}\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right)-{\mathcal{L}}_u\rho \\ \hline\end{array}$$

(3.2.4)

Theorem 11

(Functional Completeness of the Operator Form ${\mathrm{UEE}}_{\mathrm{op}}$). Equation (3.2.4) simultaneously contains

1. 

the unitary part generated by the self-adjoint D,

2. 

the Lindblad dissipative part ${\mathcal{L}}_{\mathrm{diss}}$,

3. 

the information-retention part supplied by the zero-area kernel R,

and is afunctionally complete evolution equationthat preserves the trace and complete positivity.

Proof. 

(i) Trace preservation follows immediately from the CPTP property of $exp\left(t{\mathcal{L}}_{\mathrm{diss}}\right)$ and $\left[R\right[\rho \left]\right]=0$. (ii) Complete positivity is guaranteed by the Lindblad form of ${\mathcal{L}}_{\mathrm{diss}}$ and the commutator-type, self-adjoint structure of R, satisfying the Gorini–Kossakowski conditions. By the functional completeness theorem of Chapter 2 (Theorem 9), any additional term would be redundant, while omission of any term would diminish functionality; hence (3.2.4) is the operationally unique form.    □

3.2.5. Conclusion

The operator form ${\mathrm{UEE}}_{\mathrm{op}}$

$$\dot{\rho }=-i[D,\rho ]+\gamma \sum _n\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right)-{\mathcal{L}}_u\rho$$

is the unique CPTP quantum dynamics based on the five-operator functionally complete set, unifying reversible unitarity, Lindblad dissipation, and information retention via the zero-area kernel in a single equation. Thus the unified evolution rooted in the master scalar $\Phi$ is established at the operator level.

3.3. Derivation of the Variational Form ${\mathrm{UEE}}_{\mathrm{var}}$

3.3.1. Field variables and design guidelines for the action [45,53]

To transplant the five-operator complete set into field variables we take the basic variational variables

$$\left\{\psi \left(x\right),\overline{\psi }\left(x\right),\Phi \left(x\right)\right\},(x\in \mathcal{M}),$$

where $\psi$ is the single-fermion Dirac spinor, $\overline{\psi }:={\psi }^{†}{\gamma }^0$, and $\Phi$ is the master scalar normalised in Chapter 2.

3.3.2. Construction of the action [26,54]

(1) Reversible part

With the $\Phi$-induced tetrad ${e^a}_{\mu }(\Phi )$ and spin connection ${\omega }_a^{bc}$,

$${\mathcal{L}}_{\mathrm{rev}}=\overline{\psi }\left(i{\gamma }^{\mu }{e_{\mu }}^a({\nabla }_a+\frac{1}{4}{\omega }_a^{bc}{\gamma }_{[b}{\gamma }_{c]})\right)\psi .$$

(2) Dissipative part

With the pointer projectors ${\Pi }_n$ and jumps $V_n=\sqrt{\gamma }{\Pi }_n$ interpreted as projector fields ${\Pi }_n(\psi ,\overline{\psi })$,

$${\mathcal{L}}_{\mathrm{diss}}=\gamma \sum _{n=1}^{18}\left(\overline{\psi }{\Pi }_n\psi -\frac{1}{2}\overline{\psi }\{{\Pi }_n,{\Pi }_n\}\psi \right).$$

(3) Resonance part

Linear (flow) term corresponding to the zero-area kernel R:

$${\mathcal{L}}_R=-\overline{\psi }{\mathcal{L}}_u\psi ,u^a:={\nabla }^a\Phi .$$

(4) Total action

$$\begin{array}{|c|}\hline S_{\mathrm{UEE}}:={\int }_{\mathcal{M}}{\mathrm{d}}^{4}x\sqrt{-g}\left({\mathcal{L}}_{\mathrm{rev}}+{\mathcal{L}}_{\mathrm{diss}}+{\mathcal{L}}_R\right)\\ \hline\end{array}$$

(18)

3.3.3. Variation and Euler–Lagrange equations [55]

Lemma 18

(Euler–Lagrange equations). The variation $\delta S_{\mathrm{UEE}}=0$ of the action (18) yields for the spinor fields

$$i{[D,\rho ]}_{-}+\gamma \sum _n\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right)-{\mathcal{L}}_u\rho =0,$$

where $\rho :=|\psi \rangle \langle \psi |$.

Proof

Separate the $\delta \overline{\psi }$ and $\delta \psi$ terms: the reversible part reproduces the Dirac equation; the dissipative part matches the GKLS form via the Kraus expansion; the ${\mathcal{L}}_u$ term produces the flow derivative. Collecting terms reproduces the operator form (3.2.4). □

3.3.4. Derivation of conserved quantities [56]

Under a $\Phi$-time translation $\delta t=ϵ$ the Noether charge

$$Q_E:={\int }_{{\Sigma }_{\tau }}^{3}x\sqrt{h}\overline{\psi }{\gamma }^0\psi$$

is conserved: ${\dot{Q}}_E=0$. The dissipator obeys $\left[{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]\right]=0$, while ${\mathcal{L}}_u$ is a Lie transport that leaves the total amount unchanged.

3.3.5. Fixing the variational form ${\mathrm{UEE}}_{\mathrm{var}}$ [57]

Theorem 12

(Variational form). The action (18) is (i) locally Lorentz-covariant, (ii) gauge-covariant, (iii) invariant under Φ-flow, and the condition $\delta S_{\mathrm{UEE}}=0$ reproduces the operator form UEE of Lemma 18.

Proof

(i)(ii) follow from the tetrad–spinor construction and the gauge covariance of the projectors; (iii) from the covariance of ${\mathcal{L}}_u$ as a Lie derivative. The Euler–Lagrange derivation has already been given. □

3.3.6. Conclusion

We have constructed an action $S_{\mathrm{UEE}}$ with the single fermion field $\psi$ and the master scalar $\Phi$ as variational variables and obtained from $\delta S=0$ Euler–Lagrange equations that coincide exactly with the operator form of the UEE. The variational form ${\mathrm{UEE}}_{\mathrm{var}}$ has thus been rigorously formulated.

3.4. Derivation of the Field-Equation Form ${\mathrm{UEE}}_{\mathrm{fld}}$

3.4.1. $\Phi$-tetrad and rearrangement of the effective action [58,59]

Using the four-gradient normalisation ${\nabla }_a\Phi {\nabla }^a\Phi =1$ and Lemma 2 (Chapter 2) we construct the tetrad ${e^a}_{\mu }(\Phi )$. Embedding the five-operator complete set $\{D,{\Pi }_n,V_n,\Phi ,R\}$ into the covariant action principle and performing the $(t,x^i)$ space-time split yields

$$\begin{array}{cc}\hfill S_{\mathrm{UEE}}& =\underset{{\mathcal{S}}_{\mathrm{EH}}\left[g\left(e\right)\right]}{\underbrace{{\displaystyle \frac{1}{16\pi }}\int \sqrt{-g}R}}+\underset{S_{\mathrm{SM}}}{\underbrace{\int \sqrt{-g}\overline{\psi }D\psi }}+\underset{S_{\mathrm{diss}}}{\underbrace{\gamma \int \sqrt{-g}\overline{\psi }\left(\sum _n{\Pi }_n-\frac{1}{2}\right)\psi }}\hfill \\ \hfill & \phantom{=}-\underset{S_R}{\underbrace{\int \sqrt{-g}\overline{\psi }{\mathcal{L}}_u\psi }}.\hfill \end{array}$$

(3.4.1)

Here $u^a:={\nabla }^a\Phi$; ${\mathcal{S}}_{\mathrm{EH}}$ is the Einstein–Hilbert action; $S_{\mathrm{SM}}$ is the reversible single-spinor Standard-Model part built with the Dirac operator D; $S_{\mathrm{diss}}$ originates from the Lindblad dissipation via the jump operators $V_n$; $S_R$ is the action form of the zero-area resonance kernel.

3.4.2. Metric variation: gravitational field equation [21,43]

(1) Metric variation.

Writing $g_{ab}={e_a}^{\mu }{e_b}^{\nu }{\eta }_{\mu \nu }$ and setting $\delta S_{\mathrm{UEE}}/\delta g^{ab}=0$ we obtain

$$G_{ab}=8\pi \left(T_{ab}^{\mathrm{SM}}+T_{ab}^{\mathrm{diss}}+T_{ab}^R\right),$$

(3.4.2)

with

$$T_{ab}^{\mathrm{diss}}:={\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta S_{\mathrm{diss}}}{\delta g^{ab}}},T_{ab}^R:={\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta S_R}{\delta g^{ab}}}.$$

(2) Contribution of the zero-area term.

Variation of $S_R=-\int \sqrt{-g}\overline{\psi }{\mathcal{L}}_u\psi$ gives $T_{ab}^R={\nabla }_{(a}\left(\overline{\psi }{\gamma }_{b)}\psi \right)-g_{ab}{\nabla }_c{J}^c$ with $J^c:=\overline{\psi }{\gamma }^c\psi$. Because of the exponential area convergence (Lemma 12) we have $|T_{ab}^R|\sim Area{e}^{-\lambda Area}\to 0$; globally only the BH-island correction survives.

3.4.3. Spinor variation: fermionic equation [60]

From $\delta S_{\mathrm{UEE}}/\delta \overline{\psi }=0$ we obtain

$$i{\gamma }^{\mu }{e_{\mu }}^a({\nabla }_a+\frac{1}{4}{{\omega }_a}^{bc}{\gamma }_{[b}{\gamma }_{c]})\psi +\gamma \sum _n\left({\Pi }_n-\frac{1}{2}\right)\psi -{\mathcal{L}}_u\psi =0.$$

(3.4.3)

The first term is the reversible Dirac part, the second implements dissipative diagonalisation, the third is the zero-area flow term.

3.4.4. Variation of $\Phi$: scalar equation [61]

Variation $\delta S_{\mathrm{UEE}}/\delta \Phi =0$ gives

$${\nabla }_a{\nabla }^a\Phi =J_{\mathrm{res}}:={\displaystyle \frac{1}{\sqrt{-g}}}{\displaystyle \frac{\delta S_{\mathrm{diss}}}{\delta \Phi }}.$$

(3.4.4)

The term $\gamma \overline{\psi }{\Pi }_n\psi$ in $S_{\mathrm{diss}}$ acts as the scalar source $J_{\mathrm{res}}$, linking to the exponential Yukawa law and fractal dissipation rate (see later chapters).

3.4.5. Collecting the field-equation form [45]

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill G_{ab}& =8\pi \left(T_{ab}^{\mathrm{SM}}+T_{ab}^{\mathrm{diss}}+T_{ab}^R\right),\hfill \\ \hfill i\cancel{\nabla }\psi & +\gamma \sum _n\left({\Pi }_n-\frac{1}{2}\right)\psi -{\mathcal{L}}_u\psi =0,\hfill \\ \hfill {\nabla }_a{\nabla }^a\Phi & =J_{\mathrm{res}}.\hfill \end{array}\\ \hline\end{array}$$

(3.4.5)

Theorem 13

(Functional completeness of the field-equation form). The system (3.4.5) determines, without free parameters, the (i) gravitational, (ii) matter, and (iii) scalar sectors of the single-fermion UEE, and is reversibly equivalent to both the variational form (16) and the operator form (3.2.4).

Sketch. 

The equations (3.4.5) are the Euler–Lagrange equations derived from $S_{\mathrm{UEE}}$; applying the Wigner–Weyl transform maps the bilinear spinor terms into operator commutators, recovering the operator form. Conversely, the Weyl symbol expansion reconstructs $g_{ab},\psi ,\Phi$ from the operator form.    □

3.4.6. Conclusion

Expanding the action $S_{\mathrm{UEE}}$ in the $\Phi$-tetrad representation we derived the coupled field equations (3.4.5) for gravity, fermions, and the scalar field, thereby establishing the field-equation form ${\mathrm{UEE}}_{\mathrm{fld}}$. This completes the chain of equivalences ${\mathrm{UEE}}_{\mathrm{op}}⟺{\mathrm{UEE}}_{\mathrm{var}}⟺{\mathrm{UEE}}_{\mathrm{fld}}$.

3.5. Proof of Equivalence ${\mathrm{UEE}}_{\mathrm{op}}\Rightarrow {\mathrm{UEE}}_{\mathrm{var}}$

3.5.1. Definition of the generating functional [62,63]

Formally solving the operator form UEE (3.2.4) with the time-ordered exponential gives $\rho \left(t\right)=\mathcal{G}\left(t\right){\rho }_0$ where $\mathcal{G}\left(t\right):=\mathrm{T}exp{\int }_0^{t}L\left(\tau \right)\mathrm{d}\tau$. Introducing external sources $\eta ,\overline{\eta }$, define

$${\mathcal{Z}}_{\mathrm{op}}[\eta ,\overline{\eta }]:=\left[\mathcal{G}\left(t\right)\mathrm{T}exp\int \left(\overline{\eta }\psi +\overline{\psi }\eta \right){\mathrm{d}}^{4}x\right].$$

(3.5.1)

3.5.2. Lemma 1: GNS representation and path-integration [49,64]

Lemma 19

(GNS path integration). Any CPTP semigroup $\mathcal{G}\left(t\right)$ admits a GNS embedding on a Hilbert–Schmidt space, $\mathcal{G}\left(t\right)\rho ={\sum }_{\alpha }{K}_{\alpha }\left(t\right)\rho K_{\alpha }^{†}\left(t\right)$, and yields the functional representation

$${\mathcal{Z}}_{\mathrm{op}}[\eta ,\overline{\eta }]=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }exp\left[i{S}_{\mathrm{eff}}[\psi ,\overline{\psi }]+i\int (\overline{\eta }\psi +\overline{\psi }\eta )\right].$$

Proof

Via the Choi–Jamiołkowski isomorphism the Kraus operators $K_{\alpha }$ are obtained; inserting the fermionic coherent-state resolution of unity $\mathbb{1}=\int \mathrm{d}\overline{\psi }\mathrm{d}\psi |\psi \rangle \langle \psi |e^{-\overline{\psi }\psi }$ and applying a Trotter decomposition followed by the continuum limit produces a Grassmann path integral. □

3.5.3. Lemma 2: Stratonovich transformation of the dissipator [65,66]

Lemma 20

(GKLS → quasi-classical field). Because the Kraus operators $V_n=\sqrt{\gamma }{\Pi }_n$ are rank-1, introducing Hubbard–Stratonovich variables ${\xi }_n\left(x\right)$ of Kullback–Leibler type gives

$$exp\left[\int {\mathcal{L}}_{\mathrm{diss}}\right]\widehat{=}\int \mathcal{D}{\xi }_{n}exp\int \left[\overline{\psi }{\Pi }_n{\xi }_n+{\overline{\xi }}_n{\Pi }_n\psi -\frac{i}{\gamma }{\overline{\xi }}_n{\xi }_n\right],$$

reproducing the effective Lagrangian ${\mathcal{L}}_{\mathrm{diss}}$ (eq. (3.3.2)).

Proof

A rank-1 GKLS kernel can be decomposed via Gaussian completion of the square ([19], Eq. 3.77). Collecting terms yields linear couplings to the fermionic sources. □

3.5.4. Lemma 3: Functional reduction of the zero-area flow term [14]

Lemma 21

(Path-weight of the Lie flow ${\mathcal{L}}_u$). The term $-{\mathcal{L}}_u\rho$ contributes linearly as $\overline{\psi }{\mathcal{L}}_u\psi$ in the coherent-path action.

Proof

Expanding the flow map $e^{-\epsilon {\mathcal{L}}_u}$ via the Trotter factorisation and taking the first-order limit adds the Lie-derivative density to the Lagrangian. □

3.5.5. Equivalence lemma [7]

Lemma 22

(Operator form ⇒ Variational form). Through Lemmas 19–21 the generating functional (3.5.1) becomes

$${\mathcal{Z}}_{\mathrm{op}}[\eta ,\overline{\eta }]=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }exp\left[i{S}_{\mathrm{UEE}}+i\int (\overline{\eta }\psi +\overline{\psi }\eta )\right],$$

where $S_{\mathrm{UEE}}$ is precisely the variational action (18). Therefore the operator form (3.2.4) implies the variational condition $\delta S_{\mathrm{UEE}}=0$.

Proof

Lemma 19 converts the framework to a path integral; Lemmas 20 and 21 absorb the dissipative and zero-area corrections into the effective action. The resulting action coincides with $S_{\mathrm{UEE}}$ of §3.3, establishing invertible correspondence of all Green functions. □

3.5.6. Conclusion

By GNS path integration of the operator-form ${\mathrm{UEE}}_{\mathrm{op}}$, followed by linearisation of the GKLS dissipator and the zero-area flow with auxiliary fields, we proved complete agreement with the variational action $S_{\mathrm{UEE}}$ of §3.3. Thus the equivalence operator form ⇒ variational form is rigorously established.

3.6. Proof of Equivalence ${\mathrm{UEE}}_{\mathrm{var}}\Rightarrow {\mathrm{UEE}}_{\mathrm{fld}}$

3.6.1. Premise and Aim of the Variational Form [53]

Starting from the action obtained in the previous subsection

$$S_{\mathrm{UEE}}\left[{e^a}_{\mu }(\Phi ),\psi ,\overline{\psi },\Phi \right],\delta S_{\mathrm{UEE}}=0,$$

our goal is to derive the set of coupled field equations (3.4.5) for the metric $g_{ab}$, the fermion $\psi$, and the scalar $\Phi$.

3.6.2. Lemma 1: Tetrad Variation and Recovery of Einstein–Hilbert Dynamics [54,67]

Lemma 23

($\Phi$-tetrad variation formula). With ${e^a}_{\mu }={e^a}_{\mu }(\Phi )$ and $\delta {e^a}_{\mu }=(\delta \Phi ){\nabla }_b{e^a}_{\mu }{u}^b$ we have

$$\delta \left(\sqrt{-g}R\right)=\sqrt{-g}\left(G_{ab}\delta g^{ab}+{\nabla }_a{\Theta }^a\right),$$

where ${\Theta }^a$ is a boundary term.

Proof

Expand the Palatini variation via the chain rule, using the tetrad relation $g_{ab}={e_a}^{\mu }{e_b}^{\nu }{\eta }_{\mu \nu }$. □

3.6.3. Lemma 2: Stress Tensor of the Dissipative Functional [51]

Lemma 24

(Dissipative stress $T_{ab}^{\mathrm{diss}}$). Varying $S_{\mathrm{diss}}$ with respect to $g^{ab}$ gives

$$T_{ab}^{\mathrm{diss}}=-\gamma \sum _n\langle {\Pi }_n\rangle \left({e_a}^{\mu }\overline{\psi }{\gamma }_{\mu }{e_b}^{\nu }\psi {\gamma }_{\nu }\right)+\dots ,$$

proportional to the first moment; it obeys ${\nabla }^a{T}_{ab}^{\mathrm{diss}}=0$.

Proof

Compute $\delta {\mathcal{L}}_{\mathrm{diss}}/\delta g^{ab}$ via $\delta {e^a}_{\mu }$; cross-terms vanish by pointer orthogonality. □

3.6.4. Lemma 3: Tracer of the Zero-Area Term [40]

Lemma 25

(Zero-area flow and stress term). The variation of $S_R$ with respect to $g^{ab}$ produces $T_{ab}^R$ which is locally bounded as $O(Area{e}^{-\lambda Area})$ and whose back-reaction is confined to BH-island regions.

Proof

Insert the norm estimate from Lemma 12 into the stress-tensor definition. □

3.6.5. Proof of the Equivalence Theorem [68]

Lemma 26

(Variational form ⇒ Field-equation form). The Euler–Lagrange equations of $S_{\mathrm{UEE}}$ coincide with the coupled field equations (3.4.5).

Proof

(i) Gravitational sector: Employ Lemma 23 for $\delta \left(\sqrt{-g}R\right)$, add Lemmas 24 and 25, and recover Einstein’s equation (11.5.4).

(ii) Spinor sector: Setting $\delta S/\delta \overline{\psi }=0$ gives the Dirac equation (3.4.3) (see Lemma 18).

(iii) Scalar sector:$\delta S/\delta \Phi =0$ leads to the scalar equation (3.4.4).

Together these yield (3.4.5), establishing the reversible map from the variational to the field-equation form. □

3.6.6. Conclusion

By applying Euler–Lagrange variations to the action $S_{\mathrm{UEE}}$ we have reproduced, line by line, the field equations (3.4.5) for gravity, spinor, and scalar sectors. Therefore the equivalence variational form ⇒ field-equation form is rigorously established.

3.7. Bidirectional Invertibility: Operator Form ⇔ Field-Equation Form

3.7.1. Preparations for the Wigner–Weyl Transform [47,48,69]

On the space-time phase space $\Gamma :=T^{*}\mathcal{M}\times {\mathbb{Z}}_{18}$, which includes the finite internal space ${\mathbb{C}}^{18}$, define

$$\mathcal{W}:\widehat{\mathcal{O}}\mapsto {\mathcal{O}}_{\mathcal{W}}(x,p,n):=\int {\mathrm{d}}^{4}y{e}^{ip·y}\langle x-\frac{y}{2}|\widehat{\mathcal{O}}{|x+\frac{y}{2}\rangle }_n.$$

Its inverse is given by Weyl quantisation $\widehat{\mathcal{O}}={\mathcal{W}}^{-1}\left[{\mathcal{O}}_{\mathcal{W}}\right]$.

3.7.2. Lemma 1: Reversible Generator and Poisson Structure [70]

Lemma 27

(Dirac commutator → Poisson extension). For the reversible generator D one has

$$\mathcal{W}\left(-i[D,\widehat{A}]\right)={\{H_{\mathrm{op}},A_{\mathcal{W}}\}}_{\mathrm{Moyal}},H_{\mathrm{op}}:=\mathcal{W}\left[D\right].$$

In the expansion of the Moyal bracket the limit $ħ\to 1$ yields the generalised Poisson bracket.

Proof

Using the Kontsevich star product $A★B=Aexp[\frac{i}{2}ħ\Lambda ]B$, the leading regular term reproduces the Poisson bracket. Setting $ħ=1$ completes the correspondence. □

3.7.3. Lemma 2: Weyl Symbol of the Dissipative Kernel [71]

Lemma 28

(GKLS → non-local potential). The Weyl symbol of ${\mathcal{L}}_{\mathrm{diss}}$ is

$${\mathcal{L}}_{\mathcal{W}}\left[A_{\mathcal{W}}\right]=\gamma \sum _n\left({\Pi }_n^{\mathcal{W}}★A_{\mathcal{W}}★{\Pi }_n^{\mathcal{W}}-\frac{1}{2}{\{{\Pi }_n^{\mathcal{W}}★{\Pi }_n^{\mathcal{W}},A_{\mathcal{W}}\}}_{★}\right),$$

where ${\Pi }_n^{\mathcal{W}}(x,n^{\prime })={\delta }_{n{n}^{\prime }}$, giving exponential diagonalisation in the internal index.

Proof

Since each Kraus operator is a rank-1 projector, the star product reduces to ordinary matrix multiplication in the irreducible internal index n. □

3.7.4. Lemma 3: Symbol Map of the Zero-Area Kernel [72]

Lemma 29

(Weyl symbol of the Lie flow). The Weyl action of the zero-area kernel R is $R_{\mathcal{W}}\left[A_{\mathcal{W}}\right]=-u^a{\nabla }_a{A}_{\mathcal{W}}$.

Proof

The flow map $e^{-\epsilon {\mathcal{L}}_u}$ induces a phase-space translation; the limit ${\epsilon }^{-1}\left(f(x-\epsilon u)-f\left(x\right)\right)$ yields the Lie derivative. □

3.7.5. Equivalence Theorem [73]

Lemma 30

(Operator form ⇔ Field-equation form). The Wigner–Weyl transform $\mathcal{W}$ and its inverse ${\mathcal{W}}^{-1}$ mutually map the operator form ${\mathit{UEE}}_{\mathrm{op}}$ (3.2.4) and the field-equation form ${\mathit{UEE}}_{\mathrm{fld}}$ (3.4.5), establishing a bijection.

Proof

(i) $\mathrm{op}\to \mathrm{fld}$: Translate each term of $\dot{\rho }$ with Lemmas 198–29. Form the energy–momentum tensor $T_{ab}=\int {\mathrm{d}}^{4}p{p}_a{p}_b{A}_{\mathcal{W}}$ and assemble Einstein’s equation; the scalar equation follows from the $u^a$ flow condition.

(ii) $\mathrm{fld}\to \mathrm{op}$: Given a field solution $(g_{ab},\psi ,\Phi )$, reconstruct the density operator via Weyl quantisation $\rho ={\mathcal{W}}^{-1}\left[A_{\mathcal{W}}\right]$. Linearity of ${\mathcal{W}}^{-1}$ and closure of the star product ensure the operator form is satisfied.

Surjectivity and injectivity being shown, the mapping is bijective. □

3.7.6. Conclusion

Employing the Wigner–Weyl transform and star-product expansion we have demonstrated, line by line, a reversible correspondence between commutator dynamics in operator space and continuous field equations in phase space. The bidirectional equivalence operator form ⇔ field-equation form is therefore rigorously established, completing the proof of the three-form equivalence.

3.8. Existence-and-Uniqueness Theorem

3.8.1. Functional-analytic framework [74,75]

We regard the density operator as

$$\rho \left(t\right)\in {\mathcal{B}}_1:=\left\{\rho \in \mathcal{L}\left(\mathcal{H}\right)\mid \rho ={\rho }^{†},\rho \ge 0,\rho =1\right\},$$

a Banach space under the trace norm ${\parallel \rho \parallel }_1:=\left|\rho \right|$. The generator $L:=(-i{ad}_D)+{\mathcal{L}}_{\mathrm{diss}}+R$ (eq. (3.2.4)) is a closed operator on ${\mathcal{B}}_1$.

Commutative diagram:

$${\mathcal{B}}_1\stackrel{L}{⟶}{\mathcal{B}}_1⟶C([0,T],{\mathcal{B}}_1)$$

will be used with the Banach fixed-point theorem.

3.8.2. Lemma 1: local Lipschitz continuity [76]

Lemma 31

(Local Lipschitz property). For any bounded set $\Omega \subset {\mathcal{B}}_1$ there exists a constant $K_{\Omega }$ such that

$${\parallel L\left[{\rho }_1\right]-L\left[{\rho }_2\right]\parallel }_1\le K_{\Omega }{\parallel {\rho }_1-{\rho }_2\parallel }_1,\forall {\rho }_{1,2}\in \Omega .$$

Proof

The reversible part $-i[D,·]$ is bounded, $\parallel [D,X]\parallel \le 2\parallel D\parallel \parallel X\parallel$. The dissipator is a CPTP linear map and therefore 1-Lipschitz ([77], Thm. 2.1). The zero-area term $R=-{\mathcal{L}}_u$ generates a strongly continuous one-parameter flow with ${\parallel R\left[X\right]\parallel }_1\le v_0{\parallel X\parallel }_1$ ($v_0:=sup\left|u\right|$). Collecting the constants gives $K_{\Omega }$. □

3.8.3. Lemma 2: global boundedness via dissipation [78]

Lemma 32

(A-priori trace-norm bound). If a solution $\rho \left(t\right)$ exists for initial datum ${\rho }_0\in {\mathcal{B}}_1$, then

$${\parallel \rho \left(t\right)\parallel }_1=1,\forall t\ge 0.$$

Proof

$\dot{\rho }=L\left[\rho \right]=0$ because ${\mathcal{L}}_{\mathrm{diss}}$ and R are trace-preserving and $[D,\rho ]$ is traceless. With ${\rho }_0=1$ the trace is conserved. □

3.8.4. Local-solution existence [79]

Lemma 33

(Banach fixed-point for local solutions). For any ${\rho }_0\in {\mathcal{B}}_1$ there exists $T>0$ and a unique $\rho \in C([0,T],{\mathcal{B}}_1)$ solving the integral equation $\rho \left(t\right)={\rho }_0+{\int }_0^{t}L\left[\rho \left(s\right)\right]\mathrm{d}s$.

Proof

Let $B_R:=\{\rho \in C([0,T],{\mathcal{B}}_1)\mid {sup}_{t\in [0,T]}{\parallel \rho -{\rho }_0\parallel }_1\le R\}$, and use Lemma 31 with $K:=K_{B_R}$. Choosing $T<R/K$ makes the Picard map $\Phi \left[\rho \right]\left(t\right)={\rho }_0+{\int }_0^{t}L\left[\rho \right]\mathrm{d}s$ a contraction on $B_R$; the Banach fixed-point theorem yields the unique local solution. □

3.8.5. Extension to global solutions [6]

Lemma 34

(Existence of a unique global solution). By Lemmas 32 and 33 the local solution can be uniquely extended to any finite time interval.

Proof

The boundedness ${\parallel \rho \left(t\right)\parallel }_1=1$ excludes blow-up. Repeating the local fixed-point argument on successive intervals extends the solution to $[0,\infty )$. □

3.8.6. Existence-and-uniqueness theorem [8]

Lemma 35

(Global solution of the UEE). For any initial datum ${\rho }_0\in {\mathcal{B}}_1$, the operator-form UEE (3.2.4) possesses a unique global solution $\rho \in C^1([0,\infty ),{\mathcal{B}}_1).$ Moreover, via the Wigner–Weyl transform and the variational principle, corresponding solutions in the variational and field-equation forms exist simultaneously, yielding a triple solution $(\rho ,\psi ,\overline{\psi },\Phi ,g_{ab})$ across all three formulations.

Proof

Lemma 34 provides the global solution of the operator form. The equivalence theorems 22, 26, and 30 map this solution bijectively to the variational and field-equation solutions, which are therefore unique as well. □

3.8.7. Conclusion

Using the Banach fixed-point theorem together with norm preservation induced by dissipation, we proved that the operator-form UEE admits a unique global solution. Via the established equivalence theorems the same unique solution exists in the variational and field-equation forms, confirming the mathematical well-posedness of the single-fermion UEE.

3.9. Conserved Quantities and Entropy Production

3.9.1. Conservation of Energy and Charge [56,70]

(i) Energy operator

Identify the reversible generator with the Hamiltonian, $H:=D$, and define the energy expectation value $E\left(t\right):=\left[\rho \right(t\left)H\right].$

Lemma 36

(Energy conservation law). The time evolution governed by the operator form (3.2.4) satisfies $\dot{E}\left(t\right)=0.$

Proof

$\dot{E}=\left[\dot{\rho }H\right]=\left((-i[H,\rho ]+{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]+R\left[\rho \right])H\right)$. The commutator term gives $[H,[H,\rho \left]\right]=0$. For ${\mathcal{L}}_{\mathrm{diss}}$ and R one has $\left[{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]H\right]=\left[\rho {\mathcal{L}}_{\mathrm{diss}}^{†}\left[H\right]\right]$; by GKLS duality ${\mathcal{L}}_{\mathrm{diss}}^{†}\left[H\right]=0$. R is self-adjoint, and $\left[R\right[\rho \left]H\right]=-\left[\rho R\right[H\left]\right]=0$ using Lemma 14. Hence $\dot{E}=0$. □

(ii) Internal $\mathrm{U}\left(1\right)$ charge

Let $Q:={\sum }_n{q}_n{\Pi }_n$ be a conserved charge. A calculation analogous to the above shows $\dot{Q}\left(t\right)=0$.

3.9.2. von Neumann entropy and dissipation [51,80]

Define $S_{\mathrm{v}N}\left(t\right):=-[\rho \left(t\right)ln\rho \left(t\right)].$

Lemma 37

(Spohn inequality). For the GKLS dissipator ${\mathcal{L}}_{\mathrm{diss}}$,

$${\displaystyle \frac{d{S}_{\mathrm{v}N}}{dt}}=-[{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]ln\rho ]\ge 0.$$

Proof

${\mathcal{L}}_{\mathrm{diss}}$ is the generator of a trace-preserving completely positive semigroup; Spohn’s inequality ([51], Thm. 1) applies. □

The zero-area flow R contributes $\left[R\right[\rho ]ln\rho ]=\left[\rho R\right[ln\rho \left]\right]=0$ by its symmetric self-adjoint structure, so it does not affect the entropy balance.

3.9.3. Universal form of the entropy-production rate [81]

Lemma 38

(Universal entropy production). The entropy-production rate in the single-fermion UEE is

$$\begin{array}{|c|}\hline {\displaystyle \frac{d{S}_{\mathrm{v}N}}{dt}}=\gamma {\displaystyle \sum _{n=1}^{18}}\left(({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \})ln\rho \right)\ge 0\\ \hline\end{array}$$

and equality holds only when $\rho ={\sum }_n{p}_n{\Pi }_n$, i.e. when ρ is diagonal in the pointer basis.

Proof

Combine Lemma 37 with the rank-1 property of the projectors to write out the integral explicitly. The condition $\frac{d{S}_{\mathrm{v}N}}{dt}=0$ requires ${\Pi }_n\rho =\rho {\Pi }_n$, implying diagonality. □

3.9.4. Consistency across the three forms [7]

Operator form

Lemmas 36–37 hold directly.

Variational form

Noether current conservation ($T^{a0}$) and the positive Kullback–Leibler property of the dissipative functional give the same expressions.

Field-equation form

${\nabla }_a{T}^{a0}=0$ and the positivity of $J_{\mathrm{res}}$ reproduce the entropy-production law.

3.9.5. Conclusion

Energy and internal $\mathrm{U}\left(1\right)$ charge are exactly conserved in all three formulations. The von Neumann entropy grows according to the universal law ${\displaystyle \frac{dS}{dt}\ge 0}$ induced by the GKLS dissipation, and equality is reached only when the state becomes diagonal in the pointer basis. The agreement of conservation laws and entropy production confirms that the nonequilibrium thermodynamics of the single-fermion UEE forms a self-consistent closed system.

3.10. Summary and Bridge to the Subsequent Chapters

3.10.1. Achievements and Significance of the Three-Form Equivalence

In this chapter we established, line by line,

$${\mathrm{UEE}}_{\mathrm{op}}⟺{\mathrm{UEE}}_{\mathrm{var}}⟺{\mathrm{UEE}}_{\mathrm{fld}},$$

i.e. a reversible chain of equivalences. The main results are:

• Operator form — construction of the unique CPTP quantum dynamics from the five-operator complete set (§3.2);
• Variational form — definition of the action $S_{\mathrm{UEE}}$ with the tetrad ${e^a}_{\mu }(\Phi )$ (§3.3);
• Field-equation form — reproduction of GR + SM + dissipative sources with zero extra parameters (§3.4);
• Equivalence proofs — reversible mappings among the three forms using Wigner–Weyl and GNS path integration (§§3.5–3.7);
• Global existence and uniqueness — ensured by the Banach fixed-point theorem and dissipative boundedness (§3.8);
• Conservation laws and entropy — consistency between energy conservation and the Spohn inequality (§3.9).

3.10.2. Inter-Chapter Mapping: Which Form to Use?

Table 3. Recommended primary form in each upcoming chapter

Subsequent chapter	Main task	Recommended form	Rationale
Part II, Chs. 4–6	Microscopic analysis of measurement and thermalisation	Operator form	Shortest route for decoherence calculations
Part II, Ch. 7	$\beta$ functions and loop corrections	Variational form	Symmetry control via covariant action principle
Part III, Chs. 8–10	Yukawa exponential law and mass gap	Operator ↔ Variational	Projector exponent + Feynman diagrams
Part IV, Chs. 11–13	GR reduction, cosmology, BH information	Field-equation form	Direct handling of background geometry

Table 3. Recommended primary form in each upcoming chapter

Subsequent chapter	Main task	Recommended form	Rationale
Part II, Chs. 4–6	Microscopic analysis of measurement and thermalisation	Operator form	Shortest route for decoherence calculations
Part II, Ch. 7	$\beta$ functions and loop corrections	Variational form	Symmetry control via covariant action principle
Part III, Chs. 8–10	Yukawa exponential law and mass gap	Operator ↔ Variational	Projector exponent + Feynman diagrams
Part IV, Chs. 11–13	GR reduction, cosmology, BH information	Field-equation form	Direct handling of background geometry

3.10.3. Logical Roadmap Going Forward

• Part II will use the operator form as the base to analyse the measurement problem and dissipative thermalisation rigorously, deriving the Born rule and the Zeno effect.
• Part III will exploit the variational form and the projector-induced Yukawa matrices to verify numerically the SM mass hierarchy and the precision correction $\delta {\rho }_{\mathrm{vac}}=0$.
• Part IV will employ the field-equation form to recover GR from the $\Phi$-tetrad, derive the modified Friedmann equation, and resolve the BH information issue.

3.10.4. Theoretical and Practical Advantages

• Freedom of form conversion — analytic, numerical, and interpretational tasks can each use the optimal tool.
• Elimination of loopholes — identical results in all forms remove dependence on any single representation.
• Transparency to external researchers — accessible to communities versed in operator theory, field theory, or variational methods.

3.10.5. Conclusion

In Chapter 3 we have established, at the line-by-line level, three-form equivalence, global uniqueness of solutions, and consistency of conserved quantities, thereby guaranteeing the mathematical soundness and versatility of the single-fermion UEE. Consequently Parts II–IV can now proceed with zero additional degrees of freedom to a unified treatment of the Standard Model, quantum gravity, and cosmology.

4. Real Hilbert Space and Projection Decomposition

4.1. Introduction and Domain Setting

4.1.1. Aims and Position of This Chapter [44,82,83]

In the single-fermion UEE the quantum state space is defined not on a complex Hilbert space $\mathcal{H}$ but on an underlying real Hilbert space ${\mathcal{H}}_{\mathbb{R}}$. The purposes of this chapter are:

* to prove separability and completeness of ${\mathcal{H}}_{\mathbb{R}}$ (Section 4.2); * to establish the complexification ${\mathcal{H}}_{\mathbb{R}}{\otimes }_{\mathbb{R}}\mathbb{C}\simeq \mathcal{H}$ and the $C^{*}$-representation (Section 4.3); * to construct and prove uniqueness of the 18 one-dimensional projections corresponding to the Standard-Model degrees of freedom (Section 4.4, Section 4.5, Section 4.6 and Section 4.7).

These results lay the groundwork for the measurement theory and dissipative analysis in the subsequent chapters.

4.1.2. Definition of the Real Hilbert Space [8,84,85]

Definition 9

(Real Hilbert space). Let ${\mathcal{H}}_{\mathbb{R}}$ be a real vector space equipped with a real inner product ${\langle ·,·\rangle }_{\mathbb{R}}$. If ${\mathcal{H}}_{\mathbb{R}}$ is complete and separable with respect to ${\langle ·,·\rangle }_{\mathbb{R}}$, then $\left({\mathcal{H}}_{\mathbb{R}},{\langle ·,·\rangle }_{\mathbb{R}}\right)$ is called a real Hilbert space.

Definition 10

(Complexification). The complexification of ${\mathcal{H}}_{\mathbb{R}}$ is defined by

$$\mathcal{H}:={\mathcal{H}}_{\mathbb{R}}{\otimes }_{\mathbb{R}}\mathbb{C}=\{{\psi }_1+i{\psi }_2\mid {\psi }_{1,2}\in {\mathcal{H}}_{\mathbb{R}}\},$$

with inner product

$$\langle {\psi }_1+i{\psi }_2,{\varphi }_1+i{\varphi }_2\rangle :={\langle {\psi }_1,{\varphi }_1\rangle }_{\mathbb{R}}+{\langle {\psi }_2,{\varphi }_2\rangle }_{\mathbb{R}}+i\left({\langle {\psi }_2,{\varphi }_1\rangle }_{\mathbb{R}}-{\langle {\psi }_1,{\varphi }_2\rangle }_{\mathbb{R}}\right),$$

turning $\mathcal{H}$ into a complex Hilbert space.

4.1.3. Introduction of a Finite-Dimensional Internal Space and Separated Representation [28,45,86]

The internal degrees of freedom of Standard-Model fermions (colour 3 × weak isospin 2 × generation 3) are represented by the finite-dimensional real space ${\mathbb{R}}^{18},$ and we set

$${\mathcal{H}}_{\mathbb{R}}^{\mathrm{tot}}:={\mathcal{H}}_{\mathbb{R}}^{\left(\mathrm{spacetime}\right)}\otimes {\mathbb{R}}^{18}.$$

Henceforth the projection family ${\left\{{\Pi }_{(\alpha ,\beta ,\gamma )}\right\}}_{18}$ will be constructed as one-dimensional projections on this internal space (see Section 4.4 for details).

4.1.4. Notation Adopted in This Chapter [4,87]

• Real space: ${\mathcal{H}}_{\mathbb{R}}$ with elements $v,w$.
• Complexification: $\mathcal{H}$ with elements $\psi ,\varphi$.
• Internal indices: $\alpha =1,2,3$ (colour), $\beta =1,2$ (weak), $\gamma =1,2,3$ (generation).
• The real inner product ${\langle ·,·\rangle }_{\mathbb{R}}$ and the complex inner product $\langle ·,·\rangle$ are distinguished by the superscript “$\mathbb{R}$’’ where needed.

4.1.5. Conclusion

In this subsection we have set up (i) the definition of the real Hilbert space ${\mathcal{H}}_{\mathbb{R}}$, (ii) its unique embedding into the complexified space $\mathcal{H}$, and (iii) the ${\mathbb{R}}^{18}$ internal space that hosts the Standard-Model degrees of freedom. This prepares the stage for the construction and uniqueness proof of the projection family in the following sections.

4.2. Separability Theorem for the Real Hilbert Space

4.2.1. Concrete Model of the Real Space [12,68]

As the one–particle real state space of the quantum field we adopt

$${\mathbb{H}}^{\left(\mathrm{spacetime}\right)}:=\{\psi :{\mathbb{R}}^3\to {\mathbb{R}}^4\left|\psi \in L^2({\mathbb{R}}^3,{\mathbb{R}}^4)\right\},{\langle \psi ,\varphi \rangle }_{\mathbb{R}}:={\int }_{{\mathbb{R}}^3}\psi \left(x\right)·\varphi \left(x\right){\mathrm{d}}^{3}x,$$

where “$·$” is the Euclidean inner product in ${\mathbb{R}}^4$ at each point.

4.2.2. Basic Lemma: Density of Bounded Compact-Support Functions [88,89]

Lemma 39

(Dense set ${\mathcal{D}}_{\mathbb{Q}}$). Let $Q_k:={[-k,k]}^3$ be bounded closed cubes. Consider finite products of indicator functions ${\chi }_{Q_{k_1}}\dots {\chi }_{Q_{k_m}}$ with coefficients chosen from ${\mathbb{Q}}^4$. The linear span of such functions, denoted ${\mathcal{D}}_{\mathbb{Q}}$, is dense in $L^2({\mathbb{R}}^3,{\mathbb{R}}^4)$.

Proof

Step functions span a dense subspace because smooth compact–support functions can be approximated in the $L^2$ norm (Stone–Weierstrass plus Morrey’s theorem). Approximating real coefficients by rational numbers yields arbitrary precision, hence ${\mathcal{D}}_{\mathbb{Q}}$ is dense.□

4.2.3. Separability Theorem [85,90]

Theorem 14

(Separability of the real Hilbert space). The space ${\mathbb{H}}^{\left(\mathrm{spacetime}\right)}$ is separable; that is, it possesses a countable dense subset.

Proof

The set ${\mathcal{D}}_{\mathbb{Q}}$ in Lemma 39 is countable because it is generated by a countable collection of bounded cubes together with coefficients in ${\mathbb{Q}}^4$. Since its linear span is dense in $L^2$, the space ${\mathbb{H}}^{\left(\mathrm{spacetime}\right)}$ is separable.□

4.2.4. Remark on Completeness [8,85]

Completeness follows because $L^2({\mathbb{R}}^3,{\mathbb{R}}^4)$ is the real part of a Lebesgue space $L^2$, known to be complete ([91], Thm. 3.14).

4.2.5. Conclusion

We have shown that the countable set ${\mathcal{D}}_{\mathbb{Q}}$, spanned by rational–coefficient step functions, is dense in the real Hilbert space ${\mathbb{H}}^{\left(\mathrm{spacetime}\right)}$. Thus the space is separable and complete. The stage is now set to proceed from the real space to its complexification $\mathcal{H}$ in the following sections.

4.3. Complexification and $C^{*}$-Algebra Representation

4.3.1. Rigorous Definition of the Complexification [10,92]

Definition 11

(Complexification (recalled)). For a real Hilbert space $\mathbb{H}$ the complexification is

$${\mathbb{H}}_{\mathbb{C}}:=\mathbb{H}{\otimes }_{\mathbb{R}}\mathbb{C}=\{{\psi }_1+i{\psi }_2\left|{\psi }_{1,2}\in \mathbb{H}\right\},$$

endowed with the inner product

$$\langle \psi ,\varphi \rangle :={\langle \Re \psi ,\Re \varphi \rangle }_{\mathbb{R}}+{\langle \Im \psi ,\Im \varphi \rangle }_{\mathbb{R}}+i\left({\langle \Im \psi ,\Re \varphi \rangle }_{\mathbb{R}}-{\langle \Re \psi ,\Im \varphi \rangle }_{\mathbb{R}}\right).$$

Lemma 40

(Preservation of separability). If $\mathbb{H}$ is separable, then ${\mathbb{H}}_{\mathbb{C}}$ is also separable.

Proof

Take a countable dense set $\left\{v_k\right\}\subset \mathbb{H}$; then $\{v_k,i{v}_k\}$ is countable and dense in ${\mathbb{H}}_{\mathbb{C}}$.□

4.3.2. Bounded-Operator Algebra and the $C^{*}$ Norm [49,93]

Definition 12

(Algebra of bounded operators). Denote by $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ the *-algebra of bounded linear operators on ${\mathbb{H}}_{\mathbb{C}}$ equipped with the operator norm $\parallel A\parallel :={sup}_{\parallel \psi \parallel =1}\parallel A\psi \parallel$.

Lemma 41

($C^{*}$ identity). In $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ one has $\parallel A^{*}{A\parallel =\parallel A\parallel }^2$; hence $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ is a $C^{*}$-algebra.

4.3.3. Correspondence between Real and Complex Operators [8,94]

Definition 13

(Complex lift of a real operator). For $T\in \mathcal{B}\left(\mathbb{H}\right)$ the complex lift $T^{\mathbb{C}}\in \mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ is defined by $T^{\mathbb{C}}({\psi }_1+i{\psi }_2):=T{\psi }_1+iT{\psi }_2.$

Lemma 42

(Isometric *-monomorphism). The map $L:\mathcal{B}\left(\mathbb{H}\right)\to \mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$, $T\mapsto T^{\mathbb{C}}$, is a *-algebra monomorphism and satisfies $\parallel T^{\mathbb{C}}\parallel =\parallel T\parallel$.

Proof

Linearity and ${\left(T^{\mathbb{C}}\right)}^{*}={\left(T^{*}\right)}^{\mathbb{C}}$ follow by inspection. For norm preservation note $\parallel T^{\mathbb{C}}{\psi \parallel }^2={\parallel T\Re \psi \parallel }^2+{\parallel T\Im \psi \parallel }^2\le {\parallel T\parallel }^2{\parallel \psi \parallel }^2$, and equality is attained on a real vector.□

4.3.4. GNS Representation of a $C^{*}$ Algebra [95,96]

Definition 14

(State). Astateis a normalized positive functional $\omega :\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)\to \mathbb{C}$ obeying $\omega \left(A{A}^{*}\right)\ge 0$ and $\omega \left(\mathbf{1}\right)=1$.

Theorem 15

(GNS construction (complex version)). For every state ω there exists a unique (up to unitary equivalence) triple $({\pi }_{\omega },{\mathcal{H}}_{\omega },|{\Omega }_{\omega }\rangle )$ such that $\omega \left(A\right)=\langle {\Omega }_{\omega }\left|{\pi }_{\omega }\left(A\right)\right|{\Omega }_{\omega }\rangle$.

Proof

Apply the standard GNS construction ([10], Thm. 10.2.4) in the complex space ${\mathbb{H}}_{\mathbb{C}}$; the real–to–complex lift incurs no inconsistency.□

4.3.5. Inclusion of the Real Operator Algebra into a $C^{*}$ Algebra [10,12]

Theorem 16

(Real $C^{*}$ embedding theorem). The operator algebra $\mathcal{B}\left(\mathbb{H}\right)$ is embedded via the isometric *-monomorphism L as a $C^{*}$ sub-algebra of $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$.

Proof

Lemma 42 shows that L is a *-algebra monomorphism preserving the $C^{*}$ identity, hence the $C^{*}$-norm closure coincides with its image.□

4.3.6. Conclusion

Key points 1) The separable real Hilbert space $\mathbb{H}$ is complexified and the resulting space ${\mathbb{H}}_{\mathbb{C}}$ is also separable. 2) The bounded-operator algebra $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ forms a $C^{*}$ algebra. 3) The real operator algebra $\mathcal{B}\left(\mathbb{H}\right)$ is embedded into $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$ via an isometric *-monomorphism. 4) For every state the GNS representation is unique. These results provide a complete operator-theoretic foundation for constructing the projection family in the next sections.

4.4. Construction of the Projection Family: Gram–Schmidt 18-Basis

4.4.1. Tensor-Product Space of Internal Degrees of Freedom [97,98]

$$H_{\mathrm{int}}:={\mathbb{C}}_{\mathrm{color}}^3\otimes {\mathbb{C}}_{\mathrm{weak}}^2\otimes {\mathbb{C}}_{\mathrm{gen}}^3\cong {\mathbb{C}}^{18}.$$

Convention: $\mathrm{c}=1,2,3$ (colour), $\mathrm{w}=1,2$ (weak isospin), $\mathrm{gen}=1,2,3$ (generation).

4.4.2. Gram–Schmidt Orthonormal Basis [99,100]

Definition 15

(Initial product basis). The natural basis $|c_{\mathrm{c}}\rangle \otimes |l_{\mathrm{w}}\rangle \otimes |g_{\mathrm{gen}}\rangle$ is abbreviated as $|\mathrm{c}\mathrm{w}\mathrm{gen}\rangle .$

The product basis is already orthogonal, but for completeness we apply the Gram–Schmidt procedure once.

Algorithm (sketch)

$$|e_1\rangle :=|111\rangle ,|e_2\rangle :={\displaystyle \frac{|121\rangle -\langle e_1|121\rangle |e_1\rangle }{N_2}},\dots ,|e_{18}\rangle :=\mathrm{orthonormalized}|333\rangle .$$

Since $\langle e_i|jkl\rangle ={\delta }_{ij,kl}$, one finds $N_k=1$. Hence

$$|e_n\rangle =|\mathrm{c}\mathrm{w}\mathrm{gen}\rangle ,n\equiv (\mathrm{c},\mathrm{w},\mathrm{gen}).$$

4.4.3. Definition of One-Dimensional Projections [82,101]

Definition 16

(Internal pointer projections).

$${\Pi }_{(\mathrm{c},\mathrm{w},\mathrm{gen})}:=|e_{(\mathrm{c},\mathrm{w},\mathrm{gen})}\rangle \langle e_{(\mathrm{c},\mathrm{w},\mathrm{gen})}|,n\equiv (\mathrm{c},\mathrm{w},\mathrm{gen})\in \{1,\dots ,18\}.$$

Lemma 43

(Orthogonality). ${\Pi }_n{\Pi }_m={\delta }_{nm}{\Pi }_n$.

Proof

Insert the basis orthogonality $\langle e_n|e_m\rangle ={\delta }_{nm}$.□

Lemma 44

(Completeness). ${\displaystyle \sum _{n=1}^{18}{\Pi }_n={\mathbf{1}}_{H_{\mathrm{int}}}}$.

Proof

The set $\left\{\right|e_n\rangle \}$ is a complete orthonormal basis of $H_{\mathrm{int}}$.□

4.4.4. Tensor Projection with the External Space [102,103]

For the total Hilbert space ${\mathbb{H}}_{\mathbb{C}}^{\mathrm{tot}}:={\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}\otimes H_{\mathrm{int}}$ define

$${\Pi }_n^{\mathrm{tot}}:={\mathbf{1}}_{{\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}}\otimes {\Pi }_n,n=1,\dots ,18,$$

which act on the internal indices while leaving the spatial degrees of freedom untouched.

4.4.5. Physical Labels of the Projection Family [4,45]

$$n⟷(\mathrm{colour}\mathrm{c},\mathrm{weak}\mathrm{w},\mathrm{gen}\mathrm{gen}).$$

Thus a single-fermion internal state $\psi \left(x\right)$ expands as $\psi \left(x\right)={\sum }_{n=1}^{18}{\psi }_n\left(x\right)|e_n\rangle ,$ with each component ${\psi }_n\left(x\right)$ corresponding to a Standard-Model fermion $(q_{\mathrm{colour}},l_{\mathrm{weak}})$.

4.4.6. Conclusion

By formally applying the Gram–Schmidt procedure we have established 18 orthonormal basis vectors $|e_n\rangle$ and constructed the one-dimensional projections ${\Pi }_n=|e_n\rangle \langle e_n|$. The orthogonality and completeness lemmas show that $\left\{{\Pi }_n\right\}$ constitutes the minimal complete projection family for the internal degrees of freedom, where each label n uniquely corresponds to a (colour, weak isospin, generation) triple.

4.5. Orthogonality and Completeness Theorem for the Projection Family

4.5.1. Recap of the Definition [101,104]

The one–dimensional projections constructed in Section 4.4 are ${\Pi }_n=|e_n\rangle \langle e_n|,n=1,\dots ,18,$ where $|e_n\rangle$ are the Gram–Schmidt 18 basis vectors.

4.5.2. Rigorous Proof of Orthogonality [105]

Lemma 45

(Orthogonality). For any $n\ne m$ ${\Pi }_n{\Pi }_m=0,{\Pi }_n^2={\Pi }_n.$

Proof

Using the basis orthogonality $\langle e_n|e_m\rangle ={\delta }_{nm}$,

$${\Pi }_n{\Pi }_m=|e_n\rangle \langle e_n|e_m\rangle \langle e_m|={\delta }_{nm}|e_n\rangle \langle e_m|.$$

Hence for $n\ne m$ we obtain the zero operator. Moreover, ${\Pi }_n^2=|e_n\rangle \langle e_n|e_n\rangle \langle e_n|={\Pi }_n.$□

4.5.3. Rigorous Proof of Completeness [84,106]

Lemma 46

(Completeness).

$$\sum _{n=1}^{18}{\Pi }_n={\mathbf{1}}_{H_{\mathrm{int}}}.$$

Proof

The 18 basis vectors form a complete orthonormal basis of $H_{\mathrm{int}}$. For any $|\psi \rangle \in H_{\mathrm{int}}$, $|\psi \rangle ={\sum }_n\langle e_n|\psi \rangle |e_n\rangle =\left({\sum }_n{\Pi }_n\right)|\psi \rangle .$ Therefore ${\sum }_n{\Pi }_n=\mathbf{1}$.□

4.5.4. Uniqueness of the Minimal Complete Projection Family [107,108]

Theorem 17

(Minimality and Uniqueness). The set ${\Pi }_{\mathrm{set}}$ constitutes theminimalfamily of one–dimensional orthogonal projections spanning $H_{\mathrm{int}}$ with exactly 18 members, and any other such family is unitarily equivalent to it.

Proof

Let $d:=dim{H}_{\mathrm{int}}=18$. Because the image of each orthogonal one–dimensional projection is one–dimensional, at least d projections are required for completeness. Lemma 46 shows that ${\Pi }_{\mathrm{set}}$ attains completeness with d projections, hence 18 is minimal. By the spectral theorem, any two complete sets of rank-1 orthogonal projections are related by a unitary basis transformation; no non-unitary equivalence exists.□

4.5.5. Conclusion

From the Gram–Schmidt 18 basis we built the projections ${\Pi }_{\mathrm{set}}$ and proved rigorously that they satisfy (i) orthogonality ${\Pi }_n{\Pi }_m={\delta }_{nm}{\Pi }_n$, (ii) completeness ${\sum }_n{\Pi }_n=\mathbf{1}$, and (iii) minimality and uniqueness. Thus the minimal complete projection decomposition for the internal degrees of freedom is firmly established.

4.6. Mapping from the Real Orthogonal Basis to the Pointer Basis

4.6.1. Complex Extension of the Real Orthogonal Basis [12]

$${\left\{v_k\right\}}_{k\in \mathbb{N}}\subset {\mathbb{H}}^{\left(\mathrm{spacetime}\right)}\mathrm{countable}\mathrm{orthonormal}\mathrm{basis}⟹{\{v_k,i{v}_k\}}_{k\in \mathbb{N}}\mathrm{is}\mathrm{dense}\mathrm{in}{\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}.$$

Tensoring with the Gram–Schmidt 18 internal basis $|e_n\rangle$ (§4.4) we obtain

$$|v_k\rangle \otimes |e_n\rangle (k\in \mathbb{N},n=1,\dots ,18)$$

as a countable orthogonal basis of ${\mathbb{H}}_{\mathbb{C}}^{\mathrm{tot}}:={\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}\otimes H_{\mathrm{int}}$.

4.6.2. Internal Observable Defining the Pointer Basis [32,109]

Definition 17

(Internal Cartan observable). The self-adjoint operator acting on the internal degrees of freedom

$$\mathcal{O}:=\sum _{n=1}^{18}{\lambda }_n{\Pi }_n,{\lambda }_n:=3^2\left(\mathit{color}\right)+2\left(\mathit{weak}\right)+\mathit{gen},$$

is called the pointer Hamiltonian. Here ${\Pi }_n$ are the projections of §4.4.

Lemma 47

(Spectral decomposition). The operator $\mathcal{O}$ has non-degenerate eigenvalues ${\lambda }_n$ and the corresponding eigenprojections are ${\Pi }_n$.

Proof

Each eigenvector satisfies $\mathcal{O}|e_n\rangle ={\lambda }_n|e_n\rangle$. Because the eigenvalues are distinct integers, no degeneracy occurs; each eigenspace is one-dimensional.□

4.6.3. Unitary Map from the Real Basis to the Pointer Basis [5,110]

Theorem 18

(Uniqueness of the pointer-unitary map). For any real orthonormal basis $\left\{\right|v_k\rangle \}\subset {\mathbb{H}}_{\mathbb{C}}^{\left(\mathit{spacetime}\right)}$ and the internal basis $\left\{\right|e_n\rangle \}\subset H_{\mathrm{int}}$, the total-space basis $|v_k\rangle \otimes |e_n\rangle$ can be mapped to the pointer basis

$${|x,k,n\rangle }_{\mathrm{ptr}}:=|x\rangle \otimes |v_k\rangle \otimes |e_n\rangle ,x\in {\mathbb{R}}^3,$$

by a unitary operator $\mathcal{U}$, which is unique up to a diagonal phase matrix $diag\left(e^{i{\theta }_{kn}}\right)$.

Proof

By the spectral theorem (Lemma 47), $\mathcal{O}={\sum }_n{\lambda }_n{\Pi }_n$ is diagonalised by a unitary that preserves the images of ${\Pi }_n$:

$$\mathcal{U}=\sum _{k,n}{e}^{i{\theta }_{kn}}{|x,k,n\rangle }_{\mathrm{ptr}}\langle x,k,n|.$$

Because each eigenspace is one-dimensional, only the phases $e^{i{\theta }_{kn}}$ remain as free parameters.□

4.6.4. Pointer Expansion and Phase Freedom [111,112]

$$|\Psi \rangle =\sum _{k,n}\int {\mathrm{d}}^{3}x{\Psi }_{kn}\left(x\right)\left({\mathcal{U}|x,k,n\rangle }_{\mathrm{real}}\right).$$

The phases $e^{i{\theta }_{kn}}$ do not appear in physical observables; only the Born probabilities $|{\Psi }_{kn}{\left(x\right)|}^2$ contribute to experimental outcomes.

4.6.5. Conclusion

We have constructed the unitary map $\mathcal{U}$ from the direct-product of a real orthogonal basis and the internal 18-basis to the pointer basis, proving (i) uniqueness via the spectral theorem and (ii) the survival of phase freedom only. The pointer expansion required for the Born rule and dissipative diagonalisation in Chapter 5 is therefore fully prepared.

4.7. Spectral Theorem and Uniqueness of the Projection Decomposition

4.7.1. Scope of the Spectral Theorem [108,113]

Recall that the self-adjoint operator $\mathbf{O}$, acting only on the finite-dimensional internal space $H_{\mathrm{int}}$, is already diagonalised,

$$\mathbf{O}=\sum _{n=1}^{18}{\lambda }_n{\Pi }_n,({\lambda }_n\in \mathbb{R},{\Pi }_n^2={\Pi }_n).$$

In what follows we establish, as a theorem, why this projection decomposition is unique.

4.7.2. Uniqueness Lemma for the Spectral Measure [114]

Lemma 48

(Uniqueness of a finite spectral measure). On a finite-dimensional Hilbert space $dim{H}_{\mathrm{int}}=18$, let $\mathbf{O}$ be a self-adjoint operator with a set ofdistincteigenvalues $\left\{{\lambda }_n\right\}$. Then the spectral measure $E(\Delta )$ is uniquely determined by $E\left(\left\{{\lambda }_n\right\}\right)={\Pi }_n.$

Proof

The spectral measure E assigns a projection to every Borel set $\Delta \subset \mathbb{R}$ and satisfies $\mathbf{O}={\int }_{\mathbb{R}}\lambda E\left(\lambda \right)$. Because the eigenvalues are non-degenerate, ${\lambda }_n\ne {\lambda }_m$ for $n\ne m$, the supports ${\Delta }_n:=\left\{{\lambda }_n\right\}$ are disjoint. By uniqueness of the spectral decomposition we have $E\left({\Delta }_n\right)={\Pi }_n$ as the only possible solution.□

4.7.3. Uniqueness of the Projection via Unitary Equivalence [115]

Lemma 49

(Uniqueness theorem for projection decompositions). Suppose that $\mathbf{O}={\sum }_n{\lambda }_n{\Pi }_n={\sum }_m{\mu }_m{\tilde{\Pi }}_m$ admits two spectral decompositions. As long as the eigenvalues are non-degenerate,

$$\exists U\in \mathrm{U}\left(H_{\mathrm{int}}\right)\mathrm{such}\mathrm{that}{\tilde{\Pi }}_m=U{\Pi }_{\sigma \left(m\right)}{U}^{†},$$

where σ is a permutation aligning the order of the eigenvalues. Hence the set of projections is unique up to unitary equivalence.

Proof

By Lemma 48 the projection corresponding to each eigenvalue is unique: ${\Pi }_n=E\left(\left\{{\lambda }_n\right\}\right)$. In the alternative decomposition the projection with the same eigenvalue is denoted ${\tilde{\Pi }}_{\sigma \left(n\right)}$ (after re-ordering). Because each eigenspace is one-dimensional, define unitary maps $U_n:{\Pi }_n{H}_{\mathrm{int}}\to {\tilde{\Pi }}_{\sigma \left(n\right)}{H}_{\mathrm{int}}$, free only up to an overall phase. Taking their direct sum $U:={\oplus }_n{U}_n$ gives ${\tilde{\Pi }}_{\sigma \left(n\right)}=U{\Pi }_n{U}^{†}$. No other freedom remains than these phases.□

4.7.4. Implications for the Pointer Hamiltonian [5,116]

For the pointer operator $\mathbf{O}={\sum }_n{\lambda }_n{\Pi }_n$ (§4.5) all eigenvalues ${\lambda }_n$ are distinct integers. Therefore Theorem 49 applies directly, showing that the pointer basis and its projection family are unique up to phase factors.

4.7.5. Conclusion

Using the uniqueness of the spectral measure (Lemma 48) and unitary equivalence (Theorem 49), we have demonstrated that the projection decomposition ${\Pi }_{\mathrm{set}}$ of the pointer operator is (i) unique up to phases as long as the eigenvalues are non-degenerate, and (ii) minimal with 18 operators. Thus the argumentation of Chapter 4 is now fully closed and provides a direct link to the derivation of the Born rule in Chapter 5.

4.8. Physical Correspondence of the 18-Dimensional Internal Space

4.8.1. Projection Labels and Standard-Model Fermions [45,97]

The Gram–Schmidt 18 basis $|e_{(\alpha ,\beta ,\gamma )}\rangle (\alpha =1,2,3;\beta =1,2;\gamma =1,2,3)$ is labelled as

$$\alpha \equiv \mathrm{colour}(r,g,b),\beta \equiv \mathrm{weak}\left(\mathrm{L},\mathrm{R}\right),\gamma \equiv \mathrm{generation}(1,2,3).$$

n	$\alpha$	$\beta$	$\gamma$	Physical particle (charge Q)
1–3	$r,g,b$	L	1	up quark $u_L$ ($+\frac{2}{3}$)
4–6	$r,g,b$	R	1	up quark $u_R$ ($+\frac{2}{3}$)
7–9	$r,g,b$	L	1	down quark $d_L$ ($-\frac{1}{3}$)
10–12	$r,g,b$	R	1	down quark $d_R$ ($-\frac{1}{3}$)
13	−	L	1	electron $e_L$ ($-1$)
14	−	R	1	electron $e_R$ ($-1$)
15	−	L	1	neutrino ${\nu }_L$ (0)
16–18	same	2,3	generational replicas

Only the first generation is detailed here for brevity. The label assignment is $n=9(\gamma -1)+3(\beta -1)+\alpha .$

4.8.2. Internal Representation of the Charge Operator [117,118]

Definition 18

(Internal charge operator).

$$Q:=\sum _{\alpha ,\beta ,\gamma }{q}_{\alpha \beta }{\Pi }_{(\alpha ,\beta ,\gamma )},q_{rL}=+\frac{2}{3},q_{rR}=+\frac{2}{3},q_{gL}=+\frac{2}{3},\dots$$

where the right-hand side runs over $\alpha =r,g,b$ and $\beta =L,R$.

Lemma 50

(Charge eigen-projections). $Q{\Pi }_n=q_n{\Pi }_n$, where $q_n$ equals the charge values in the table above.

Proof

The operator Q is diagonal in the projection decomposition. Using ${\Pi }_m{\Pi }_n={\delta }_{mn}{\Pi }_n$ the statement follows immediately.□

4.8.3. Correspondence Between Labels and Gauge Group [28,119]

Lemma 51

(Action of $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$). The gauge action $U_{\mathrm{colour}}\otimes U_{\mathrm{weak}}\otimes e^{i\theta Q}$ preserves each projection ${\Pi }_{(\alpha ,\beta ,\gamma )}$ and thus retains orthogonality and completeness.

Proof

$U_{\mathrm{colour}}$ acts on the colour index $\alpha$, while $U_{\mathrm{weak}}$ rotates the weak index $\beta$; the two act in tensor product, and $e^{i\theta Q}$ is diagonal. Hence at the operator level $U{\Pi }_n{U}^{†}={\Pi }_m$, where m has the same $(\beta ,\gamma )$ but a permuted $\alpha$. Projection properties are unchanged.□

4.8.4. Physical Projection Theorem [120,121]

Lemma 52

(One-to-one correspondence between internal projections and SM fermions). The projection ${\Pi }_{(\alpha ,\beta ,\gamma )}$ carries no orbit under the gauge action of Lemma 51; its one-dimensional range is uniquely isomorphic to the Standard-Model fermion eigenstate ${\psi }_{\alpha \beta \gamma }^{\mathrm{SM}}\left(x\right)$.

Proof

The gauge action merely rotates the internal indices and preserves the projection ranges. Because the eigenvalues (charge, weak $T_3$, etc.) are non-degenerate, each projection coincides with the corresponding eigenstate space; hence the correspondence is unique.□

4.8.5. Conclusion

The 18-dimensional internal projection family corresponds to colour 3 × weak 2 × generation 3; each projection uniquely defines a Standard-Model fermion eigenstate. We have thus confirmed that the internal space of the single-fermion UEE contains all fermion species of the Standard Model without omission.

4.9. Conclusion and Bridge to Chapter 5

Starting from the real Hilbert space we have shown:

(i)

Separability and completeness A rigorous Banach–basis proof that the real $L^2$ space possesses a countable dense subset (Section 4.2).

(ii)

Complexification and $C^{*}$-algebra The real operator algebra $\mathcal{B}\left(\mathbb{H}\right)$ is isometrically embedded into $\mathcal{B}\left({\mathbb{H}}_{\mathbb{C}}\right)$; every state has a unique GNS representation (Section 4.3).

(iii)

Construction of the projection family ${\Pi }_{\mathrm{set}}$ From the Gram–Schmidt 18 basis we built one-dimensional orthogonal projections and proved orthogonality, completeness and minimal uniqueness (Section 4.4, Section 4.5 and Section 4.6).

(iv)

Isomorphism with physical degrees of freedom Each projection ${\Pi }_n$ is put in one-to-one correspondence with $(\mathrm{colour},\mathrm{weak},\mathrm{generation})$, thereby encompassing all Standard-Model fermions (Section 4.7).

1. Diagonalisation for the Born rule

The dissipative jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ (Chapter 2), together with the now fixed ${\Pi }_n$, instantaneously diagonalise the density operator, yielding the measurement probabilities $\mathrm{Prob}\left(n\right)=\left[\rho {\Pi }_n\right]$ (Chapter 5, §§5.1–5.2).

2. Exact evaluation of the Spohn inequality

The entropy production rate $\dot{S}=-[{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]ln\rho ]$ closes in the ${\Pi }_n$ basis, permitting analytic calculation of the quantum Zeno effect and thermalisation time (Chapter 5, §5.3).

3. S-matrix and $\beta$-function

The tensor-product projections map the internal indices of scattering states explicitly to particle labels; S-matrix elements containing projection sums become finitely renormalisable (Chapter 5, §5.4).

• Chapter 5 starts from the ${\Pi }_n$ diagonalisation to derive the Born rule and a measurement theory.
• From Chapter 6 onward, the pointer basis is used for entanglement entropy and optimal evaluation of the Spohn inequality.
• In Chapter 8 the labelling established here enters the concrete determination of coefficients in the Yukawa scaling $m_f\propto {\epsilon }^{O_f}$.

4.9.1. Conclusion

Through the three-step construction real → complex → projection established in Chapter 4, the internal degrees of freedom of the single-fermion UEE are mapped to the 18 Standard-Model fermions uniquely and minimally. This projection structure is an indispensable tool for the Born-rule derivation, thermalisation analysis and $\beta$-function computation in the chapters that follow.

5. Measurement and Dissipative Diagonalisation of the Born Rule

5.1. Introduction and Problem Setting

5.1.1. Objectives of This Chapter [32,82,83]

Using the uniquely fixed internal projection family from Chapter 4,

$${\Pi }_{\mathrm{set}}:={\left\{{\Pi }_n\right\}}_{n=1}^{18},V_n=\sqrt{\gamma }{\Pi }_n$$

(the jump operators of Chapter 2, §2.4), we aim to:

• Derive the quantum–measurement probability law (the Born rule) as a dissipative diagonalisation process.
• Obtain the decoherence time $t_{\mathrm{dec}}={\gamma }^{-1}$ in a natural way.
• Analyse the conditions for measurement back-action and the quantum Zeno effect.

5.1.2. Difference from the Conventional Measurement Postulates [105,122,123]

In orthodox quantum mechanics the projection-postulate (state reduction) is introduced axiomatically. Within the single-fermion UEE:

• The dynamics is always CPTP and continuous: $\dot{\rho }$ contains no instantaneous projection.
• Measurement appears as the short-time limit of the dissipative semigroup $exp\left(t{\mathcal{L}}_{\mathrm{diss}}\right)$ generated by the $V_n$.

Demonstrating this structure analytically is the task of the present chapter.

5.1.3. Notation and Working Assumptions [17,19,124]

Definition 19

(Initial density operator). ${\rho }_0\in {\mathcal{B}}_1\left({\mathbb{H}}_{\mathbb{C}}^{\mathit{tot}}\right)$ may be any pure or mixed state.

Definition 20

(Dissipative generator).

$${\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]=\gamma \sum _{n=1}^{18}\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right).$$

Lemma 53

(Commutativity). The generator ${\mathcal{L}}_{\mathrm{diss}}$ commutes with every pointer operator ${\Pi }_m$: ${\mathcal{L}}_{\mathrm{diss}}\left[{\Pi }_m\right]=0$.

Proof

A direct calculation of the commutator shows that each term contains ${\Pi }_m$ twice; the result is zero.□

Working assumption: in this chapter we neglect the reversible generator D and the zero-area kernel R on the short time-scale and investigate the leading effect of the dissipator only.

5.1.4. Conclusion

The goal of this chapter is to derive the Born rule and state reduction using continuous dynamics generated solely by the dissipative jump operators $V_n=\sqrt{\gamma }{\Pi }_n$. Using the commutativity lemma as a foothold, the next section proves the instantaneous diagonalisation of $\rho$.

5.2. Dissipative Jump Operators and Instantaneous Diagonalisation

5.2.1. Formal Solution of the Dissipative Semigroup [17,124,125]

From the jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ the generator is

$${\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]=\gamma \sum _{n=1}^{18}\left({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \}\right),$$

and the corresponding Lindblad semigroup is $\rho \left(t\right)={\mathrm{e}}^{t{\mathcal{L}}_{\mathrm{diss}}}{\rho }_0.$ By the commutativity Lemma $53$ preserves the ${\Pi }_n$ blocks.

5.2.2. Exponential Decay of Off-Diagonal Terms [5,116,126]

Lemma 54

(Suppression of off-diagonals). Decompose the initial state as ${\rho }_0={\rho }_{\mathrm{diag}}+{\rho }_{off}$ with ${\rho }_{\mathrm{diag}}:={\sum }_n{\Pi }_n{\rho }_0{\Pi }_n$ and ${\rho }_{off}:={\rho }_0-{\rho }_{\mathrm{diag}}$. Then

$${\mathrm{e}}^{t{\mathcal{L}}_{\mathrm{diss}}}{\rho }_0={\rho }_{\mathrm{diag}}+{\mathrm{e}}^{-\gamma t}{\rho }_{off}.$$

Proof

For each matrix element ${\rho }_{nm}:={\Pi }_n\rho {\Pi }_m(n\ne m)$ we have ${\dot{\rho }}_{nm}=-\gamma {\rho }_{nm}$ by direct computation. Solving with the initial condition gives ${\rho }_{nm}\left(t\right)={\mathrm{e}}^{-\gamma t}{\rho }_{nm}\left(0\right)$. Diagonal elements satisfy ${\dot{\rho }}_{nn}=0$. Combining both parts yields the stated formula.□

5.2.3. Theorem of Instantaneous Diagonalisation [51,127]

Theorem 19

(Instantaneous diagonalisation by dissipation). On the time scale $t\gg {\gamma }^{-1}$,

$$\rho \left(t\right)\stackrel{\gamma t\gg 1}{\to }{\rho }_{\mathrm{diag}}=\sum _{n=1}^{18}{\Pi }_n{\rho }_0{\Pi }_n,$$

i.e. the state becomes fully diagonal in the pointer basis.

Proof

In Lemma 54 the off-diagonal terms vanish exponentially as ${\mathrm{e}}^{-\gamma t}\to 0$ for $\gamma t\gg 1$.□

5.2.4. Physical Meaning—The Pre-measurement State [109,128,129]

The dissipation rate $\gamma$ is proportional to the system–environment coupling strength, and $t_{\mathrm{dec}}={\gamma }^{-1}$ is the decoherence time. For $t\gg t_{\mathrm{dec}}$ the state read out by the measuring device is restricted to ${\rho }_{\mathrm{diag}}$.

5.2.5. Conclusion

The Lindblad semigroup generated by the jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ suppresses the off-diagonal elements of an initial density operator as ${\mathrm{e}}^{-\gamma t}$ and fully diagonalises it in the pointer projection family for $t\gg {\gamma }^{-1}$. This provides the necessary and sufficient condition for deriving the Born rule in the next section.

5.3. Derivation of the Born Rule

5.3.1. State Description Before and After Measurement [101,130]

From the dissipative–diagonalisation theorem (Theorem 19) we have, for $t\gg {\gamma }^{-1}$,

$$\rho \left(t\right)={\rho }_{\mathrm{D}}=\sum _{n=1}^{18}{p}_n{\Pi }_n,p_n:=\left[{\Pi }_n{\rho }_0\right].$$

The set $\left\{p_n\right\}$ is positive and satisfies ${\sum }_n{p}_n=1$ by trace preservation.

5.3.2. Proof of the Probability Law [105,131,132]

Lemma 55

(Normalisation of probabilities). One has $p_n\ge 0$ and ${\sum }_n{p}_n=1$.

Proof

Because ${\Pi }_n$ is a positive projection, ${\Pi }_n{\rho }_0{\Pi }_n\ge 0$; trace positivity yields $p_n\ge 0$. Completeness ${\sum }_n{\Pi }_n=\mathbf{1}$ together with ${\rho }_0=1$ implies ${\sum }_n{p}_n=1$.□

Theorem 20

(Born rule (UEE version)). The probability of obtaining the measurement outcome n in the pointer basis is

$$\begin{array}{|c|}\hline \mathbb{P}\left(n\right)=\left[{\rho }_0{\Pi }_n\right]\\ \hline\end{array}$$

Proof

Immediately before read-out the state is ${\rho }_{\mathrm{D}}$; for a projective measurement the probability is $\mathbb{P}\left(n\right)=\left[{\rho }_{\mathrm{D}}{\Pi }_n\right]$. Since ${\rho }_{\mathrm{D}}{\Pi }_n=p_n{\Pi }_n$ and $\left[{\Pi }_n\right]=1$ (one–dimensional projection), $\mathbb{P}\left(n\right)=p_n=\left[{\Pi }_n{\rho }_0\right]$.□

5.3.3. Post-Measurement State (Lüders Update) [101,133]

Stopping the dissipative semigroup at a small time $\delta t$ before $t\to \infty$ gives the conditional state

$${\rho }_{n|\delta t}={\displaystyle \frac{{\Pi }_n\rho \left(\delta t\right){\Pi }_n}{\left[{\Pi }_n\rho \left(\delta t\right)\right]}}\stackrel{\delta t\to 0}{\to }{\displaystyle \frac{{\Pi }_n{\rho }_0{\Pi }_n}{p_n}},$$

which coincides with the standard Lüders rule.

5.3.4. Recovery of Expectation Values [134,135]

For any observable A commuting with all ${\Pi }_n$

$${\langle A\rangle }_{\mathrm{after}}=\sum _n{p}_n\left[A{\Pi }_n\right]=\left[A{\rho }_0\right],$$

showing that no statistical bias is introduced by the measurement.

5.3.5. Conclusion

From the pointer-diagonal state ${\rho }_{\mathrm{diag}}$ obtained through dissipative diagonalisation we derived the measurement probabilities $\mathbb{P}\left(n\right)=\left[{\rho }_0{\Pi }_n\right]$, reproducing the axiomatic Born rule. Moreover, the Lüders update emerges naturally as the continuous-dynamics limit of the same process.

5.4. Dissipative Time-scale and Decoherence

5.4.1. Time Evolution of the Off-Diagonal Fidelity [5,116]

Tracing the result of Theorem 19 at the level of matrix elements, for indices $n\ne m$ we have

$$C_{nm}\left(t\right):=\left[{\Pi }_n\rho \left(t\right){\Pi }_m\right]=C_{nm}\left(0\right){\mathrm{e}}^{-\gamma t},$$

(5.3.1)

where $C_{nm}\left(0\right)$ is the initial coherence.

5.4.2. Definition of the Decoherence Time [19,126]

Definition 21

(Decoherence time).

$$t_{\mathrm{dec}}:={\gamma }^{-1}ln\left(\frac{1}{ϵ}\right),$$

with a small threshold $ϵ\ll 1$ such that coherence is deemed practically vanished if $|C_{nm}\left(t_{\mathrm{dec}}\right)|\le ϵ|C_{nm}\left(0\right)|$.

Choosing, in particular, $ϵ={\mathrm{e}}^{-1}$ yields the natural-unit decoherence time $t_{\mathrm{dec}}={\gamma }^{-1}$.

5.4.3. Diverging Entropy and the Spohn Inequality [51,136]

Lemma 56

(Growth rate of the linear entropy). For the linear entropy $S_2:=1-\left[{\rho }^2\right]$ one has

$${\displaystyle \frac{\mathrm{d}{S}_2}{\mathrm{d}t}}=2\gamma \sum _{n\ne m}{\left|C_{nm}\left(t\right)\right|}^2\ge 0.$$

Proof

Using $\dot{\rho }=-\gamma {\rho }_{off}+\dots$ and evaluating $\left[\rho \dot{\rho }\right]$. Only off-diagonal elements contribute; insert equation (5.3.1).□

The result is compatible with the Spohn inequality ${\dot{S}}_{\mathrm{vN}}\ge 0$ (Chapter 3, §3.9); $S_2$ saturates rapidly on the scale $t_{\mathrm{dec}}$.

5.4.4. Physical Model for the Parameter $\gamma$ [137,138]

For a weakly coupled linear system–environment model

$$V_{\mathrm{int}}=\sum _n{A}_n\otimes B_n,A_n=|e_n\rangle \langle e_n|,$$

a Redfield/GKLS reduction gives $\gamma ={2\pi J(\omega =0)\left|g\right|}^2,$ where $J\left(\omega \right)$ is the environmental spectral density and g the coupling constant. Hence

$$t_{\mathrm{dec}}\propto {\displaystyle \frac{1}{{\left|g\right|}^{2}J\left(0\right)}}.$$

5.4.5. Illustrative Experimental Values [128,139,140]

In laser-cooled atomic systems with $\left|g\right|\sim {10}^{-2}\mathrm{MHz}$ and $J\left(0\right)\sim {10}^3\mathrm{Hz}$, $t_{\mathrm{dec}}\sim {10}^{-5}\mathrm{s}.$ In high-temperature solids the time can shrink down to the femtosecond regime.

5.4.6. Conclusion

The jump-induced dissipation suppresses the off-diagonal components of the density operator in the pointer basis as ${\mathrm{e}}^{-\gamma t}$ and sets the decoherence time ${\displaystyle t_{\mathrm{dec}}={\gamma }^{-1}ln(1/ϵ)}$. The rate $\gamma$ is fixed by the environmental spectral density and the coupling constant and ranges from ${10}^{-15}$ s to ${10}^{-5}$ s in typical experiments. This time-scale constitutes the fundamental constant governing the dynamics of thermalisation and entropy production studied in Chapter 6.

5.5. Quantum-Zeno Effect and the Continuous-Measurement Limit

5.5.1. Set-up of the Discrete-Measurement Protocol [141,142]

Definition 22

(Discrete measurement sequence). The total observation time T is divided into N equal intervals, giving the inter-measurement spacing ${\tau }_{\mathrm{M}}=T/N$. During each interval we apply, in alternation,

1. 

the dissipative semigroup evolution $exp\left({\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}\right)$, and

2. 

the projective measurement $\left\{{\Pi }_n\right\}$.

We denote the overall operation by ${\mathcal{M}}_N$.

For an initial state ${\rho }_0$

$${\rho }^{\left(N\right)}\left(T\right):={\left(\sum _n{\Pi }_n{\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}\right)}^N{\rho }_0{\left(\sum _m{\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}{\Pi }_m\right)}^N.$$

(5.4.1)

5.5.2. Zeno Contraction Lemma [143,144]

Lemma 57

(Low-order transition probability). If ${\tau }_{\mathrm{M}}\ll {\gamma }^{-1}$, the off-diagonal transition probability is

$$P_{n\to m}\left({\tau }_{\mathrm{M}}\right)=\gamma {\tau }_{\mathrm{M}}+O\left({\left(\gamma {\tau }_{\mathrm{M}}\right)}^2\right),(n\ne m).$$

Proof

Expand $exp\left({\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}\right)\rho =\rho +{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]+O\left({\tau }_{\mathrm{M}}^2\right)$. For $n\ne m$, the off-diagonal component of ${\mathcal{L}}_{\mathrm{diss}}$ is $-\gamma {\rho }_{nm}$ (Lemma 54), so the leading transition probability is $\gamma {\tau }_{\mathrm{M}}$.□

5.5.3. Continuous-Measurement Limit [145,146]

Theorem 21

(Quantum-Zeno fixation theorem). In the limit $N\to \infty ,{\tau }_{\mathrm{M}}=T/N\to 0$ one obtains

$${\rho }^{\left(N\right)}\left(T\right)\stackrel{\mathrm{SOT}}{\to }\sum _n{\Pi }_n{\rho }_0{\Pi }_n\equiv {\rho }_{\mathrm{diag}},$$

i.e. the state freezes completely in the pointer-projection subspace.

Proof

The off-diagonal survival factor per measurement step is $1-\gamma {\tau }_{\mathrm{M}}+O\left({\left(\gamma {\tau }_{\mathrm{M}}\right)}^2\right)$; after N steps ${(1-\gamma {\tau }_{\mathrm{M}})}^N\stackrel{N\to \infty }{\to }{\mathrm{e}}^{-\gamma T}\to 0.$ Lemma 57 shows that the diagonal blocks are preserved while the off-diagonals decay exponentially. The convergence holds in the strong-operator topology (SOT).□

5.5.4. Implications for Measurable Quantities [142,147]

• Raising the measurement frequency (${\tau }_{\mathrm{M}}^{-1}$) prolongs the dwell time in a single projection sector; formally ${\tau }_{\mathrm{M}}\to 0$ yields complete freezing (the Zeno fixation).
• Practical limitation: if ${\tau }_{\mathrm{M}}$ becomes shorter than the detector-response time, apparatus noise effectively increases $\gamma$ and the Zeno effect is destroyed.

5.5.5. Conclusion

Applying the dissipative semigroup and projective measurements alternately with a vanishing interval ${\tau }_{\mathrm{m}}\to 0$ suppresses pointer-basis transitions to $O\left(\gamma {\tau }_{\mathrm{m}}\right)$ per step, so that after a finite time T the off-diagonal elements decay as $exp(-\gamma T)\to 0$. Thus the quantum-Zeno effect emerges naturally within the single-fermion UEE framework.

5.6. Entanglement Generation and Measurement Back-Action

5.6.1. Measurement-Apparatus Model [82,148]

Definition 23

(Apparatus Hilbert space and pointer states). The measuring device is described by a countable–dimensional Hilbert space ${\mathcal{H}}_{\mathrm{app}}$ that possesses mutually orthogonal pointer states ${\left\{\right|n\rangle }_{\mathrm{app}}{\}}_{n=1}^{18}$. The initial apparatus state is ${\rho }_{\mathrm{app}}^{\left(0\right)}=|0\rangle \langle 0|$.

Definition 24

(System–apparatus interaction). The measurement process is realised by the unitary

$$U_{\mathrm{meas}}=\sum _n{\Pi }_n\otimes U_n,U_n{|0\rangle }_{\mathrm{app}}={|n\rangle }_{\mathrm{app}},$$

(5.5.1)

i.e. a von-Neumann–type pre-measurement.

5.6.2. Entanglement–Generation Lemma [149]

Lemma 58

(System–apparatus entangled state). For an initial product state ${\rho }_{\mathrm{sys}}\otimes {\rho }_{\mathrm{app}}^{\left(0\right)}$, the interaction (5.5.1) produces

$${\rho }_{\mathrm{sys}A}=\sum _{n,m}{\Pi }_n{\rho }_{\mathrm{sys}}{\Pi }_m\otimes {|n\rangle \langle m|}_{\mathrm{app}}.$$

(5.5.2)

Proof

Insert $U_{\mathrm{meas}}$ explicitly: $U_{\mathrm{meas}}({\rho }_{\mathrm{sys}}\otimes |0\rangle \langle 0\left|\right)U_{\mathrm{meas}}^{†}={\sum }_{n,m}{\Pi }_n{\rho }_{\mathrm{sys}}{\Pi }_m\otimes {|n\rangle \langle m|}_{\mathrm{app}}$.□

5.6.3. Measurement Back-Action and the Lüders Update [135,150]

Theorem 22

(Conditional state update). If the apparatus registers the outcome n, the conditional state of the system is

$${\rho }_{\mathrm{sys}|n}={\displaystyle \frac{{\Pi }_n{\rho }_{\mathrm{sys}}{\Pi }_n}{\left[{\Pi }_n{\rho }_{\mathrm{sys}}\right]}},$$

i.e. exactly Lüders’ rule.

Proof

The conditional state is ${\rho }_{\mathrm{sys}|n}{=}_{\mathrm{app}}\left[(\mathbf{1}\otimes |n\rangle \langle n\left|\right){\rho }_{\mathrm{sys}A}\right]/Pr\left(n\right)$. Substituting (5.5.2) and using $Pr\left(n\right)=\left[{\Pi }_n{\rho }_{\mathrm{sys}}\right]$ gives the stated expression.□

5.6.4. Consistency with Dissipative Diagonalisation [109,151]

In the short-time limit of the dissipative semigroup the system density operator becomes ${\rho }_{\mathrm{sys}}\mapsto {\rho }_{\mathrm{diag}}$ (Section 5.2). Applying $U_{\mathrm{meas}}$ afterwards one has ${\Pi }_n{\rho }_{\mathrm{diag}}{\Pi }_n={\Pi }_n{\rho }_{\mathrm{sys}}{\Pi }_n$; the entangling unitary therefore merely transfers the classical probabilities to the pointer while leaving the already diagonalised ${\rho }_{\mathrm{diag}}$ unchanged—so the back-action is effectively null.

5.6.5. Entanglement Entropy [152,153]

After the pre-measurement, but before reading the pointer (trace over the apparatus),

$$S_{\mathrm{vN}}\left({\rho }_{\mathrm{sys}}\right)\le S_{\mathrm{vN}}\left({\rho }_{\mathrm{sys}A}\right)=H\left(\left\{p_n\right\}\right),$$

where H is the Shannon entropy. Thus the measurement transfers information to the pointer and can decrease the entropy of the system alone.

5.6.6. Conclusion

The unitary interaction $U_{\mathrm{meas}}$ entangles the system with the measuring device into a one-dimensional, pointer-labelled state ${\sum }_n{\Pi }_n|\psi \rangle \otimes |n\rangle$. Upon obtaining the outcome n, the system state collapses to ${\displaystyle \rho \to {\Pi }_n\rho {\Pi }_n/p_n}$—the Lüders update. When the system has already been dissipatively diagonalised, this measurement induces virtually no additional back-action, consistent with the framework developed in previous sections.

5.7. Extension to General POVMs

5.7.1. Construction Principle for POVM Elements [13,14]

Starting from the pointer projection family $\left\{{\Pi }_n\right\}$ we form linear combinations with an Orthon–type coefficient matrix $C=\left(c_{\mu n}\right)$:

$$\begin{array}{|c|}\hline E_{\mu }:={\displaystyle \sum _{n=1}^{18}}{c}_{\mu n}{\Pi }_n,c_{\mu n}\ge 0\\ \hline\end{array}$$

(5.6.1)

Definition 25

(Projection-sum POVM). If the coefficient matrix satisfies ${\sum }_{\mu }{c}_{\mu n}=1$ for every n, the collection ${\left\{E_{\mu }\right\}}_{\mu =1}^M$ is called aprojection-sum POVM.

5.7.2. Completeness and Positivity [106,134]

Lemma 59

(POVM completeness).

$$\sum _{\mu =1}^M{E}_{\mu }=\sum _n\left(\sum _{\mu }{c}_{\mu n}\right){\Pi }_n=\sum _n{\Pi }_n=\mathbf{1}.$$

Proof

The first equality is the definition, the second follows from ${\sum }_{\mu }{c}_{\mu n}=1$, and the third from the completeness of $\left\{{\Pi }_n\right\}$.□

Because each $E_{\mu }$ is a positive linear combination of projections, one has $E_{\mu }\ge 0$ automatically.

5.7.3. Choice of Kraus Operators [13,154]

$$M_{\mu n}:=\sqrt{c_{\mu n}}{\Pi }_n⟹E_{\mu }=\sum _n{M}_{\mu n}^{†}{M}_{\mu n}.$$

This “visible’’ dilation is completed entirely within the internal index space—no additional Hilbert space for an environment is required (no Naimark extension).

5.7.4. Measurement Probabilities and Lüders Update [101,133]

Theorem 23

(POVM probability and state update). For a system state ρ one has

$$Pr\left(\mu \right)=\left[\rho E_{\mu }\right],\rho ⟼{\rho }_{\mu }={\displaystyle \frac{{\sum }_n{M}_{\mu n}\rho M_{\mu n}^{†}}{Pr\left(\mu \right)}}={\displaystyle \frac{{\sum }_n{c}_{\mu n}{\Pi }_n\rho {\Pi }_n}{Pr\left(\mu \right)}}.$$

In particular, choosing $c_{\mu n}={\delta }_{\mu n}$ recovers projective measurement and the usual Born rule.

Proof

Standard GKLS/Kraus construction. Off-diagonal terms ${\Pi }_n\rho {\Pi }_m(n\ne m)$ vanish because ${\Pi }_n{M}_{\mu k}=0$ unless $n=k$. Consequently the update involves only projection sums and preserves the pointer-diagonal structure.□

5.7.5. Information–Theoretic Implications [155,156]

A POVM coarsens the projection information ${\Pi }_n$ to produce a classical probability distribution $p_{\mu }={\sum }_n{c}_{\mu n}{p}_n$, whose Shannon entropy satisfies $H\left(\left\{p_{\mu }\right\}\right)\ge H\left(\left\{p_n\right\}\right)$. The information loss is governed by the mixing properties of the coefficient matrix.

5.7.6. Conclusion

Any POVM can be realised as a non-negative coefficient sum of the pointer projections, $E_{\mu }={\sum }_n{c}_{\mu n}{\Pi }_n$, provided completeness and positivity are respected—no extra Naimark dilation is necessary. Hence the projection structure obtained within the UEE framework suffices to encompass the entire theory of general quantum measurements.

5.8. Summary and Bridge to Chapter 6

• Dissipative–diagonalisation theorem (Sec. 5.2): The jump operators $V_n=\sqrt{\gamma }{\Pi }_n$ exponentially diagonalise the density operator $\rho$ in the pointer basis within the time scale $t_{\mathrm{dec}}={\gamma }^{-1}$.
• Born rule (Sec. 5.3): After diagonalisation the measurement probabilities appear automatically as $\mathbb{P}\left(n\right)=\left[{\rho }_0{\Pi }_n\right]$; the post–measurement state reproduces the Lüders rule.
• Quantum Zeno effect (Sec. 5.4): In the limit of vanishing measurement interval ${\tau }_{\mathrm{m}}\to 0$ the off–diagonal transition amplitudes are suppressed to $\mathcal{O}\left(\gamma {\tau }_{\mathrm{m}}\right)$, freezing the evolution within the pointer subspace.
• POVM extension (Sec. 5.6): Any general measurement can be realised as a non–negative coefficient sum $E_{\mu }={\sum }_n{c}_{\mu n}{\Pi }_n$ that satisfies completeness and positivity, thus eliminating the need for an additional Naimark dilation.

Deterministic core vs. stochastic output

The UEE equation of motion

$$\dot{\rho }=-i[D,\rho ]+{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]-{\mathcal{L}}_u\rho$$

is fully deterministic once the five–operator complete set is specified. Probabilities emerge only at the instant of observation through the two–step mechanism “dissipative diagonalisation $⟹$ projection read-out.’’ Thus quantum probabilities are not intrinsic to the dynamics but are a by–product of the measurement process.

From the Spohn inequality to the area law

The pointer–diagonal state ${\rho }_{\mathrm{diag}}$ obtained after measurement represents a “classicalised’’ quantum state; during thermalisation one has the monotonic approach $S_{\mathrm{v}N}\left(\rho \right)\stackrel{t}{\to }H\left(\left\{p_n\right\}\right)$ governed by the Spohn inequality. Chapter 6 will analyse

• the entanglement entropy obeying the area law $S_{\mathrm{e}nt}\sim \mathcal{A}$;
• the hierarchy between the decoherence time $t_{\mathrm{dec}}$ and the thermalisation time $t_{\mathrm{th}}$;
• the conditions under which the Zeno effect slows down the thermalisation rate.

Conclusion

Chapter 5 established quantitatively that “the UEE is intrinsically deterministic, while probabilities appear only at measurement.’’ Dissipative diagonalisation by pointer projections unifies the Born rule, the Zeno effect, and POVMs as dynamical consequences, thereby providing the groundwork for the analysis of thermalisation and entropy production in the following chapter.

6. Entanglement, Thermalisation, and the Quantum Zeno Effect

6.1. Introduction and Scope

6.1.1. Aims of This Chapter [5,32,51]

Building on the dissipative diagonalisation $\rho \to {\mathcal{P}}_{\mathrm{ptr}}$ and the probabilistic measurement framework established in Chapter 5, the goals of the present chapter are:

• to give a rigorous proof of the area law for the entanglement entropy generated by a pointer–diagonal state, ${\displaystyle S_{\mathrm{ent}}\propto \mathcal{A}}$ (Sec. 6.2);
• to derive a finite–time thermalisation theorem from the Spohn inequality ${\dot{S}}_{\mathrm{v}N}\ge 0$ (Sec. 6.3);
• to evaluate the hierarchy between the decoherence time $t_{\mathrm{dec}}$ and the thermalisation time $t_{\mathrm{th}}$, and to analyse the parameter region in which Zeno-frequency measurements suppress thermalisation (Secs. 6.4–6.5);
• to ensure that no violation of the area law occurs by invoking bounds on information propagation based on the Lieb–Robinson velocity (Sec. 6.6).

6.1.2. Definitions of the Relevant Time Scales [19,137]

Definition 26

(Decoherence time). Via the dissipative rate γ we set

$$t_{\mathrm{dec}}:={\gamma }^{-1}ln\left(1/ϵ\right),$$

where $ϵ\ll 1$ denotes the threshold below which coherence is regarded as practically lost (Sec. 5.4). With the representative choice $ϵ={\mathrm{e}}^{-1}$ one has $t_{\mathrm{dec}}={\gamma }^{-1}$.

Definition 27

(Thermalisation time). Depending on the system–environment coupling constant g and on the environmental spectral density $J\left(0\right)$, we define

$$t_{\mathrm{th}}:={\displaystyle \frac{1}{{\left|g\right|}^{2}J\left(0\right)}}.$$

For many physical systems one finds the hierarchy $t_{\mathrm{dec}}\ll t_{\mathrm{th}}$ (UEE_02 §9). The analyses in this chapter are carried out under this assumption.

6.1.3. Area Law and the Pointer Basis [116,157,158,159]

Definition 28

(Area law for entanglement entropy). For a spatial region Ω with boundary area $\mathcal{A}$, the entanglement entropy of the pointer–diagonal state ${\mathcal{P}}_{\mathrm{ptr}}$ is said to obey the “area law’’ if

$$S_{\mathrm{ent}}(\Omega )=\kappa \mathcal{A}+o\left(\mathcal{A}\right),$$

where the constant κ coincides with the exponential decay rate of the zero-area resonance kernel R and with the structure-formation constant (UEE_02 §9).

6.1.4. Methodological Tools Employed in This Chapter [17,160,161,162]

• Dissipative master equation: Redfield → GKLS coarse-graining is used to obtain analytic expressions for $\rho \left(t\right)$.
• Information measures: We employ the von Neumann entropy $S_{\mathrm{v}N}$ and the relative-entropy production rate.
• Lieb–Robinson bound: A finite velocity $v_{\mathrm{LR}}$ for information propagation is used to control correlation spread.

Conclusion

In this chapter we analyse, under the hierarchy ${\displaystyle t_{\mathrm{dec}}\ll t_{\mathrm{th}}}$, how pointer–diagonalisation gives rise to entanglement growth, thermalisation, and Zeno suppression. The aim is to exhibit explicitly how thermodynamic behaviour emerges from the deterministic UEE dynamics by means of the area law and the Spohn inequality.

6.2. Entanglement Structure of the Pointer-Diagonal State

6.2.1. Form of the Pointer-Diagonal State [32,126]

From Chapter 5 the pointer-diagonalised state is

$${\mathcal{P}}_{\mathrm{ptr}}=\sum _{n=1}^{18}\int \mathcal{D}\left[{\psi }_n\right]P_n\left[{\psi }_n\right]|{\psi }_n\rangle \langle {\psi }_n|\otimes {\Pi }_n,$$

(6.2.1)

where the set $\left\{\right|{\psi }_n\rangle \}$ lives in the spatial sector ${\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}$ and is tensored with the internal projection ${\Pi }_n$.

6.2.2. Definition of the Entanglement Entropy [4,163]

Definition 29

(Bipartition and entanglement entropy). For a finite spatial region ${\Omega }_R\subset {\mathbb{R}}^3$ with complement ${\Omega }_C$ we introduce the tensor decomposition ${\mathbb{H}}_{\mathbb{C}}^{\left(\mathrm{spacetime}\right)}={\mathbb{H}}_{\mathbb{C},{\Omega }_R}\otimes {\mathbb{H}}_{\mathbb{C},{\Omega }_C}.$ Because the pointer projectors act only on the internal space they commute with this split. Tracing over ${\Omega }_C$ gives the reduced state ${\mathcal{P}}_{\mathrm{ptr},{\Omega }_R}:{=}_{{\Omega }_C}{\mathcal{P}}_{\mathrm{ptr}}.$ Its von Neumann entropy $S_{\mathrm{ent}}\left({\Omega }_R\right):={-}_{{\Omega }_R}[{\mathcal{P}}_{\mathrm{ptr},{\Omega }_R}ln{\mathcal{P}}_{\mathrm{ptr},{\Omega }_R}]$ is called the entanglement entropy.

6.2.3. Clustering Lemma [164,165]

Lemma 60

(Exponential clustering induced by the zero-area kernel). The zero-area resonance kernel R induces a finite correlation length ξ such that for two points $x,y$ at distance $d\gg \xi$ one has

$$〈{\Pi }_n\left(x\right){\Pi }_m\left(y\right)〉-\langle {\Pi }_n\left(x\right)\rangle \langle {\Pi }_m\left(y\right)\rangle \le C_0{\mathrm{e}}^{-d/\xi }.$$

Proof

The exponential suppression $R\sim {\mathrm{e}}^{-\mathcal{A}/{\ell }_R}$ generates in the Euler–Lagrange equations a mass term $m\propto {\xi }^{-1}$, leading to a Yukawa-type decay of the two-point function. □

6.2.4. Area-Law Theorem [157,158,159]

Theorem 24

(Area law for the pointer-diagonal state). Provided the correlation length ξ is finite, the entanglement entropy of the region ${\Omega }_R$ satisfies

$$S_{\mathrm{ent}}\left({\Omega }_R\right)=\kappa \mathcal{A}(\partial {\Omega }_R)+O\left(\xi \partial \mathcal{A}\right),$$

with ${\displaystyle \kappa =-\sum _n{p}_{n}ln{p}_n,p_n=\left[{\Pi }_n{\mathcal{P}}_{\mathrm{ptr}}\right].}$

Proof

Apply the strong sub-additivity $S_{AB}+S_{BC}-S_{ABC}-S_B\ge 0$ to adjacent blocks $(A,B,C)$. Lemma 60 bounds long-range contributions by $\mathcal{O}\left({\mathrm{e}}^{-d/\xi }\right)$. Tiling the global region with cells of width $\xi$ reduces the entropy to a sum over boundary cells; the number of such cells is proportional to $\mathcal{A}/{\xi }^2$, hence the leading area term. Curvature-related corrections are bounded by $O(\xi \partial \mathcal{A})$. □

6.2.5. Physical Meaning of the Constant $\kappa$ [40,166]

The constant $\kappa$ equals the Shannon entropy density of the pointer probabilities,

$$\kappa =-\sum _n{p}_{n}ln{p}_n=H\left(\left\{p_n\right\}\right),$$

quantifying the local degree of mixing. Throughout this chapter the distribution $\left\{p_n\right\}$ is assumed to have been equilibrated by the zero-area kernel, so that $\kappa$ behaves as a universal constant.

Conclusion

Because the zero-area kernel introduces a finite correlation length, the pointer-diagonal state rigorously obeys the area law ${\displaystyle S_{\mathrm{ent}}=\kappa \mathcal{A}+o\left(\mathcal{A}\right)}$. The prefactor $\kappa =-{\sum }_n{p}_{n}ln{p}_n$ is the Shannon entropy density of the pointer probabilities, here established as a universal constant.

6.3. Spohn’s Inequality and the Thermalisation Theorem

6.3.1. Recap of Spohn’s Inequality [17,51]

Definition 30

(Spohn’s inequality). Let a Lindblad semigroup $\dot{\rho }=L\left[\rho \right]$ admit a stationary state ${\rho }_{\mathrm{eq}}$ with $L\left[{\rho }_{\mathrm{eq}}\right]=0$. Then the relative entropy $S(\rho \parallel {\rho }_{\mathrm{eq}})=\left[\rho \left(ln\rho -ln{\rho }_{\mathrm{eq}}\right)\right]$ satisfies

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S\left(\rho \parallel {\rho }_{\mathrm{eq}}\right)=-\left[L\left[\rho \right]\left(ln\rho -ln{\rho }_{\mathrm{eq}}\right)\right]\le 0.$$

(6.3.1)

Throughout this subsection we identify $L={\mathcal{L}}_{\mathrm{diss}}$ and ${\rho }_{\mathrm{eq}}={\mathcal{P}}_{\mathrm{ptr}}$.

6.3.2. Monotonicity of the Relative Entropy [167,168]

Lemma 61

(Monotonicity). For $\rho \left(t\right)={\mathrm{e}}^{t{\mathcal{L}}_{\mathrm{diss}}}{\rho }_0$ one has

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S\left(\rho \left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}\right)\le 0,\forall t\ge 0.$$

Proof

Since ${\mathcal{L}}_{\mathrm{diss}}\left[{\mathcal{P}}_{\mathrm{ptr}}\right]=0$ (Sec. 5.2) and ${\mathcal{L}}_{\mathrm{diss}}$ is a GKLS generator, the statement follows directly from (6.3.1). □

6.3.3. Thermalisation Theorem [169,170,171]

Theorem 25

(Finite-time thermalisation). The relative entropy satisfies

$$S\left(\rho \left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}\right)\le S\left({\rho }_0\parallel {\mathcal{P}}_{\mathrm{ptr}}\right){\mathrm{e}}^{-2\gamma t},$$

so that ${\displaystyle \underset{t\to \infty }{lim}\rho \left(t\right)={\mathcal{P}}_{\mathrm{ptr}}}$ with exponential rate γ.

Proof

Using the off-diagonal suppression ${\rho }_{\mathrm{off}}\left(t\right)={\mathrm{e}}^{-\gamma t}{\rho }_{\mathrm{off}}\left(0\right)$ (Lemma 5.2) we split the relative entropy into diagonal/off-diagonal parts:

$$S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}})=\left[{\rho }_{\mathrm{diag}}ln{\rho }_{\mathrm{diag}}-\rho ln{\mathcal{P}}_{\mathrm{ptr}}\right]+\left[{\rho }_{\mathrm{off}}ln{\rho }_{\mathrm{diag}}\right].$$

The off-diagonal contribution decays as $\parallel {\rho }_{\mathrm{off}}{\left(t\right)\parallel }_1\le {\mathrm{e}}^{-\gamma t}{\parallel {\rho }_{\mathrm{off}}\left(0\right)\parallel }_1$. With Pinsker’s inequality $S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}})\ge {\displaystyle \frac{1}{2}}{\parallel \rho -{\mathcal{P}}_{\mathrm{ptr}}\parallel }_1^2$ we obtain $\parallel \rho -{\mathcal{P}}_{\mathrm{ptr}}{\parallel }_1\le c{\mathrm{e}}^{-\gamma t}$, where c is bounded by the initial relative entropy. Hence thermalisation is exponential. □

6.3.4. Thermalisation Time and the Entropy-Production Rate [19,137]

The entropy-production rate

$$\sigma \left(t\right):=-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S\left(\rho \left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}\right)\ge 2\gamma S\left(\rho \left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}\right)$$

implies that $t_{\mathrm{th}}={\displaystyle \frac{1}{2\gamma }}ln\left(S({\rho }_0\parallel {\mathcal{P}}_{\mathrm{ptr}})/\delta \right)$ is sufficient to reach $S(\rho \left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}})\le \delta$.

Conclusion

Applying Spohn’s inequality to the pointer-diagonal stationary state ${\mathcal{P}}_{\mathrm{ptr}}$ shows that the relative entropy decreases as ${\mathrm{e}}^{-2\gamma t}$. Consequently the finite-time thermalisation theorem holds, yielding an explicit thermalisation time $t_{\mathrm{th}}\propto {\gamma }^{-1}$.

6.4. Evaluation of the Thermalisation Time Scale

6.4.1. System–Environment Interaction Model [137,138]

Definition 31

(Generic weak–coupling model). For a system Hilbert space ${\mathcal{H}}_{\mathrm{sys}}$ and an environment ${\mathcal{H}}_{\mathrm{env}}$,

$${\mathcal{H}}_{\mathrm{SE}}:=H_{\mathrm{sys}}+H_{\mathrm{env}}+g\sum _{\alpha }{A}_{\alpha }\otimes B_{\alpha },$$

(6.4.1)

with system observables $A_{\alpha }$, environment operators $B_{\alpha }$, and a dimensionless coupling constant $g\ll 1$.

The environment is assumed to be in equilibrium ${\rho }_{\mathrm{env}}^{\beta }\propto {\mathrm{e}}^{-\beta H_{\mathrm{env}}}$. Its bath correlations are $C_{\alpha \beta }\left(t\right):{=}_{\mathrm{env}}\left[B_{\alpha }\left(t\right)B_{\beta }{\rho }_{\mathrm{env}}^{\beta }\right].$

6.4.2. Born–Markov Reduction and the Dissipation Rate [18,19]

Lemma 62

(Redfield → GKLS dissipation rate). The dissipation rate associated with an energy transition ω is

$$\gamma \left(\omega \right)={2\pi \left|g\right|}^2\sum _{\alpha \beta }\langle e|A_{\alpha }|e^{\prime }\rangle \langle e^{\prime }|A_{\beta }^{†}|e\rangle J_{\alpha \beta }\left(\omega \right),$$

where the spectral density is $J_{\alpha \beta }\left(\omega \right):={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{i\omega t}{C}_{\alpha \beta }\left(t\right)\mathrm{d}t.$

Proof

Apply the standard Born–Markov expansion ([19], Ch. 3) in the pointer–diagonal basis. Principal-value terms are absorbed into the Lamb shift. Fermi’s golden rule then yields the stated rate. □

6.4.3. Effective Dissipation Rate and Thermalisation Time [138,172]

Define the minimum positive rate ${\gamma }_{\mathrm{eff}}:={min}_{\omega \ne 0}\gamma \left(\omega \right)>0$ (for a gapless bath $J\left(0\right)>0$).

Definition 32

(Thermalisation time). The minimal time $t_{\mathrm{th}}\left(\delta \right)$ such that the relative entropy satisfies $S\left({\rho }_{\mathrm{sys}}\left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}\right)\le \delta$ is called thethermalisation time.

Theorem 26

(Upper bound on the thermalisation time). For an arbitrary initial state ${\rho }_{\mathrm{sys}}\left(0\right)$,

$$t_{\mathrm{th}}\left(\delta \right)\le {\displaystyle \frac{1}{2{\gamma }_{\mathrm{eff}}}}ln\left[\frac{S({\rho }_{\mathrm{sys}}\left(0\right)\parallel {\mathcal{P}}_{\mathrm{ptr}})}{\delta }\right].$$

Proof

Using Spohn’s inequality (Sec. 6.3) with the lower bound $\gamma \ge {\gamma }_{\mathrm{eff}}$ one finds $S({\rho }_{\mathrm{sys}}\left(t\right)\parallel {\mathcal{P}}_{\mathrm{ptr}})\le S({\rho }_{\mathrm{sys}}\left(0\right)\parallel {\mathcal{P}}_{\mathrm{ptr}}){\mathrm{e}}^{-2{\gamma }_{\mathrm{eff}}t}$. Setting the r.h.s. equal to $\delta$ and solving for t gives the claimed bound. □

6.4.4. Scaling in $\left|g\right|$ and $J\left(0\right)$

From Lemma 62 at $\omega \simeq 0$ ${\gamma }_{\mathrm{eff}}=2\pi {\left|g\right|}^{2}J\left(0\right).$ Hence

$$t_{\mathrm{th}}\left(\delta \right)\propto {\displaystyle \frac{1}{{\left|g\right|}^{2}J\left(0\right)}}ln\left[S({\rho }_{\mathrm{sys}}\left(0\right)\parallel {\mathcal{P}}_{\mathrm{ptr}})/\delta \right].$$

Weak coupling (${\left|g\right|}^2\ll 1$) or low temperature with $J\left(0\right)\to 0$ enlarges the thermalisation time, approaching the Quantum-Zeno regime.

6.4.5. Examples: Cold Atoms vs. Solids [173,174]

• Optical-lattice cold atoms: $\left|g\right|\sim {10}^{-2}$, $J\left(0\right)\sim {10}^3$Hz $\Rightarrow {\gamma }_{\mathrm{eff}}\sim 0.6$kHz $\Rightarrow t_{\mathrm{th}}\sim 1$ms.
• High-temperature solid: $\left|g\right|\sim 1$, $J\left(0\right)\sim {10}^{12}$Hz $\Rightarrow t_{\mathrm{th}}\sim {10}^{-12}$s.

Thus experimental conditions realise a broad range ${10}^{-12}-{10}^{-3}$s.

Conclusion

From the Born–Markov reduction the dissipation rate is $\gamma \left(\omega \right)={2\pi \left|g\right|}^{2}J\left(\omega \right)$; its minimum ${\gamma }_{\mathrm{eff}}$ controls the thermalisation speed. The relative-entropy bound gives

$$t_{\mathrm{th}}\lesssim {\displaystyle \frac{1}{2{\gamma }_{\mathrm{eff}}}}ln\left[\frac{S({\rho }_0\parallel {\mathcal{P}}_{\mathrm{ptr}})}{\delta }\right],$$

i.e. $t_{\mathrm{th}}\propto {\left(\right|g|}^2{J\left(0\right))}^{-1}$. Weak coupling or low temperature therefore delays thermalisation and moves the system into the Zeno-suppressed domain discussed in Sec. 6.5.

6.5. Thermalisation Suppression via the Quantum–Zeno Effect

6.5.1. Continuous Measurement and the Effective Generator [141,144,175]

Definition 33

(Measurement frequency and interval). The observation time T is divided into N equal slices; the measurement interval is ${\tau }_{\mathrm{M}}:=T/N$ and the frequency is $f:={\tau }_{\mathrm{M}}^{-1}$. In Stinespring form the sequence “dissipative semigroup ${\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}$ followed by the projective measurement $\left\{{\Pi }_n\right\}$’’ repeated N times is denoted ${\mathcal{M}}_N$.

Lemma 63

(Effective GKLS generator). In the limit $N\to \infty$, ${\tau }_{\mathrm{M}}\to 0$, ${\mathcal{M}}_N$ approaches

$${\displaystyle \frac{\mathrm{d}\rho }{\mathrm{d}t}}={\mathcal{L}}_{\mathrm{Z}}\left[\rho \right],{\mathcal{L}}_{\mathrm{Z}}:={\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}+\mathcal{O}\left(\gamma {\tau }_{\mathrm{M}}\right),$$

where ${\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}\left[\rho \right]=\gamma {\sum }_n({\Pi }_n\rho {\Pi }_n-\frac{1}{2}\{{\Pi }_n,\rho \})$.

Proof

One step acts as $\rho \mapsto {\sum }_n{\Pi }_n{\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}\rho {\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}{\Pi }_n$. The BCH expansion gives ${\mathrm{e}}^{{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}}\rho =\rho +{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss}}\left[\rho \right]+\mathcal{O}\left({\tau }_{\mathrm{M}}^2\right)$. The projection removes off–diagonal terms to $\mathcal{O}\left({\tau }_{\mathrm{M}}\right)$. Repeating N times, ${(1+{\tau }_{\mathrm{M}}{\mathcal{L}}_{\mathrm{diss},\mathrm{diag}})}^N\stackrel{N\to \infty }{\to }{\mathrm{e}}^{T{\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}}$, while the remainder scales as $\mathcal{O}\left(N{\tau }_{\mathrm{M}}^2\right)=\mathcal{O}\left({\tau }_{\mathrm{M}}\right)$. □

6.5.2. Suppression Rate of Entropy Production [135,176]

Lemma 64

(Spohn inequality (Zeno version)). For the relative entropy $S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}})$,

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}})=-\left[{\mathcal{L}}_{\mathrm{Z}}\left[\rho \right](ln\rho -ln{\mathcal{P}}_{\mathrm{ptr}})\right]\le -2\gamma (1-ϵ)S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}}),ϵ:=\gamma {\tau }_{\mathrm{M}}\ll 1.$$

Proof

Decompose ${\mathcal{L}}_{\mathrm{Z}}={\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}+\delta L$ with $\delta L=\mathcal{O}\left(ϵ\gamma \right)$. ${\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}$ alone yields the entropy–decay rate $2\gamma$ (Sec. 6.3, Eq. (6.3.2)). Since $\parallel \delta L\parallel \le ϵ\gamma$, the coefficient is reduced to $(1-ϵ)$. □

6.5.3. Thermalisation–Suppression Theorem [144,177]

Theorem 27

(Quantum-Zeno suppression of thermalisation). If the measurement interval satisfies ${\tau }_{\mathrm{M}}<{\tau }_{\mathrm{Z}}:={\gamma }^{-1},$ the thermalisation time obeys

$$t_{\mathrm{th}}^{\left(\mathrm{Z}\right)}\ge {\displaystyle \frac{1}{2\gamma (1-\gamma {\tau }_{\mathrm{M}})}}ln\left[\frac{S({\rho }_0\parallel {\mathcal{P}}_{\mathrm{ptr}})}{\delta }\right],$$

i.e. $t_{\mathrm{th}}^{\left(\mathrm{Z}\right)}$ is longer by the factor ${(1-\gamma {\tau }_{\mathrm{M}})}^{-1}$ than without measurements. In the extreme limit ${\tau }_{\mathrm{M}}\to 0$, $t_{\mathrm{th}}^{\left(\mathrm{Z}\right)}\to \infty$: thermalisation is frozen.

Proof

Lemma 64 shows that the decay rate of the relative entropy is suppressed to $2\gamma (1-ϵ)$. Re-doing the estimate of Sec. 6.4 with this rate yields the stated bound. □

6.5.4. Phase Diagram: Thermalisation vs. Zeno [156,178]

Taking the measurement interval ${\tau }_{\mathrm{M}}$ and the environment parameters $\left(\right|g{|}^{2}J\left(0\right))$ as axes,

$$\left\{\begin{array}{cc}{\tau }_{\mathrm{M}}<{\tau }_{\mathrm{Z}}\mathrm{and}{\left|g\right|}^{2}J\left(0\right)<{\displaystyle \frac{\gamma \left(1-\gamma {\tau }_{\mathrm{M}}\right)}{ln(S/\delta )}}\hfill & ⟹\mathrm{Zeno}\mathrm{regime},\hfill \\ {\tau }_{\mathrm{M}}>{\tau }_{\mathrm{Z}}\hfill & ⟹\mathrm{ordinary}\mathrm{thermalisation}.\hfill \end{array}\right.$$

Thus, by increasing the measurement frequency one can suppress thermalisation even in weakly-coupled systems.

Conclusion

Repeating projective measurements at interval ${\tau }_{\mathrm{m}}$ renormalises the dissipator to ${\mathcal{L}}_{\mathrm{diss}}\to {\mathcal{L}}_{\mathrm{Z}}={\mathcal{L}}_{\mathrm{diss},\mathrm{diag}}+\mathcal{O}\left(\gamma {\tau }_{\mathrm{m}}\right)$, reducing the entropy–decay rate by the factor $(1-\gamma {\tau }_{\mathrm{m}})$. For ${\tau }_{\mathrm{m}}\ll {\gamma }^{-1}$ the thermalisation time diverges and the state is frozen in the pointer subspace: a quantitative demonstration of the Quantum–Zeno suppression of thermalisation.

6.6. Entanglement Velocity and the Lieb–Robinson Bound

6.6.1. Lattice Partition and Distance Function [162,179]

Embed physical space into a cubic lattice ${\mathbb{Z}}^3$ with spacing a and measure the distance between two regions $X,Y$ by

$$d(X,Y):=\underset{x\in X,y\in Y}{min}{\parallel x-y\parallel }_1,$$

i.e. the Manhattan distance.

6.6.2. Operational form of the Lieb–Robinson Bound [161,180]

Definition 34

(Lieb–Robinson velocity [161]). For a local Hamiltonian $H={\sum }_Z{h}_Z$ with interaction range $\mathrm{diam}\left(Z\right)\le R_0$ and bounded norm $\parallel h_Z\parallel \le h_0$, any two local operators $A_X,B_Y$ satisfy

$$∥[A_X\left(t\right),B_Y]∥\le C\parallel A_X\parallel \parallel B_Y\parallel exp\left(-{\displaystyle \frac{d(X,Y)-v_{\mathrm{LR}}\left|t\right|}{{\xi }_{\mathrm{LR}}}}\right),$$

(6.6.1)

where $v_{\mathrm{LR}}$ is the Lieb–Robinson velocity, ${\xi }_{\mathrm{LR}}$ a correlation length, and C a geometric constant.

The reversible generator D of the single-fermion UEE is produced by a local Hamiltonian; hence $R_0\sim a$, $h_0\sim 1/a$, and a finite $v_{\mathrm{LR}}$ exists.

6.6.3. Upper Bound on Entanglement Growth [159,179]

Lemma 65

(Entropy growth rate under a velocity constraint). For a spatial region ${\Omega }_R$ the von Neumann entropy $S_{{\Omega }_R}\left(t\right):=S\left({\rho }_{{\Omega }_R}\left(t\right)\right)$ obeys

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{S}_{{\Omega }_R}\left(t\right)\le s_{max}{v}_{\mathrm{LR}}\mathcal{A}\left(\partial {\Omega }_R\right),$$

where $s_{max}:=ln{d}_{\mathrm{loc}}$ is the logarithm of the local Hilbert-space dimension.

Proof

Apply the Hastings–Koma method [181] to the time evolution $\rho \left(t\right)={\mathrm{e}}^{-itD}{\mathcal{P}}_{\mathrm{ptr}}{\mathrm{e}}^{itD}$ starting from the pointer-diagonal state ${\mathcal{P}}_{\mathrm{ptr}}$. The entropy increase is limited by the flux of information that crosses the boundary; smoothing the bound (6.6.1) in space–time yields a growth rate bounded by $v_{\mathrm{LR}}\mathcal{A}(\partial {\Omega }_R)$. □

6.6.4. Theorem Excluding Violations of the Area Law [164,182]

Theorem 28

(Preservation of the area law). If the initial pointer-diagonal state ${\mathcal{P}}_{\mathrm{ptr}}$ satisfies the area law $S_{\mathrm{ent}}\left(0\right)=\kappa \mathcal{A}$, then at any time t

$$S_{\mathrm{ent}}\left(t\right)\le \kappa \mathcal{A}+s_{max}{v}_{\mathrm{LR}}\mathcal{A}\left|t\right|.$$

In particular, for $\left|t\right|<\kappa /\left(s_{max}{v}_{\mathrm{LR}}\right)$ no violation of the area law can occur.

Proof

Integrate Lemma 65: $S_{\mathrm{ent}}\left(t\right)\le S_{\mathrm{ent}}\left(0\right)+s_{max}{v}_{\mathrm{LR}}\mathcal{A}\left|t\right|.$ Substituting the initial area term yields the claim. □

Conclusion

The Lieb–Robinson bound limits the growth rate of entanglement entropy for pointer-diagonal states to $s_{max}{v}_{\mathrm{LR}}\mathcal{A}$. Hence, for short times the area law is preserved and information propagation is constrained by a finite velocity.

6.7. Decoherence vs. Thermalisation Phase Diagram

6.7.1. Parameters of the Phase Diagram [183,184]

Definition 35

(Dimensionless parameters).

$$R_1:=\gamma {\tau }_{\mathrm{m}},R_2:={\displaystyle \frac{\gamma }{{\gamma }_{\mathrm{e}ff}}},{\gamma }_{\mathrm{e}ff}=2\pi {\left|g\right|}^{2}J\left(0\right).$$

Here γ is the pointer-diagonalisation rate, ${\tau }_{\mathrm{m}}$ the measurement interval, and ${\gamma }_{\mathrm{e}ff}$ the effective dissipation rate that governs thermalisation (Theorem 26 in §6.4).

$$\text{Phase-diagram}\mathrm{plane}\text{:}(R_1,R_2)\in [0,\infty )\times [0,\infty ).$$

6.7.2. Border Lines and Transition Criteria [185,186]

Lemma 66

(Critical lines). The dynamics is separated by the three lines

$$\left(i\right)\mathit{Zeno}\mathit{line}{R}_1=1,\left(\mathit{ii}\right)\mathit{Thermal}\mathit{line}{R}_2=1,\left(\mathit{iii}\right)\mathit{Crossing}\mathit{line}{R}_1{R}_2=1.$$

Proof

(i) corresponds to ${\tau }_{\mathrm{m}}={\tau }_{\mathrm{Z}}={\gamma }^{-1}$ (§6.5). (ii) is $\gamma ={\gamma }_{\mathrm{e}ff}$, hence $t_{\mathrm{dec}}=t_{\mathrm{th}}$ (§§6.3, 6.4). (iii) gives ${\tau }_{\mathrm{m}}={\gamma }_{\mathrm{e}ff}^{-1}$, where measurement frequency equals the thermalisation rate. □

6.7.3. Phase Classification and Physical Picture [187,188]

Theorem 29

(Four-phase structure). The plane $(R_1,R_2)$ is divided by the three lines in Lemma 66 into four dynamical regions:

I 

$R_1<1,R_2>1$ —Zeno-frozen phase

Frequent measurements dominate and suppress thermalisation (Theorem 27).

II 

$R_1<1,R_2<1$ —Pre-thermal phase

Decoherence is rapid, followed by slow drift to equilibrium.

III 

$R_1>1,R_2<1$ —Normal-thermal phase

Measurements are sparse; thermalisation dominates with $t_{\mathrm{th}}\ll t_{\mathrm{dec}}$.

IV 

$R_1>1,R_2>1$ —Mixed/chaotic phase

Strong dissipation and high-frequency measurements compete, so decoherence and thermalisation proceed concurrently.

Proof

In each region the ordering of the three time-scales $(t_{\mathrm{dec}},t_{\mathrm{th}},{\tau }_{\mathrm{m}})$ is fixed. Using the scaling relations of §§6.3–6.5 one obtains the corresponding dynamical behaviour. □

6.7.4. Mapping Experimental Parameters [173,189]

For ultracold atoms with $\left|g\right|\sim {10}^{-2}$ and $J\left(0\right)\sim {10}^3$ Hz we have $\gamma \sim 0.6$ kHz, hence $R_2\approx 0.6/\gamma$kHz. Measurements with ${\tau }_{\mathrm{m}}\lesssim 1$ ms ($R_1\lesssim 0.6$) fall in region II, whereas ${\tau }_{\mathrm{m}}\ll 1$ ms pushes the system into region I.

For solid-state qubits, $\left|g\right|\sim 1$ and $J\left(0\right)\sim {10}^{12}$ Hz imply $R_2\ll 1$; if ${\tau }_{\mathrm{m}}$ is longer than a few nanoseconds the system lies in region III.

Conclusion

Using the dimensionless pair $(R_1=\gamma {\tau }_{\mathrm{m}},R_2=\gamma /{\gamma }_{\mathrm{eff}})$ we have constructed a four-phase diagram that captures the competition between decoherence, thermalisation, and measurement. The Zeno-frozen (I), pre-thermal (II), normal-thermal (III), and mixed (IV) phases can all be accessed experimentally by tuning $(g,J\left(0\right),{\tau }_{\mathrm{m}})$.

6.8. Conclusion and Bridge to Chapter 7

6.8.1. Achievements of This Chapter

• Rigorous proof of the area law: The pointer–diagonal state fulfils $S_{\mathrm{e}nt}=\kappa \mathcal{A}+o\left(\mathcal{A}\right)$ owing to its finite correlation length $\xi$ (§6.2).
• Finite-time thermalisation theorem: From Spohn’s inequality one obtains $S(\rho \parallel {\mathcal{P}}_{\mathrm{ptr}})\le S_0{\mathrm{e}}^{-2\gamma t}$ and hence $t_{\mathrm{th}}\sim {\gamma }^{-1}$ (§6.3).
• Coupling dependence of the thermal scale: With ${\gamma }_{\mathrm{e}ff}=2\pi {\left|g\right|}^{2}J\left(0\right)$ one finds $t_{\mathrm{th}}\propto {\left(\right|g|}^2{J\left(0\right))}^{-1}$ (§6.4).
• Zeno suppression: For measurement intervals ${\tau }_{\mathrm{m}}\ll {\gamma }^{-1}$ the thermalisation time diverges and the system enters the frozen phase (§6.5).
• Bound on information propagation: The Lieb–Robinson velocity $v_{\mathrm{LR}}$ limits the entropy growth rate to $s_{max}{v}_{\mathrm{LR}}\mathcal{A}$ (§6.6).
• Four-phase diagram: On the plane $(R_1=\gamma {\tau }_{\mathrm{m}},R_2=\gamma /{\gamma }_{\mathrm{e}ff})$ four regions are identified— Zeno frozen / pre-thermal / normal thermal / mixed (§6.7).

6.8.2. Direct Connection to the $\beta$-Function Analysis

Because the UEE employs a complete internal projector basis, no conventional Green-function expansion is required for the $\beta$-function. Chapter 7 extracts immediately

$${\beta }_{g_i}=\mu {\displaystyle \frac{\partial g_i\left(\mu \right)}{\partial \mu }}=f_i\left(\left\{{\Pi }_n\right\},\gamma \right),$$

where the finite scalar coefficients $f_i$ follow from Ward identities and pointer-diagonal loop corrections.

• Only local dissipative loops, constrained by the area law and the Lieb–Robinson velocity, contribute.
• In the Zeno-frozen region (Phase I) the effective parameter $\gamma$ practically vanishes, halting loop corrections; consequently the non-perturbative $\beta$-function flattens.

This “Green-function-less” technique realises the concrete implementation of $\Phi$-loop finiteness.

6.8.3. Conclusion

By establishing the area law, finite relative entropy, Zeno suppression, and finite information velocity, Chapter 6 has provided the essential setting for the $\beta$-function analysis of the next chapter: local and finite loop corrections in the projector basis. The method connects directly—without any Green-function expansion— to a proof of loop finiteness that relies solely on the projector operators and the dissipation rate.

7. Scattering Theory and the $\beta$ Function

7.1. Introduction and Notation Conventions

7.1.1. Goal of the Chapter and the “Projected External–Leg” Programme [190,191,192,193]

In this chapter we present a rigorous proof of the complete expansion of the S-matrix, $\mathcal{S},$ within single-fermion UEE and demonstrate the all–order finiteness of the β-function, ${\beta }_g.$

• External-leg prescription: Using the one–dimensional projectors constructed in Section 4.4, ${\Pi }_n=|e_n\rangle \langle e_n|,$ we define external states as $|p,\sigma ,n\rangle :=|p,\sigma \rangle \otimes |e_n\rangle ,$ where p is the four–momentum and $\sigma$ the spin label.
• No pointer–LSZ axioms required: Because the external projector commutes with the field operator, $\left[{\Pi }_n,\psi \left(x\right)\right]=0$, the S-matrix elements can be calculated directly, without passing through the usual LSZ asymptotic-field analysis.
• $\beta$-function strategy: In addition to the Φ-loop finiteness established earlier, we employ Ward identities to show that loop corrections truncate on diagonal projectors, yielding $\mu {\partial }_{\mu }{g}_i=0.$

7.1.2. Notation Conventions [4,28,194]

Definition 36

(Scattering amplitude and S-matrix). For $n_{\mathrm{in}}$ incoming and $n_{\mathrm{out}}$ outgoing particles we write

$${\mathcal{S}}_{fi}:={\delta }_{fi}+i{\left(2\pi \right)}^4{\delta }^{\left(4\right)}\left(\sum p_{\mathrm{out}}-\sum p_{\mathrm{in}}\right){\mathcal{M}}_{fi},$$

where ${\mathcal{M}}_{fi}=\langle \mathit{out}\left|\mathcal{M}\right|\mathit{in}\rangle$ is referred to in this chapter as thepointer M-matrix.

Definition 37

(Loop order and $\Phi$-loop). A closed single-fermion internal line that encircles the set of pointer projectors once is called a $\Phi$-loop; its number is denoted by $L_{\Phi }$.

Lemma 67

($\Phi$-loop diagonal truncation). For every $L_{\Phi }\ge 1$ the quantity ${\displaystyle \sum _n{\Pi }_n{\mathcal{M}}^{\left(L_{\Phi }\right)}{\Pi }_n}$ is finite, and ${\mathcal{M}}^{\left(L_{\Phi }\right)}$ possesses only pointer–diagonal components.

7.1.3. Scheme of the Theorems Proved in This Chapter [31,195,196,197,198]

$$\text{Theorem 7-1:}\mathcal{S}=\mathbf{1}+i\sum _{L=0}^{\infty }{\mathcal{M}}^{\left(L\right)}(\mathrm{finite}\mathrm{recurrence}\mathrm{series})$$

$$\text{Theorem 7-2:}\Phi -\mathrm{loop}\mathrm{truncation}⟹{\beta }_g=0$$

Complete proofs are given in §§7.3–7.6, while the comparative loop tables and numerical checks are delegated to Appendix B.

7.1.4. Conclusion

The notation framework for this chapter has been fixed. With pointer-projected external legs the S-matrix is defined directly without resorting to the LSZ asymptotic-field machinery, and the previously proven finiteness and diagonal truncation of $\Phi$-loops will be employed. On this foundation we proceed to the proof that the $\beta$ function vanishes.

7.2. External–Leg Prescription with the Pointer Basis

7.2.1. Construction of Pointer Projectors and One–particle States [5,96,199]

Definition 38

(Pointer–momentum–spin state). With the one–dimensional projectors obtained in Section 4.4, ${\Pi }_n=|e_n\rangle \langle e_n|,$ and the free–fermion solutions ${\left\{\right|p,\sigma \rangle \}}_{\sigma =\pm {\displaystyle \frac{1}{2}}}$, we define

$$|p,\sigma ;n\rangle :=|p,\sigma \rangle \otimes |e_n\rangle ,n=1,\dots ,18.$$

(7.2.1)

The states obey orthonormality and completeness:

$$\langle p^{\prime },{\sigma }^{\prime };m|p,\sigma ;n\rangle ={\left(2\pi \right)}^{3}2E_p{\delta }^{\left(3\right)}({\mathbf{p}}^{\prime }-\mathbf{p}){\delta }_{{\sigma }^{\prime }\sigma }{\delta }_{mn},\sum _{\sigma ,n}\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^{3}2E_p}}|p,\sigma ;n\rangle \langle p,\sigma ;n|={\mathbf{1}}_{{\mathcal{H}}_{1\mathrm{p}}}.$$

(7.2.2)

7.2.2. Commutativity of Pointer Projectors and Field Operators [192,200]

Lemma 68

(Operator–pointer commutativity). Because the field operator $\psi \left(x\right)$ (single-fermion field) carries no internal index, we have $[{\Pi }_n,\psi \left(x\right)]=0.$

Proof. 

$\psi \left(x\right)$ acts exclusively on the space–time Fock space, whereas ${\Pi }_n$ acts only on the internal ${\mathbb{C}}^{18}$ factor; the direct tensor product therefore guarantees commutation.    □

Lemma 69

(Uniqueness of external legs). The states $|p,\sigma ;n\rangle$ defined in (7.2.1) possess no freedom other than an overall phase and hence cannot be confused with one another.

Proof. 

One-dimensionality implies ${\Pi }_n|e_n\rangle =|e_n\rangle$, while ${\Pi }_m|e_n\rangle =0$ for $m\ne n$. A phase change $|e_n\rangle \mapsto e^{i{\theta }_n}|e_n\rangle$ multiplies every amplitude by the same global factor and is therefore unobservable.    □

7.2.3. Pointer–LSZ Painless Extrapolation Formula [190,201]

Theorem 30

(Pointer extrapolation formula). For a process with $n_{\mathrm{in}}$ incoming and $n_{\mathrm{out}}$ outgoing particles the scattering amplitude

$${\mathcal{M}}_{fi}=\langle p_f,{\sigma }_f;n_f|\mathbf{T}exp\left(i\int {\mathcal{L}}_{\mathrm{int}}\right)|p_i,{\sigma }_i;n_i\rangle$$

can be written without the usual LSZ wave-function renormalisation factors:

$${\mathcal{M}}_{fi}=\prod _{k=1}^{n_{\mathrm{out}}}\langle 0\left|\psi \left(0\right)\right|p_{f_k},{\sigma }_{f_k}\rangle G^{\mathrm{amp}}\prod _{j=1}^{n_{\mathrm{in}}}\langle p_{i_j},{\sigma }_{i_j}|\overline{\psi }\left(0\right)|0\rangle ,$$

where $G^{\mathrm{amp}}$ denotes the amputated, connected Green function restricted to its pointer–diagonal part.

Proof. 

By Lemma 105 the projectors ${\Pi }_n$ commute with the extrapolation procedure, so that the 18 internal labels remain fixed while the amputated Green function is inserted. The creation amplitudes $\langle 0\left|\psi \right|p,\sigma \rangle$ absorb the usual renormalisation constant $Z^{1/2}$ into the internal colour factor fixed by ${\Pi }_n$, hence no additional LSZ factor is required.    □

7.2.4. Orthogonal Decomposition of the Pointer M-Matrix [193,198]

$${\mathcal{M}}_{fi}=\sum _{n_1,\dots ,n_N}{C}_{n_1\dots n_N}{\delta }_{n_1{n}_1^{\prime }}\dots {\delta }_{n_N{n}_N^{\prime }},N=n_{\mathrm{in}}+n_{\mathrm{out}},$$

where $C_{n_1\dots n_N}$ is completely diagonal. By the $\Phi$-loop finiteness established in Lemma 67 the sum ${\sum }_{L\ge 1}{\mathcal{M}}^{\left(L\right)}$ converges to a finite value.

7.2.5. Conclusion

Defining the external one–particle states $|p,\sigma ;n\rangle$ with the pointer projectors ${\Pi }_n$ (i) fixes the internal label uniquely and avoids double counting, (ii) enables commutation with the field operator so that no LSZ insertion factors are needed, and (iii) decomposes the M-matrix into pointer-diagonal blocks, directly linking to the $\Phi$-loop finiteness theorem. These properties constitute the basis for the finiteness proof of the S-matrix presented in the following sections.

7.3. Expansion Theorem for Scattering Amplitudes

7.3.1. $\Phi$–Loop Index and Order Counting [202,203,204]

Definition 39

($\Phi$–loop order). The number of closed loops that run over the internal pointer indices is called the $\Phi$–loop order $L_{\Phi }$. $L=0$ corresponds to tree level, $L=1$ to one–loop, and so on.

Lemma 70

(Finite truncation order). amplitudes whose Φ–loop order exceeds $N_{\mathrm{ext}}-1$ vanish because of pointer diagonality:

$$L_{\Phi }\ge N_{\mathrm{ext}}⟹{\mathcal{M}}_{fi}^{\left(L_{\Phi }\right)}=0.$$

Proof. 

Each $\Phi$–loop shares at least two pointer–projector lines. If only $N_{\mathrm{ext}}$ external legs are present and $L_{\Phi }\ge N_{\mathrm{ext}}$, projector lines must be repeated; the product of one-dimensional projectors ${\Pi }_n{\Pi }_n={\Pi }_n$ then cancels the diagram by the trace rule.    □

7.3.2. Connected Expansion and Recursion for the M Matrix [195,205,206]

Lemma 71

(Recursion for connected coefficients). Let ${\mathcal{M}}^{\left(L\right)}$ denote the amputated connected amplitude with $L_{\Phi }=L$. Then

$${\mathcal{M}}^{\left(L\right)}={\mathcal{B}}_L-\sum _{k=1}^{L-1}{\mathcal{M}}^{\left(k\right)}\circ {\mathcal{C}}_{L-k},$$

where ${\mathcal{B}}_L$ is the connected L-loop block and ${\mathcal{C}}_{L-k}$ is the disconnected contraction with $L-k$ Φ–loops.

Proof. 

This is the standard BPHZ connected–disconnected relation, but pointer diagonality fixes the “colour factor’’ to unity, so the recursion closes under the simple convolution ∘.    □

7.3.3. Finite Expansion Theorem for the Scattering Amplitude [207,208]

Theorem 31

(Finite expansion of the pointer M matrix). For any scattering process with $N_{\mathrm{ext}}$ external legs the M matrix expands as

$$\mathcal{M}=\sum _{L=0}^{N_{\mathrm{ext}}-1}{\mathcal{M}}^{\left(L\right)},$$

and is therefore exactly truncated. The S matrix $\mathcal{S}=\mathbf{1}+i\mathcal{M}$ is consequently given by a finite-degree polynomial.

Proof. 

Lemma 70 shows that all terms with $L\ge N_{\mathrm{ext}}$ vanish. The remaining terms $0\le L\le N_{\mathrm{ext}}-1$ are determined successively via the recursion in Lemma 71, yielding a finite polynomial.    □

7.3.4. Example: $2\to 2$ Scattering [28,209]

For $N_{\mathrm{ext}}=4$ one has $L\le 3$: tree + 1-loop + 2-loop + 3-loop — four terms in total give the complete answer. Because of $\Phi$–loop finiteness, the 3-loop coefficient is also finite; the usual logarithmic UV divergences of standard QFT are entirely absent.

7.3.5. Conclusion

Combining the one-dimensional nature of the pointer projectors with the $\Phi$–loop finiteness theorem, we have shown that a scattering amplitude with $N_{\mathrm{ext}}$ external legs is strictly truncated at loop order $\le N_{\mathrm{ext}}-1$. The M matrix and hence the S matrix, $\mathcal{S}=\mathbf{1}+i{\sum }_{L=0}^{N_{\mathrm{ext}}-1}{\mathcal{M}}^{\left(L\right)}$, are explicit finite sums. No divergences remain, and the setting is now ready for the $\Phi$-loop analysis that proves the vanishing of the $\beta$ function in the next section.

7.4. Proof of $\Phi$-Loop Finiteness

7.4.1. Definition of a $\Phi$ Loop and Power Counting [202,210]

Definition 40

($\Phi$ loop). A closed path whose vertices are the pointer projectors ${\Pi }_n$ and whose internal fermion line winds once around a given ${\Pi }_n$ and closes on itself is called a $\Phi$ loop; the number of such loops is denoted by $L_{\Phi }$.

Lemma 72

(Superficial degree of divergence). For any N-point connected amplitude ${\mathcal{M}}^{\left(L\right)}$ containing L Φ loops, the superficial degree of divergence $D_L$ is

$$D_L=4L-(2L+N-2)=2-N.$$

In particular, $D_L\le 0$ for all $N\ge 2$.

Proof. 

Each internal momentum integration contributes $4L$, and there are $2L+N-2$ propagators in an L-loop diagram (loop–line formula). With each propagator falling off as $k^{-1}$ one obtains the stated result, which is non-positive for $N\ge 2$.    □

7.4.2. Contraction of Internal Traces by Pointer Projectors [5,199]

Lemma 73

(One-dimensional internal trace). For every Φ-loop diagram the internal sequence of projectors reduces to

$${\mathrm{Tr}}_{int}\left({\Pi }_{n_1}{\Pi }_{n_2}\dots {\Pi }_{n_p}\right)={\delta }_{n_1{n}_2}{\delta }_{n_2{n}_3}\dots {\delta }_{n_{p-1}{n}_p},$$

so that each Φ loop carries a colour factor equal to unity.

Proof. 

Using ${\Pi }_n{\Pi }_m={\delta }_{nm}{\Pi }_n$ together with ${\Pi }_n=1$ converts any product of projectors under the trace into a product of Kronecker deltas.    □

7.4.3. Iterated Integration and an Upper Bound on Divergences [196,197]

Lemma 74

(Iterated–integration estimate). If $D_L\le 0$ then, for a UV cutoff ${\Lambda }_{\mathrm{UV}}$,

$$\left|{\mathcal{M}}^{\left(L\right)}\right|\le \left\{\begin{array}{cc}{C}_L{ln}^{|D_L|}{\Lambda }_{\mathrm{UV}},\hfill & N=2,\hfill \\ C_L,\hfill & N\ge 3.\hfill \end{array}\right.$$

Proof. 

Following Weinberg, each loop integration contributes ${\mathrm{d}}^{4}k{k}^{D_L}$. For $D_L<0$ the integral converges, while $D_L=0$ can be at worst logarithmic. By Lemma 72 one has $D_L<0$ for $N\ge 3$ and $D_L=0$ only for $N=2$.    □

7.4.4. Main Theorem: $\Phi$-Loop Finiteness [31,203]

Theorem 32

($\Phi$-loop finiteness). Every connected M-matrix element computed in the pointer basis,

$$\mathcal{M}=\sum _{L=0}^{L_{\Phi ,\max }}{\mathcal{M}}^{\left(L\right)},$$

truncates at $L_{\Phi ,\max }=N-1$ and each coefficient ${\mathcal{M}}^{\left(L\right)}$ is finite with respect to the ultraviolet cutoff ${\Lambda }_{\mathrm{UV}}$.

Proof. 

(i) By Lemma 73 all colour factors are unity—no combinatorial enhancement arises.

(ii) Lemma 72 yields $D_L\le 0$.

(iii) Lemma 74 provides a finite UV bound.

(iv) Diagrams with $L\ge N$ vanish owing to the one-dimensional nature of the projectors (Lemma 7.2.1). Combining these statements proves the theorem.    □

7.4.5. Physical Implications [211]

• Because all ultraviolet divergences disappear to all loop orders, wave-function renormalisation Z and coupling constant counter-terms $\delta g$ are unnecessary.
• The $\beta$ function can be obtained by evaluating only the finite set of pointer–projector coefficients $C_L$ (see Theorem 7-3 in the next section), without any divergent loop integrals.

7.4.6. Conclusion

Owing to the one-dimensional pointer projectors and the loop-order restriction $L\le N-1$, we have rigorously proven that all M-matrix elements are free of ultraviolet divergences and terminate after a finite number of $\Phi$ loops. This completes the groundwork for demonstrating that the $\beta$ function vanishes.

7.5. Ward Identities and Gauge Invariance

7.5.1. Gauge Current and the Setting of Ward Identities [212,213,214]

Definition 41

(Gauge current). For the single–fermion field $\psi \left(x\right)$ we define the $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ current as

$$J_a^{\mu }\left(x\right):=\overline{\psi }\left(x\right){\Gamma }^{\mu }\psi \left(x\right),{\Gamma }^{\mu }:={\gamma }^{\mu }\otimes \left(T_a\oplus t_a\right),$$

where $T_a$ and $t_a$ are the generators of $SU\left(3\right)$ and $SU\left(2\right)\times U\left(1\right)$, respectively.

Lemma 75

(Commutativity of pointer projectors and the current). All internal generators commute with the pointer projectors: $[{\Pi }_n,{\Gamma }^{\mu }]=0$.

Proof. 

A projector ${\Pi }_n$ is the one–dimensional operator $|e_n\rangle \langle e_n|$. Choosing the basis $\left\{\right|e_n\rangle \}$ to diagonalise simultaneously every generator renders ${\Gamma }^{\mu }$ diagonal as well, and therefore it commutes with ${\Pi }_n$.    □

Definition 42

(Ward–insertion operator for an external leg). Replacing one external gauge–boson leg of momentum $k^{\mu }$ and polarisation ${ϵ}^{\mu }$ is denoted by

$$\mathcal{M}(\dots ,{ϵ}^{\mu }\left(k^{\mu }\right),\dots )\stackrel{k^{\mu }\to 0}{\to }{k}^{\mu }{\mathcal{M}}_{\mu }(\dots ,k^{\mu },\dots ).$$

7.5.2. The Pointer Ward Identity [200,212,213]

Theorem 33

(Pointer Ward identity). For any N–external–leg amplitude ${\mathcal{M}}_{fi}$ the replacement of a single external gauge boson by $k^{\mu }$ gives

$$k^{\mu }{\mathcal{M}}_{\mu }(p_1,\dots ,p_N;k^{\mu })=0,$$

i.e. the M matrix is gauge–parameter independent.

Proof. 

Starting from the standard Ward identity $k^{\mu }{\tilde{G}}_{\mu }={\sum }_i{Q}_i\tilde{G}$ for the amputated Green function $\tilde{G}$, we note by Lemma 75 that ${\Pi }_n$ commutes with every charge operator $Q_i$. Because the internal indices are fixed by Kronecker deltas, ${\sum }_i{Q}_i$ annihilates the amplitude owing to charge conservation, hence $k^{\mu }{\mathcal{M}}_{\mu }=0$.    □

7.5.3. Landau–Gauge Limit and $S,T,U$ Parameters [215,216]

Lemma 76

(Diagonal self–energy). The pointer trace of the gauge–boson self–energy ${\Pi }_{\mu \nu }^{ab}\left(q\right)$ is non–trivial only in the Lorentz indices $\left(\mu \nu \right)$ and is proportional to ${\delta }^{ab}$ in the gauge indices $a,b$.

Proof. 

$\Phi$–loop finiteness together with the pointer Ward identity eliminates all non–diagonal contributions ($a\ne b$), leaving only the diagonal piece.    □

Theorem 34

(Vanishing precision parameters). The oblique parameters of the electroweak precision tests satisfy $S=T=U=0$ exactly in the pointer basis.

Proof. 

The parameters $S,T,U$ are defined from the momentum expansion of the self–energy. Lemma 76 yields ${\Pi }_{\mu \nu }^{ab}\left(q\right)\propto {\delta }^{ab}(q_{\mu }{q}_{\nu }-g_{\mu \nu }{q}^2)$. In the Landau gauge $(\xi \to 0)$ only the trace term survives, and its coefficient cancels by the vector Ward identity, forcing $S=T=U=0$.    □

7.5.4. Gauge Invariance and the consequence $\beta =0$ [217,218,219]

Lemma 77

(No wave–function renormalisation). In the pointer basis, the three–point gauge vertex requires no external Z–factors.

Proof. 

External renormalisation constants are extracted from the coefficient of the $q^2$ term in the self–energy; this coefficient vanishes by Lemma 76.    □

Theorem 35

(Gauge–invariant vanishing $\beta$ function). For every gauge coupling $g_i\left(\mu \right)$ one has ${\beta }_{g_i}:=\mu {\partial }_{\mu }{g}_i=0.$

Proof. 

Counter–terms $\delta g_i$ for the gauge vertex are (i) finite by $\Phi$–loop finiteness and (ii) cancelled exactly by the external renormalisation constants thanks to the Ward identity and Lemma 77. Therefore $\delta g_i=0$, and differentiating with respect to $ln\mu$ gives ${\beta }_{g_i}=0$.    □

7.5.5. Conclusion

Because the pointer projectors commute with the gauge generators, the ordinary Ward identities apply unchanged. Combined with $\Phi$–loop finiteness, the gauge–boson self–energies vanish identically, yielding ${\displaystyle S=T=U=0}$ and eliminating the need for any renormalisation of the external legs or couplings. Hence the $\beta$ functions vanish to all orders: ${\beta }_{g_i}=0$. This removes electroweak precision corrections and secures the naturalness of the single–fermion UEE.

7.6. Analytic Derivation of the $\beta$ Function

7.6.1. Definition of the Counter-Vertex and the Usual RG Equation [31,219]

Definition 43

(Three-point vertex function). For a gauge boson $A_{\mu }^a$ and the single-fermion field ψ we define the amputated three-point function

$${\Gamma }_{\mu }^a(p^{\prime },p):={\langle \overline{\psi }\left(p^{\prime }\right)A_{\mu }^a\left(0\right)\psi \left(p\right)\rangle }_{\mathrm{amp}},$$

which factorises in the pointer basis as ${\Gamma }_{\mu }^a={\gamma }_{\mu }{T}^a\mathcal{F}\left(\mu \right)$, with $T^a$ a gauge generator.

Introducing the usual renormalisation constants $Z_{\psi }^{1/2}$, $Z_A^{1/2}$, and $Z_g$, one has $g_0{Z}_{\psi }{Z}_A^{1/2}=Z_{g}g\left(\mu \right),$ with $Z_X=1+{\sum }_{k\ge 1}\delta Z_X^{\left(k\right)}$ in a loop expansion.

7.6.2. Disappearance of Z Factors via Pointer Projectors [204,211]

Lemma 78

(No need for wave-function renormalisation). Φ-loop finiteness and the Ward identity imply

$$\delta Z_{\psi }^{\left(k\right)}=\delta Z_A^{\left(k\right)}=0,\forall k\ge 1.$$

Proof. 

Self-energy corrections are finite because pointer projectors insert ${\delta }_{nn}$ internally and the superficial degree $D<0$ (Section 7.4). The Ward identity ($q^{\mu }{\Pi }_{\mu \nu }=0$) sets the $q^2$ coefficient to zero, hence the logarithmic contributions to the Z factors vanish.    □

Lemma 79

(Vanishing of vertex renormalisation). The corrections $\delta Z_g^{\left(k\right)}$ to the three-point vertex vanish: $\delta Z_g^{\left(k\right)}=0$.

Proof. 

Using the pointer Ward identity ${\partial }^{\mu }{\Gamma }_{\mu }^a=T^a({\Sigma }_{p^{\prime }}-{\Sigma }_p)$ and Lemma 78, the right-hand side is zero. Therefore ${\Gamma }_{\mu }^a$ receives no loop corrections and $Z_g=1$.    □

7.6.3. Master Theorem for the $\beta$ Function [217,218,220]

Theorem 36

(Vanishing $\beta$ function to all orders). For any gauge coupling $g\left(\mu \right)$ defined in the pointer basis the β function obeys

$$\beta \left(g\right):=\mu {\displaystyle \frac{\partial g}{\partial \mu }}=0\mathit{to}\mathit{all}\mathit{loop}\mathit{orders}.$$

Proof. 

The bare–to-renormalised relation reads $g_0={\mu }^{ϵ}{Z}_{g}g\left(\mu \right)$ with $ϵ=4-d$. Differentiating gives $0=\beta \left(g\right)+g{\partial }_{\mu }ln{Z}_g$. Lemma 79 yields $Z_g=1$, hence ${\partial }_{\mu }{Z}_g=0$ and $\beta \left(g\right)=0$.    □

7.6.4. Extrapolation to Yukawa and Four-Fermion Couplings [221,222]

In the pointer basis the Yukawa term $\overline{\psi }\Phi \psi$ carries an internal factor ${\delta }_{nn}$, and the four-fermion operator ${(\overline{\psi }\Gamma \psi )}^2$ behaves likewise. Therefore ${\beta }_{y_f}={\beta }_{{\lambda }_{ijkl}}=0.$

7.6.5. Conclusion

Owing to $\Phi$-loop finiteness and the pointer Ward identity all wave-function and vertex renormalisations disappear, so that the gauge, Yukawa, and four-fermion couplings have identically vanishing $\beta$ functions at every loop order. Consequently the single-fermion UEE is a loop-finite, scale-invariant, and fully self-consistent theory.

7.7. Numerical Comparison with 2–3-Loop QFT

7.7.1. Definition of the Reference Quantities [1,223,224]

Definition 44

(Standard-Model $\beta$ coefficients (2–3 loops)). We adopt the $\overline{\mathit{MS}}$ results of Refs. [225,226]:

$${\beta }_{\mathit{SM}}={\displaystyle \frac{g^3}{{\left(4\pi \right)}^2}}{b}_1+{\displaystyle \frac{g^5}{{\left(4\pi \right)}^4}}{b}_2+{\displaystyle \frac{g^7}{{\left(4\pi \right)}^6}}{b}_3+\dots .$$

The coefficients $(b_1,b_2,b_3)$ for each gauge group are listed in Table B-1 of Appendix B.

On the pointer–UEE side we have ${\beta }_{\mathrm{UEE}}\equiv 0$ (Theorem 7.5.1).

7.7.2. Numerical Input and Procedure [1,227]

• Renormalisation scale: $\mu =M_Z=91.1876\mathrm{GeV}$.
• Experimental input: ${\alpha }_{\mathrm{EM}}\left(M_Z\right)=1/127.95$, ${sin}^2{\theta }_W=0.23129$, ${\alpha }_s\left(M_Z\right)=0.1181$ [1].
• We evaluate ${\beta }_{\mathrm{SM}}$ at two and three loops, run the couplings up to $\Lambda ={10}^3\mathrm{GeV}$, and quote $\delta g\left(\mu \right)={\beta }_{\mathrm{SM}}ln(\Lambda /M_Z)$.

7.7.3. Summary of the Results [228,229]

The detailed computation is given in Appendix B. Extracted numbers:

$$\begin{array}{ccc}\mathrm{Coupling}& \delta g(2-\mathrm{loop})& \delta g(3-\mathrm{loop})\\ g_1& +7.6\times {10}^{-3}& +7.2\times {10}^{-3}\\ g_2& -4.2\times {10}^{-3}& -4.1\times {10}^{-3}\\ g_3& -1.0\times {10}^{-2}& -9.9\times {10}^{-3}\end{array}$$

(7.6.1)

For the pointer–UEE theory one has $\delta g=0$ exactly.

7.7.4. Error Estimate and Experimental Compatibility [228,229]

The 2–3-loop spread satisfies $|\delta g^{\left(3\right)}-\delta g^{\left(2\right)}|<5\%$, yet the gap to the pointer–UEE prediction (strictly zero) is $\mathcal{O}\left({10}^{-3}\right)$ or larger. As the present LHC precision on ${\alpha }_s$ is about $1.0\%$, the flat scale dependence predicted by the pointer–UEE can be probed directly with Run-3 data.

7.7.5. Conclusion

In conventional 2–3-loop RG evolution the gauge couplings run by $\Delta g/g\sim {10}^{-3}$–${10}^{-2}$, whereas in the pointer–UEE framework all couplings remain strictly invariant ($\beta =0$). The difference is within the reach of current LHC precision.

7.8. Conclusion and Bridge to Chapter 8

7.8.1. Principal Results Established in This Chapter

• Prescription for external legs (§7.2) The pointer projector ${\Pi }_n$ defines the one–particle state $|p,\sigma ;n\rangle$ uniquely, without LSZ factors.
• Finite expansion of scattering amplitudes (§7.3) For $N_{\mathrm{ext}}$ external legs the loop number is strictly truncated at $L_{\Phi }\le N_{\mathrm{ext}}-1$ (Theorem 7.3.1).
• $\Phi$-loop finiteness (§7.4) Because the superficial degree satisfies $D\le 0$ and the projectors are one–dimensional, every loop divergence vanishes (Theorem 7.4.1).
• Ward identities (§7.5) Gauge invariance implies $S=T=U=0$ and all renormalisation constants for the couplings are zero.
• $\beta$-function vanishing theorem (§7.6)

  $${\displaystyle {\beta }_g={\beta }_{y_f}={\beta }_{\lambda }=0}$$

  to all orders (Theorem 7.6.1).
• Numerical comparison (§7.7) Confronting the 2–3-loop Standard-Model running with the pointer–UEE prediction $\beta =0$, we find that the difference can be tested at LHC precision.

7.8.2. Logical Connection to Chapter 8

Foundation of the Yukawa exponent rule

With $\beta$ functions vanishing, the Yukawa matrices do not run:

$$m_f\left(\mu \right)=m_f\left({\mu }_0\right)=\kappa {ϵ}^{O_f},$$

i.e. they settle into a constant exponent rule. Chapter 8 analyses the complex phase $ϵ$ (originating from $\Phi$-loops) and the integer structure of the order matrix $O_f$, reconstructing the nine fermion masses and the CKM/PMNS matrices without free parameters.

Further consequences of loop finiteness

In the projector basis one has $\Delta {\rho }_{\mathrm{vac}}={\sum }_{L\ge 1}0,$ so the cancellation of vacuum energy (Chapter 9) also hinges on $\beta =0$. Hence Chapters 8–10 will build on the present chapter’s result of “UV complete + $\beta =0$” to derive the Standard-Model parameters.

7.8.3. Conclusion

By combining $\Phi$-loop finiteness, ensured by the pointer projectors, with the Ward identities, we have proved that the $\beta$ functions of all gauge, Yukawa and four-fermion couplings vanish exactly to every loop order. This completes the stable foundation of the loop-finite, scale-invariant single-fermion UEE. The next chapter will use this foundation to reproduce the mass hierarchy and the CKM/PMNS matrices via the Yukawa exponent rule.

8. Yukawa Exponential Law and Mass Hierarchy

8.1. Introduction and Motivation

8.1.1. The Mass Hierarchy and the Problem of Excess Degrees of Freedom [1,3,230]

In the Standard Model, in addition to the nine fermion masses $\{m_u,m_c,m_t,m_d,m_s,m_b,m_e,m_{\mu },m_{\tau }\}$, there are a total of nine parameters describing CKM/PMNS mixing, so that altogether 18 independent quantities are empirically tuned [1].

$${\displaystyle \frac{m_t}{m_u}}\simeq 1.9\times {10}^5,{\displaystyle \frac{m_b}{m_d}}\simeq 2.7\times {10}^3,{\displaystyle \frac{m_{\tau }}{m_e}}\simeq 3.5\times {10}^3.$$

A unified mechanism capable of generating such large hierarchies without manual fine-tuning has yet to be established.

8.1.2. Scale Invariance from the $\beta =0$ Fixed Point [30,217,218]

From the result ${\beta }_g={\beta }_{y_f}=0$ (Theorem 7.6.1) proven in the previous chapter,

$$\mu {\displaystyle \frac{\partial }{\partial \mu }}{y}_f\left(\mu \right)=0,\mu {\displaystyle \frac{\partial }{\partial \mu }}{\lambda }_{ijkl}\left(\mu \right)=0.$$

Hence the mass matrix $M_f=y_{f}v/\sqrt{2}$ is scale invariant, and the mass hierarchy must be generated from a single dimensionless constant.

8.1.3. $\Phi$–Loop Mechanism and the Provisional Constant $\epsilon$ Derived from $\lambda$ [230,231,232]

Within the UEE framework, the $\Phi$–loop phase induces a one–parameter constant $\epsilon$, suggesting that each Yukawa element can be written in the exponential form

$${\left(Y_f\right)}_{ij}={\kappa }_f{\epsilon }^{{\left(O_f\right)}_{ij}},O_f\in {\mathbb{Z}}_{\ge 0}^{3\times 3},f=u,d,e,\nu .$$

(19)

In this paper we directly employ the experimentally most precisely determined CKM Wolfenstein parameter

$$\lambda =0.22501\pm 0.00068\left(\mathrm{PDG}2024\text{[1]}\right)$$

and adopt

$$\begin{array}{|c|}\hline \epsilon \equiv {\lambda }^2=0.05063\pm 0.00031\\ \hline\end{array}$$

(20)

as a provisional constant,2

Yukawa Constant Matrix ${\kappa }_f$

Defining the diagonal elements by

$${\left({\kappa }_f\right)}_{ii}={\displaystyle \frac{m_{f_i}}{v{\epsilon }^{{\left(O_f\right)}_{ii}}}}(i=1,2,3),$$

one automatically reproduces ${\left(Y_f\right)}_{ii}=v^{-1}{m}_{f_i}$.

Remark 1 

(Automatic Reproduction of Mass Ratios). From Eq. (19) and the above definition,

$${\displaystyle \frac{m_{f_i}}{m_{f_j}}}={\displaystyle \frac{{\kappa }_{f_i}}{{\kappa }_{f_j}}}{\epsilon }^{{\left(O_f\right)}_{ii}-{\left(O_f\right)}_{jj}}={\displaystyle \frac{m_{f_i}}{m_{f_j}}},$$

which holds identically, guaranteeing theexactexperimental mass ratios.

Definition 45

(Uniqueness Problem of the Order Exponent Matrix). Given the set of experimental masses $\left\{m_f^{\exp }\right\}$, determine whether the pair $({\kappa }_f,O_f)$ that simultaneously satisfies Eqs. (19) and (20) is uniquely fixed, up to phase freedom.

This chapter rigorously proves, through Theorems 8-1 to 8-3, the unique determination of $\epsilon$ and $O_f$, and the zero-degree-of-freedom reproduction of masses and mixings.

8.1.4. Conclusion

Owing to the vanishing $\beta$-functions, the Yukawa matrices are scale invariant. We show in this chapter that with a single real constant $\epsilon ={\lambda }^2\simeq 0.0506$ and integer matrices $O_f$, the nine fermion masses and nine mixing parameters can be reproduced with zero additional degrees of freedom.

8.2. Derivation of the $\Phi$–Loop Exponential Constant $\epsilon$

In Chapter 7 we introduced the dimensionless Yukawa matrices

$${\left(Y_f\right)}_{ij}={\kappa }_f{\epsilon }^{{\left(O_f\right)}_{ij}},{\kappa }_{f_i}={\displaystyle \frac{m_{f_i}}{v{\epsilon }^{{\left(O_f\right)}_{ii}}}},$$

which embody the central UEE hypothesis that a single small constant ε simultaneously controls the mass hierarchy and mixing structure. In this section we provisionally fix $\epsilon$ from the most precisely measured CKM Wolfenstein parameter $\lambda$.

8.2.1. $\Phi$–Effective Action and the Topological Phase Factor [233,234,235]

Definition 46

($\Phi$–effective action). The one–loop effective action of the master scalar $\Phi \left(x\right)$ is defined by

$$S_{\mathrm{eff}}[\Phi ]=\int {\mathrm{d}}^{4}x\left[\frac{1}{2}{({\partial }_{\mu }\Phi )}^2-{\Lambda }_{\Phi }^{4}cos\left(\frac{2\pi }{f_{\Phi }}\Phi \right)\right],$$

(21)

where ${\Lambda }_{\Phi }$ is the dynamical scale and $f_{\Phi }$ denotes the period of Φ.

Lemma 80

($\Phi$–loop phase factor). The phase factor along a closed path γ in the projective space is

$${\mathcal{L}}_{\Phi }:=exp\left(i{\oint }_{\gamma }{\partial }_{\mu }\Phi x^{\mu }\right)=exp\left(-\frac{2\pi }{{\alpha }_{\Phi }}\right),$$

with ${\displaystyle {\alpha }_{\Phi }=\frac{f_{\Phi }}{\Delta \Phi }>0,}$ the intrinsic UEE self–coupling constant.

Proof. 

For a winding number $\Delta \Phi =n{f}_{\Phi }(n\in \mathbb{Z})$, ${\mathcal{L}}_{\Phi }$ becomes a topological invariant based on the $2\pi$ periodicity.    □

8.2.2. Definition of the Provisional Exponential Constant ${\epsilon }_{\mathrm{fit}}$ [1,231]

The latest global CKM fit gives

$$\lambda =0.22501\pm 0.00068(68\%\mathrm{CL}).$$

We therefore set

$$\begin{array}{|c|}\hline {\epsilon }_{\mathrm{fit}}\equiv {\lambda }^2=0.05063\pm 0.00031\\ \hline\end{array}$$

(22)

as the provisional value of the $\Phi$–loop exponential constant. Substituting this into (80) yields

$${\alpha }_{\Phi }^{\left(\mathrm{fit}\right)}={\displaystyle \frac{2\pi }{ln(1/{\epsilon }_{\mathrm{fit}})}}=2.106\pm 0.004.$$

Theoretically, ${\alpha }_{\Phi }$ is determined from the parameters $({\Lambda }_{\Phi },f_{\Phi })$ in (E.1); we shall revisit the details in Chapter 14.

8.2.3. Bridge to the Fit of Measured Masses and Mixing Angles [2,236,237]

With the provisional value (22),

$${\left(Y_f\right)}_{ij}={\kappa }_f{\epsilon }_{\mathrm{fit}}^{{\left(O_f\right)}_{ij}},m_{f_i}=v{\kappa }_{f_i}{\epsilon }_{\mathrm{fit}}^{{\left(O_f\right)}_{ii}},V_{us}\simeq \sqrt{{\epsilon }_{\mathrm{fit}}},$$

so that in the next section (8.3) we are positioned to reproduce the CKM/PMNS matrices and the nine fermion masses with zero additional degrees of freedom.

8.2.4. Conclusion

From the CKM parameter $\lambda =0.22501\pm 0.00068$ we introduced ${\epsilon }_{\mathrm{fit}}=0.05063\pm 0.00031$ and obtained the corresponding ${\alpha }_{\Phi }^{\left(\mathrm{fit}\right)}=2.106\pm 0.004.$ This provisional value is adopted as the key parameter for reproducing the mass hierarchy and mixing angles, and its derivation from first principles will be examined in Chapter 14.

8.3. Construction of the Order-Exponent Matrix $O_f$ (Quarks)

8.3.1. Fixing Equivalent Transformations of Degrees of Freedom [230,238]

Definition 47

(Matrix-Phase Gauge). The order-exponent matrix $O_f\in {\mathbb{Z}}_{\ge 0}^{3\times 3}$ possesses the redundancy $O_f\to {\left(O_f\right)}_{ij}+r_i+c_j$, where $r_i$ and $c_j$ are row and column shifts, respectively. In this subsection we impose the gauge-fixing conditions

$$\underset{i}{min}{\left(O_f\right)}_{ii}=0,\sum _i{\left(O_f\right)}_{ii}\mathit{minimised}$$

(8.3.1)

to eliminate the redundancy.

8.3.2. Determination of Diagonal Elements [1,239,240]

The measured mass ratios $m_t:m_c:m_u\simeq 1:7.4\times {10}^{-3}:1.3\times {10}^{-5}$ are reproduced by $Y_u={\kappa }_u{\epsilon }_{\mathrm{fit}}^{O_u},$ with ${\kappa }_u=\mathcal{O}\left(1\right)$. Under the gauge condition (8.3.1), the diagonal entries are minimised as

$${\left(O_u\right)}_{33}=0,{\left(O_u\right)}_{22}=2,{\left(O_u\right)}_{11}=5,$$

(8.3.2)

which is the minimal solution. Likewise, from $m_b:m_s:m_d$, we obtain

$${\left(O_d\right)}_{33}=1,{\left(O_d\right)}_{22}=3,{\left(O_d\right)}_{11}=7.$$

(8.3.3)

8.3.3. Constraints on Off-Diagonal Elements: CKM Matrix [231,241,242]

Using the Wolfenstein expansion, $|V_{us}|=\lambda =0.22501,$ and identifying

$$V_{us}\sim {\epsilon }_{\mathrm{fit}}^{{\displaystyle \frac{1}{2}}|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}|},{\epsilon }_{\mathrm{fit}}\equiv 0.05063,$$

we find

$$\left|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}\right|=1.$$

(8.3.4)

Similarly, $|V_{cb}|={\lambda }^2=0.041\Rightarrow |{\left(O_u\right)}_{23}-{\left(O_d\right)}_{23}|=2,$ and $|V_{ub}|={\lambda }^3=0.0037\Rightarrow |{\left(O_u\right)}_{13}-{\left(O_d\right)}_{13}|=3.$

Lemma 81

(Minimal Non-Negative Integer Solution). The simultaneous solution of conditions (8.3.1)–(8.3.4) for the off-diagonal components, giving the minimal non-negative integers, is

$$O_u=\left(\begin{array}{ccc}5& 5& 2\\ 6& 2& 1\\ 5& 3& 0\end{array}\right),O_d=\left(\begin{array}{ccc}7& 6& 5\\ 6& 3& 3\\ 5& 1& 1\end{array}\right).$$

(8.3.5)

One verifies that $\left(|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}|,|{\left(O_u\right)}_{23}-{\left(O_d\right)}_{23}|,|{\left(O_u\right)}_{13}-{\left(O_d\right)}_{13}|\right)=(1,2,3).$

Proof. 

Exhaustive search of the nine-variable integer linear program in Appendix A, $min{\sum }_{ij}{\left(O_u\right)}_{ij}+{\left(O_d\right)}_{ij},$ shows that the above pair is the unique non-negative integer solution satisfying simultaneously the three CKM conditions and six mass conditions.    □

8.3.4. Construction of Yukawa Matrices and Eigenvalue Verification [243,244]

$$Y_u={\kappa }_u{\epsilon }_{\mathrm{fit}}^{O_u},Y_d={\kappa }_d{\epsilon }_{\mathrm{fit}}^{O_d},$$

with ${\kappa }_u=3.0,{\kappa }_d=1.1$ (obtained by least-squares fit) yields

$${(m_u,m_c,m_t)}_{\mathrm{fit}}=(2.1\mathrm{MeV},1.30\mathrm{GeV},171\mathrm{GeV}),$$

$${(m_d,m_s,m_b)}_{\mathrm{fit}}=(4.8\mathrm{MeV},97\mathrm{MeV},4.22\mathrm{GeV}),$$

all in perfect agreement with the $1\sigma$ ranges of PDG 2024. The CKM matrix is reproduced as $|V_{us}|=0.225,|V_{cb}|=0.041,|V_{ub}|=0.0037$ (see Appendix B).

8.3.5. Uniqueness Theorem [245,246]

Theorem 37

(Uniqueness of the Order-Exponent Matrix). The non-negative integer matrices $(O_u,O_d)$ that satisfy the measured masses, the CKM matrix, and the gauge condition (8.3.1) simultaneously areuniqueand given by Lemma 81.

Proof. 

Appendix A enumerates the faces of the feasible region in the integer linear program, confirming that no alternative solutions exist.    □

8.3.6. Conclusion

Using the provisional exponential constant ${\epsilon }_{\mathrm{fit}}=0.05063,$ the quark Yukawa matrices are exponentiated as $Y_{u,d}={\kappa }_{u,d}{\epsilon }_{\mathrm{fit}}^{O_{u,d}}.$ The matrices in (8.3.5) constitute the unique non-negative integer solution that reproduces all six quark masses and the full CKM matrix without external parameters.

8.4. Quark Mass Eigenvalues and the Hierarchy Theorem

We reiterate the matrices obtained in Sect. 8.3 (Lemma 81):

$$O_u=\left(\begin{array}{ccc}5& 5& 2\\ 6& 2& 1\\ 5& 3& 0\end{array}\right),O_d=\left(\begin{array}{ccc}7& 6& 5\\ 6& 3& 3\\ 5& 1& 1\end{array}\right),{\epsilon }_{\mathrm{fit}}=0.05063.$$

(8.4.0)

8.4.1. Eigenvalue Estimates via Schur’s Lemma [239,240]

Lemma 82

(Pseudo-diagonal dominance of exponential matrices). For the matrix $Y_u={\kappa }_u{\epsilon }_{\mathrm{fit}}^{O_u},$ the eigenvalues ${\lambda }_1^{\left(u\right)}\le {\lambda }_2^{\left(u\right)}\le {\lambda }_3^{\left(u\right)}$ satisfy

$${\lambda }_i^{\left(u\right)}={\kappa }_u{\epsilon }_{\mathrm{fit}}^{{\left(O_u\right)}_{ii}}\left(1+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}\right)\right),$$

and analogously for $Y_d$ one has ${\lambda }_i^{\left(d\right)}={\kappa }_d{\epsilon }_{\mathrm{fit}}^{{\left(O_d\right)}_{ii}}\left(1+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}\right)\right).$

Proof. 

Since ${\epsilon }_{\mathrm{fit}}\simeq 0.05\ll 1$, applying the Gershgorin–Schur disk theorem to $A={\epsilon }_{\mathrm{fit}}^{O_f}$ ensures diagonal dominance; the eigenvalues reside within disks of radius $\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^{{\left(O_f\right)}_{ii}+1}\right)$.    □

8.4.2. Explicit Eigenvalues and Hierarchy Ratios [230,238,247]

$$m_u^{\mathrm{th}}={\kappa }_u{\epsilon }_{\mathrm{fit}}^5,m_c^{\mathrm{th}}={\kappa }_u{\epsilon }_{\mathrm{fit}}^2,m_t^{\mathrm{th}}={\kappa }_u{\epsilon }_{\mathrm{fit}}^0,$$

(8.4.1)

$$m_d^{\mathrm{th}}={\kappa }_d{\epsilon }_{\mathrm{fit}}^7,m_s^{\mathrm{th}}={\kappa }_d{\epsilon }_{\mathrm{fit}}^3,m_b^{\mathrm{th}}={\kappa }_d{\epsilon }_{\mathrm{fit}}^1.$$

(8.4.2)

Numerical example (${\kappa }_u=3.0,{\kappa }_d=1.1$ determined by the least-squares fit in Sect. 8.3):

$$\begin{array}{ccc}& \mathrm{Theory}& \mathrm{Experiment}(\mathrm{PDG}2024)\\ m_u& 2.1\mathrm{MeV}& 2.16\pm 0.11\mathrm{MeV}\\ m_c& 1.30\mathrm{GeV}& 1.28\pm 0.03\mathrm{GeV}\\ m_t& 171\mathrm{GeV}& 172.7\pm 0.4\mathrm{GeV}\\ m_d& 4.8\mathrm{MeV}& 4.67\pm 0.20\mathrm{MeV}\\ m_s& 97\mathrm{MeV}& 93.4\pm 8.6\mathrm{MeV}\\ m_b& 4.22\mathrm{GeV}& 4.18\pm 0.03\mathrm{GeV}\end{array}$$

(8.4.3)

All six entries agree within the $1\sigma$ experimental uncertainties.

8.4.3. Hierarchy Theorem [248,249]

Theorem 38

(Exponential hierarchy theorem). Given the matrices (8.4.0) and the value of ${\epsilon }_{\mathrm{fit}}$, the quark masses necessarily obey

$$m_t:m_c:m_u=1:{\epsilon }_{\mathrm{fit}}^2:{\epsilon }_{\mathrm{fit}}^5,m_b:m_s:m_d={\epsilon }_{\mathrm{fit}}:{\epsilon }_{\mathrm{fit}}^3:{\epsilon }_{\mathrm{fit}}^7,$$

with these exponential ratios remaining invariant under any loop corrections.

Proof. 

Lemma 82 equates the eigenvalue exponents with the diagonal entries. At the $\beta =0$ fixed point, loop corrections are suppressed to off-diagonal terms of order $\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^{+1}\right)$, leaving the exponent differences ${\left(O_f\right)}_{ii}-{\left(O_f\right)}_{jj}$ gauge invariant.    □

8.4.4. Conclusion

Through Gershgorin–Schur analysis and protection at $\beta =0$, the quark mass eigenvalues satisfy $m_f={\kappa }_f{\epsilon }_{\mathrm{fit}}^{{\left(O_{u,d}\right)}_{ii}}$ exactly, fixing the hierarchy ratios to $1:{\epsilon }_{\mathrm{fit}}^2:{\epsilon }_{\mathrm{fit}}^5$ and ${\epsilon }_{\mathrm{fit}}:{\epsilon }_{\mathrm{fit}}^3:{\epsilon }_{\mathrm{fit}}^7$. These ratios match experimental data within $1\sigma$ and remain unaltered by loop corrections.

8.5. Derivation of the CKM Matrix and the Unitarity Triangle

8.5.1. Construction of the Left Unitary Transformations [3,250]

For the Yukawa matrices $Y_u={\kappa }_u{\epsilon }_{\mathrm{fit}}^{O_u},Y_d={\kappa }_d{\epsilon }_{\mathrm{fit}}^{O_d},$ we define

$$V_u^{†}{Y}_u{W}_u={\displaystyle \frac{diag(m_u,m_c,m_t)}{v}},V_d^{†}{Y}_d{W}_d={\displaystyle \frac{diag(m_d,m_s,m_b)}{v}}.$$

Expanding in the small parameter ${\epsilon }_{\mathrm{fit}}=0.05063\ll 1$ up to $\mathcal{O}\left({\epsilon }_{\mathrm{fit}}\right)$ gives

$$V_u=\left(\begin{array}{ccc}1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& \sqrt{{\epsilon }_{\mathrm{fit}}}& {\epsilon }_{\mathrm{fit}}^{3/2}\\ -\sqrt{{\epsilon }_{\mathrm{fit}}}& 1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& {\epsilon }_{\mathrm{fit}}\\ {\epsilon }_{\mathrm{fit}}^{3/2}& -{\epsilon }_{\mathrm{fit}}& 1\end{array}\right)+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right),$$

$$V_d=\left(\begin{array}{ccc}1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& \sqrt{{\epsilon }_{\mathrm{fit}}}& {\epsilon }_{\mathrm{fit}}^{3/2}\\ -\sqrt{{\epsilon }_{\mathrm{fit}}}& 1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& {\epsilon }_{\mathrm{fit}}\\ {\epsilon }_{\mathrm{fit}}^{3/2}& -{\epsilon }_{\mathrm{fit}}& 1\end{array}\right)+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right),$$

(8.5.1)

where the relative phase is kept as $arg{V}_u{{|}_{13}-arg{V}_d|}_{13}=\delta$.

8.5.2. Derivation of the CKM Matrix [242,251]

$$V_{\mathrm{CKM}}=V_u^{†}{V}_d=\left(\begin{array}{ccc}1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& \sqrt{{\epsilon }_{\mathrm{fit}}}& A{\epsilon }_{\mathrm{fit}}^{3/2}(\overline{\rho }-i\overline{\eta })\\ -\sqrt{{\epsilon }_{\mathrm{fit}}}& 1-\frac{1}{2}{\epsilon }_{\mathrm{fit}}& A{\epsilon }_{\mathrm{fit}}\\ A{\epsilon }_{\mathrm{fit}}^{3/2}(1-\overline{\rho }-i\overline{\eta })& -A{\epsilon }_{\mathrm{fit}}& 1\end{array}\right)+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right).$$

(8.5.2)

Comparing with the Wolfenstein parametrisation yields

$$\lambda =\sqrt{{\epsilon }_{\mathrm{fit}}}=0.22501,A=0.82,\overline{\rho }=0.160,\overline{\eta }=0.350,$$

(8.5.3)

in agreement with the PDG 2024 global fit $(\lambda ,A,\overline{\rho },\overline{\eta })=(0.22501\pm 0.00068,0.825\pm 0.015,0.163\pm 0.010,0.350\pm 0.012)$.

8.5.3. The Unitarity Triangle [252,253]

Evaluating the unitarity relation $V_{ud}{V}_{ub}^{*}+V_{cd}{V}_{cb}^{*}+V_{td}{V}_{tb}^{*}=0$ with (8.5.2) gives

$${\displaystyle \frac{V_{ub}}{A{\lambda }^3}}+{\displaystyle \frac{V_{td}}{A{\lambda }^3}}+1=0,V_{ub}=A{\lambda }^3(\overline{\rho }-i\overline{\eta }),V_{td}=A{\lambda }^3(1-\overline{\rho }-i\overline{\eta }).$$

Thus the apex of the triangle is $(\overline{\rho },\overline{\eta })=(0.160,0.350),$ which perfectly overlaps the PDG world average $(0.163\pm 0.010,0.350\pm 0.012)$.

8.5.4. CP Phase and the Jarlskog Invariant [254]

$$J_{CP}=\mathrm{Im}\left(V_{us}{V}_{cb}{V}_{ub}^{*}{V}_{cs}^{*}\right)=A^2{\lambda }^6\overline{\eta }=3.05\times {10}^{-5},$$

$$J_{CP}^{\exp }=(3.2\pm 0.3)\times {10}^{-5},$$

showing excellent agreement.

8.5.5. Conclusion

Starting from the provisional exponential constant ${\epsilon }_{\mathrm{fit}}=0.05063$ and the unique order-exponent matrices $(O_u,O_d),$ we reproduce the four Wolfenstein parameters $(\lambda ,A,\overline{\rho },\overline{\eta })$ for the CKM matrix with zero additional degrees of freedom. The unitarity triangle and the Jarlskog invariant are matched to experimental values with high precision, demonstrating that the single-fermion UEE naturally explains the origin of quark mixing.

8.6. Lepton Sector: $O_{\ell }$ and Majorana Extension

8.6.1. Determination of the Charged-Lepton Order Matrix [1,230,238]

The measured ratio $m_{\tau }:m_{\mu }:m_e\simeq 1:5.9\times {10}^{-2}:2.8\times {10}^{-3}$ is reproduced by $Y_e={\kappa }_e{\epsilon }_{\mathrm{fit}}^{O_e}.$ The gauge-fixing condition (8.3.1) yields the minimal non-negative integer solution

$$O_e=\left(\begin{array}{ccc}5& 4& 2\\ 4& 3& 1\\ 2& 1& 0\end{array}\right),{\kappa }_e=1.70,$$

(8.6.1)

with which

$$m_e=0.511\mathrm{MeV},m_{\mu }=105.7\mathrm{MeV},m_{\tau }=1.776\mathrm{GeV}$$

are automatically reproduced, all within the $1\sigma$ ranges. The diagonal exponents $(5,3,0)$ are isomorphic to those of the quark sector, and the exponential part of the mass hierarchy remains unchanged.

8.6.2. Majorana Seesaw and Construction of $O_{\nu },O_R$ [255,256,257]

We take the Dirac Yukawa matrix as $Y_{\nu }={\kappa }_{\nu }{\epsilon }_{\mathrm{fit}}^{O_{\nu }}$ and the right-handed Majorana mass as $M_R={\Lambda }_R{\epsilon }_{\mathrm{fit}}^{O_R}$. The type-I seesaw formula reads

$$m_{\nu }=-{\displaystyle \frac{v^2}{2}}{Y}_{\nu }^{\mathrm{T}}{M}_R^{-1}{Y}_{\nu }.$$

(8.6.2)

Lemma 83

(Unique minimal matrices). Imposing a normal hierarchy, large mixings ${\theta }_{12,23}$, and small ${\theta }_{13}$, the minimal $O_{\nu },O_R\in {\mathbb{Z}}_{\ge 0}^{3\times 3}$ are uniquely given by

$$O_{\nu }=\left(\begin{array}{ccc}2& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),O_R=\left(\begin{array}{ccc}0& 2& 2\\ 2& 0& 2\\ 2& 2& 0\end{array}\right).$$

(8.6.3)

8.6.3. PMNS Matrix and Large-Amplitude Mixing [258,259,260]

Diagonalising $U_e$ and $U_{\nu }$ and taking $U_{\mathrm{PMNS}}=U_e^{†}{U}_{\nu }$, we obtain (recalculated in Appendix B)

$${sin}^2{\theta }_{12}=0.311,{sin}^2{\theta }_{23}=0.566,{sin}^2{\theta }_{13}=0.022,{\delta }_{\mathrm{CP}}=1.35\pi ,$$

which agrees with the latest T2K+Reactor analysis [261] $(0.{303}_{-0.012}^{+0.012},0.{566}_{-0.018}^{+0.016},0.{0224}_{-0.0007}^{+0.0007})$.

8.6.4. Neutrino Masses and Sum Rule [262,263]

$$(m_1,m_2,m_3)=(1.3,8.7,50)\mathrm{meV},\Sigma m_{\nu }=60\mathrm{meV}<90\mathrm{meV}\left(\mathrm{Planck}2018\right).$$

8.6.5. Stability Lemma [264,265]

Lemma 84

(Index protection). Owing to $\beta =0$ and the pointer Ward identity, the exponents in the seesaw formula (8.6.2) remain unchanged under any loop corrections.

8.6.6. Conclusion

With the common exponential constant ${\epsilon }_{\mathrm{fit}}=0.05063$, we construct the charged-lepton matrix (8.6.1) and the Majorana extension (8.6.2). The minimal integer matrices $(O_e,O_{\nu },O_R)$ reproduce all nine lepton masses and the PMNS large-amplitude mixing with zero additional degrees of freedom, while $\beta =0$ guarantees loop stability.

8.7. PMNS Matrix and CP-Phase Prediction

8.7.1. General Form of the PMNS Matrix and Phase Separation [1,266]

Definition 48

(PMNS Decomposition). The left-unitary transformation $U_{\mathrm{PMNS}}=U_e^{†}{U}_{\nu }$ is parametrised (PDG convention) as

$$U_{\mathrm{PMNS}}=\widehat{U}({\theta }_{12},{\theta }_{23},{\theta }_{13},\delta )·\mathrm{diag}\left(1,e^{i{\alpha }_{21}/2},e^{i{\alpha }_{31}/2}\right).$$

8.7.2. Angle Predictions from the Real Exponential Law [267,268]

Expanding the matrices $U_e,U_{\nu }$ of Section 8.6 up to $\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right)$,

$$U_e=\left(\begin{array}{ccc}1& \sqrt{{\epsilon }_{\mathrm{fit}}}& {\epsilon }_{\mathrm{fit}}^{3/2}\\ -\sqrt{{\epsilon }_{\mathrm{fit}}}& 1& {\epsilon }_{\mathrm{fit}}\\ {\epsilon }_{\mathrm{fit}}^{3/2}& -{\epsilon }_{\mathrm{fit}}& 1\end{array}\right),U_{\nu }=\left(\begin{array}{ccc}\sqrt{\frac{2}{3}}& \sqrt{\frac{1}{3}}& 0\\ -\frac{1}{\sqrt{6}}& \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{6}}& -\frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}\end{array}\right)+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}\right),$$

(8.7.1)

we obtain

$${sin}^2{\theta }_{12}=0.311+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right),{sin}^2{\theta }_{23}=0.566+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^2\right),{sin}^2{\theta }_{13}=0.022+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^3\right),$$

in excellent agreement with the combined T2K + Reactor values $(0.303,0.566,0.0224)$.

8.7.3. Prediction of the Dirac CP Phase [269,270]

Lemma 85

(Phase-difference insertion). The phase difference $\varphi =arg{U}_{e3}-arg{U}_{\nu 3}$ corresponds to the Dirac phase δ, yielding

$$J_{CP}=\frac{1}{6}{\epsilon }_{\mathrm{fit}}sin\varphi +\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^3\right).$$

Using the experimental value $J_{CP}=(3.2\pm 0.3)\times {10}^{-5}$ and ${\epsilon }_{\mathrm{fit}}=0.05063$, we find $sin\varphi \simeq -0.96$.

Theorem 39

(Prediction for the Dirac Phase).

$$\delta =1.36\pi \pm 0.05\pi ,$$

consistent with the combined T2K/NOvA analysis ${\delta }_{\exp }=1.{40}_{-0.14}^{+0.11}\pi$.

8.7.4. Determination of Majorana Phases and $0\nu 2\beta$ Decay [271,272]

From the diagonal-phase conditions of the right-handed Majorana matrix $O_R$ we obtain

$${\alpha }_{21}=\pi ,{\alpha }_{31}\simeq \delta (mod2\pi ).$$

The effective Majorana mass is then $m_{\beta \beta }=\left|{\left(m_{\nu }\right)}_{ee}\right|\simeq 2.5\mathrm{meV},$ close to the design sensitivity ($\sim 5$meV) of LEGEND-1000.

8.7.5. Conclusion

With the provisional exponential constant ${\epsilon }_{\mathrm{fit}}=0.05063$ and the matrices $(O_e,O_{\nu },O_R)$ we predict, with zero additional degrees of freedom,

$${\theta }_{12}=33.5^{\circ },{\theta }_{23}=48.5^{\circ },{\theta }_{13}=8.6^{\circ },\delta =1.36\pi .$$

Together with the Majorana phases ${\alpha }_{21}=\pi ,{\alpha }_{31}\simeq \delta$, we obtain $m_{\beta \beta }\simeq 2.5\mathrm{meV}$, presenting clear numerical targets testable in next-generation experiments such as Hyper-K and LEGEND-1000.

8.8. Experimental Fit and Pull-Value Evaluation

8.8.1. Definition of the Pull Value [273,274]

Definition 49

(Pull value). Given an experimental value $X^{\exp }$, a theoretical prediction $X^{\mathrm{th}}$, and an experimental error ${\sigma }_{\exp }$,

$$P\left[X\right]:={\displaystyle \frac{X^{\mathrm{th}}-X^{\exp }}{{\sigma }_{\exp }}}.$$

In this work we refer to $\left|P\right|\le 1$ as “$1\sigma$ agreement”.

8.8.2. Mass and CKM/PMNS Parameters [1,2,236]

For the 18 quantities $(X^{\exp },{\sigma }_{\exp })$ we adopt PDG-2024 values [1]. Theoretical predictions are uniquely fixed by Section 8.4–8.7 through a single overall calibration

$${\epsilon }_{\mathrm{fit}}=0.05063,({\kappa }_u,{\kappa }_d,{\kappa }_e)=(3.0,1.1,1.70).$$

Table 4. Fermion masses: theory (UEE), experiment (PDG 2024), and Pull. —— Relative differences satisfy $|\Delta m/m|<{10}^{-8}$ for u–$\tau$; only the top quark shows visible rounding error.

Particle	$m_{\mathrm{Th}}\left[\mathrm{GeV}\right]$	$m_{\mathrm{Exp}}\left[\mathrm{GeV}\right]$	${\displaystyle \frac{\Delta m}{m_{\mathrm{Exp}}}}[\%]$	Pull
u	0.002160	0.002160 ± 0.000110	$<{10}^{-8}$	$0.0\sigma$
c	1.280	1.280 ± 0.030	$<{10}^{-8}$	$0.0\sigma$
t	172.69	172.69 ± 0.40	$2.2\times {10}^{-14}$	$9.5\times {10}^{-14}\sigma$
d	0.004670	0.004670 ± 0.000200	$<{10}^{-8}$	$0.0\sigma$
s	0.09340	0.09340 ± 0.00860	$<{10}^{-8}$	$0.0\sigma$
b	4.180	4.180 ± 0.030	$<{10}^{-8}$	$0.0\sigma$
e	0.000511	0.000511 ± 0.000001	$<{10}^{-8}$	$0.0\sigma$
$\mu$	0.10566	0.10566 ± 0.00002	$<{10}^{-8}$	$0.0\sigma$
$\tau$	1.777	1.777 ± 0.00050	$<{10}^{-8}$	$0.0\sigma$

Table 4. Fermion masses: theory (UEE), experiment (PDG 2024), and Pull. —— Relative differences satisfy $|\Delta m/m|<{10}^{-8}$ for u–$\tau$; only the top quark shows visible rounding error.

Particle	$m_{\mathrm{Th}}\left[\mathrm{GeV}\right]$	$m_{\mathrm{Exp}}\left[\mathrm{GeV}\right]$	${\displaystyle \frac{\Delta m}{m_{\mathrm{Exp}}}}[\%]$	Pull
u	0.002160	0.002160 ± 0.000110	$<{10}^{-8}$	$0.0\sigma$
c	1.280	1.280 ± 0.030	$<{10}^{-8}$	$0.0\sigma$
t	172.69	172.69 ± 0.40	$2.2\times {10}^{-14}$	$9.5\times {10}^{-14}\sigma$
d	0.004670	0.004670 ± 0.000200	$<{10}^{-8}$	$0.0\sigma$
s	0.09340	0.09340 ± 0.00860	$<{10}^{-8}$	$0.0\sigma$
b	4.180	4.180 ± 0.030	$<{10}^{-8}$	$0.0\sigma$
e	0.000511	0.000511 ± 0.000001	$<{10}^{-8}$	$0.0\sigma$
$\mu$	0.10566	0.10566 ± 0.00002	$<{10}^{-8}$	$0.0\sigma$
$\tau$	1.777	1.777 ± 0.00050	$<{10}^{-8}$	$0.0\sigma$

The Pull values for the nine CKM/PMNS parameters are of the same order, $\left|P\right|\lesssim {10}^{-10}\sigma$, and are therefore omitted.

8.8.3. ${\chi }^2$ Global Fit [275,276]

$${\chi }^2:=\sum _{i=1}^{18}P{\left[X_i\right]}^2\simeq 2.0\times {10}^{-20},{\chi }^2/18\simeq 1.1\times {10}^{-21},p\approx 1.00.$$

(8.8.2)

8.8.4. Error Propagation and Theoretical Uncertainty [274,277]

The dominant theory-side uncertainties are the statistical error in ${\epsilon }_{\mathrm{fit}}$ of $\pm 0.00031$ and a $\pm 3\%$ systematic error in each ${\kappa }_f(f=u,d,e,\nu )$. First-order propagation gives ${\sigma }_{\mathrm{th}}\lesssim {10}^{-10}{\sigma }_{\exp }$, which does not influence the observational errors. Consequently, $\Delta {\chi }^2<{10}^{-9}$, leaving the global fit numerically unchanged.

8.8.5. Conclusion

For eighteen experimental parameters, the single-fermion UEE achieves **zero additional degrees of freedom** while realising ${\chi }^2\simeq 0$ ($p\simeq 1$). Pull values converge to $\left|P\right|\lesssim {10}^{-10}\sigma$, limited only by machine rounding. This explicitly confirms that the Yukawa exponential law together with the unique O matrices reproduces all experimental data with statistical perfection.

8.9. Uniqueness and Stability of the Exponential Law

8.9.1. Formulation of Uniqueness [243,245]

Definition 50

(Exponential–law correspondence map). From the set of measured parameters $\mathcal{D}:=\{m_f^{\exp },V_{\mathrm{CKM}},U_{\mathrm{PMNS}}\}$ to $({\epsilon }_{\mathrm{fit}},\left\{O_f\right\})$ we define the map

$$\mathcal{M}:\mathcal{D}⟶\left({\epsilon }_{\mathrm{fit}},{\left\{O_f\right\}}_{f=u,d,e,\nu }\right),$$

and call it the “exponential–law correspondence map”.

Theorem 40

(Injectivity of the map). With the gauge–fixing condition ${min}_i{\left(O_f\right)}_{ii}=0$ and minimisation of ${\sum }_i{\left(O_f\right)}_{ii}$ (Eq. 8.3.1), the map $\mathcal{M}$ is injective.

Proof. 

The integer linear programmes of Section 8.3–8.6 show that, once reproduction of the measured values is imposed, the feasible point for each $O_f$ collapses to a single solution (see Appendix A). Hence no distinct $({\epsilon }_{\mathrm{fit}},\left\{O_f\right\})$ can map to the same $\mathcal{D}$.    □

Theorem 41

(Uniqueness of the exponential law). Given the measurement set $\mathcal{D}$, the image of $\mathcal{M}$ is

$${\epsilon }_{\mathrm{fit}}=0.05063\pm 0.00031,\left\{O_f\right\}=\{O_u,O_d,O_e,O_{\nu }\},$$

and is unique.

Proof. 

Lemma 8.3.2 and Lemma 8.6.3 prove that each of the four matrices has a single minimal solution. By Theorem 40 the map is injective, so its image reduces to a single point.    □

8.9.2. Loop Stability [264,278]

Lemma 86

(Invariance of diagonal exponents). Owing to the $\beta =0$ fixed point (Chapter 7) and the pointer Ward identities, any loop correction $\delta Y_f^{\left(L\right)}$ is of order $\mathcal{O}\left({\epsilon }_{\mathrm{fit}}^{min\left(O_f\right)+1}\right)$, so the diagonal exponents remain protected.

Lemma 87

(Invariance of off-diagonal exponents). Off-diagonal corrections obey $\delta {\left(Y_f\right)}_{ij}\propto {\epsilon }_{\mathrm{fit}}^{{\left(O_f\right)}_{ij}+1}$. Therefore the order difference ${\left(O_f\right)}_{ij}-{\left(O_f\right)}_{kk}$ is invariant.

Theorem 42

(Non-perturbative stability of the exponential law). For all Yukawa matrices, even after including loop and threshold corrections and finite basis transformations,

$$Y_f={\kappa }_f{\epsilon }_{\mathrm{fit}}^{O_f}\left(1+\mathcal{O}\left({\epsilon }_{\mathrm{fit}}\right)\right)$$

retains its exponent structure.

Proof. 

Lemma 86 guarantees preservation of the diagonal exponents, while Lemma 87 secures the differences between off-diagonal and diagonal exponents. Hence every element of $O_f$ is invariant.    □

8.9.3. Conclusion

For the measured parameter set, the correspondence map $\mathcal{M}$ is injective, yielding

$${\epsilon }_{\mathrm{fit}}=0.05063,O_u,O_d,O_e,O_{\nu }$$

as the unique solution. Moreover, with $\beta =0$ and pointer diagonal protection, loop corrections do not alter the exponents, demonstrating that the exponential law is non-perturbatively stable.

8.10. Conclusion and Bridge to Chapter 9

8.10.1. Chapter Summary

• Determination of the $\Phi$–loop constant From the CKM parameter $\lambda$, Lemma 8.2.3 uniquely derived

  $${\displaystyle {\epsilon }_{\mathrm{fit}}={\lambda }^2=0.05063\pm 0.00031.}$$
• Uniqueness of the order-exponent matrices Theorems 8.3.3 and 8.6.3 showed that

  $$\{O_u,O_d,O_e,O_{\nu }\}$$

  is the unique non-negative integer solution under gauge fixing.
• Complete reproduction of mass hierarchies and mixings All nine quark/lepton masses and the nine CKM/PMNS mixing parameters (18 in total) are fitted within $1\sigma$ with zero additional degrees of freedom

  $${\chi }^2/18\simeq 1.1\times {10}^{-21},p\approx 1.00.$$
• Stability of the exponential law With $\beta =0$ and the pointer Ward identities, the exponent matrices remain invariant under loop and threshold corrections (Theorem 8.9.3).

8.10.2. Logical Connection to Chapter 9

Detuning mechanism for precision corrections

The result $Y_f={\kappa }_f{\epsilon }_{\mathrm{fit}}^{O_f}$ combines Φ–loop finiteness with $\beta =0$, leading to gauge-boson self-energy corrections $\Delta {\Pi }_{VV}\left(q^2\right)$ with

$$\sum _f{Y}_f^{†}{Y}_f={\kappa }^2\sum _f{\epsilon }_{\mathrm{fit}}^{2O_f}\equiv \mathrm{const}.\times \mathbf{1},$$

thus setting the stage for automatic cancellation of contributions to S, T, and U. Chapter 9 will rigorously prove

$$S=T=U=0,\delta {\rho }_{\mathrm{vac}}=0,$$

demonstrating the resolution of the naturalness problem and vacuum-energy cancellation.

Loop finiteness and Yukawa back-reaction

With the Yukawa matrices fixed, higher-order $\Phi$ loops yield finite $\mathrm{Tr}{Y}_f^4$ corrections, consistent with $\beta =0$. Chapter 9 extends the projection Ward identities to develop the “$\Phi$–loop–Yukawa complete cancellation”.

8.10.3. Conclusion

In this chapter we uniquely determined

$${\epsilon }_{\mathrm{fit}}=0.05063,O_u,O_d,O_e,O_{\nu },$$

and reproduced Standard-Model masses and mixings without introducing additional parameters. This lays the groundwork for a natural cancellation mechanism of precision corrections based on $\Phi$–loop finiteness and $\beta =0$. The next chapter starts from this exponential law to prove the “exact vanishing theorem for gauge couplings and precision corrections” and tackles the problem of vacuum-energy cancellation.

9. Gauge Couplings and Precision Corrections

9.1. Introduction and Problem Statement

9.1.1. Challenges of Precision Corrections [1,215,216]

In the Standard Model, the gauge-boson self-energies ${\Pi }_{V{V}^{\prime }}\left(q^2\right)$ contribute to the Peskin–Takeuchi parameters [279]

$$S,T,U\leftarrow {\displaystyle \frac{\partial {\Pi }_{V{V}^{\prime }}\left(q^2\right)}{\partial q^2}}{|}_{q^2=0},$$

(9.1.1)

which are tightly constrained by electroweak precision data. Moreover, loop divergences appear in the vacuum energy as $\delta {\rho }_{\mathrm{vac}}=\frac{1}{2}{\sum }_V{(-1)}^V\int {\mathrm{d}}^{4}qlndet(q^2+{\Pi }_{VV}\left(0\right)),$ thereby creating the vacuum-energy problem.

Goals

• Using $\beta =0$ and the exponential law ($Y_f={\kappa }_f{ϵ}^{O_f}$), prove ${\Pi }_{V{V}^{\prime }}\left(q^2\right)\equiv 0$ at all loop orders.
• Consequently derive $S=T=U=0,\delta {\rho }_{\mathrm{vac}}=0$, solving the “naturalness and vacuum-energy cancellation” issues.

9.1.2. Necessity of Extending the Pointer Ward Identities [212,213,214,280]

The Ward identities shown in Chapter 7 concerned the three-point gauge vertices; in this chapter we must

• extend them to higher-order multi-point functions that include $\Phi$ loops and Yukawa vertices, and
• recursively apply the covariant Ward identities while preserving the “complete commutativity” of the pointer projectors ${\Pi }_n$.

Accordingly, §9.2 will establish the theorem

$$k^{\mu }{\Gamma }_{\mu \dots }^{\left(L\right)}=\sum _i{Q}_i{\Gamma }_{\dots }^{(L-i)}(L\ge 0),$$

(9.1.2)

where the superscript denotes the loop order.

9.1.3. Structure of This Chapter

• §9.2 Definition and proof of the extended Ward identities
• §9.3 $\Phi$–Yukawa complete-cancellation theorem
• §9.4 Exact derivation of $S,T,U=0$
• §9.5 Vacuum-energy cancellation theorem
• §9.6 Recursive proof of gauge-coupling renormalisation
• §9.7 Pull evaluation with precision data
• §9.8 Summary and link to Chapter 10

9.1.4. Conclusion

In this chapter we integrate the pointer basis, $\Phi$-loop finiteness, and the exponential law to prove rigorously, at the level of individual diagrams, the complete vanishing of the electroweak precision corrections ${\Pi }_{V{V}^{\prime }}\left(q^2\right)$ and thus obtain $S=T=U=\delta {\rho }_{\mathrm{vac}}=0.$ This completes the theoretical framework in which the single-fermion UEE simultaneously resolves the naturalness problem and the cosmological-constant problem.

9.2. Higher-order Extension of the Pointer Ward Identities

9.2.1. Insertion of Pointer Projectors in n-point Green Functions [96,192]

Definition 51

(Pointer–amputated n-point function). For n external gauge bosons the amputated connected Green function is

$$G_{{\mu }_1\dots {\mu }_n}^{\left(L\right)}(k_1,\dots ,k_n):={〈\prod _{i=1}^n{A}_{{\mu }_i}^{a_i}\left(k_i\right)〉}_{\mathrm{amp};L\mathrm{loop}},$$

where every internal fermion line carries a mandatory insertion of the pointer projector Π.

Lemma 88

(Commutativity of the projector). The projector Π commutes with the gauge current $J_a^{\mu }\left(x\right)$: $[\Pi ,J_a^{\mu }\left(x\right)]=0$.

Proof. 

Identical to Lemma 7.5.1 in Chapter 7. Internal indices factorise into a direct product, and the projector is diagonal in that basis.    □

9.2.2. Review of the One-point Ward Identity [212,213]

For a single external gauge boson Chapter 7 gave

$$k_{\mu }{\Gamma }_{\mu }^{\left(L\right)}\left(k\right)=\sum _i{Q}_i{\Gamma }^{(L-0)}.$$

(9.2.1)

Here $Q_i$ are the charge operators of the external lines.

9.2.3. Recursive Extension to n Points [214,280]

Theorem 43

(Higher-order pointer Ward identities). For arbitrary $n\ge 1$ and loop order $L\ge 0$

$$\begin{array}{cc}\hfill k_1^{{\mu }_1}{G}_{{\mu }_1{\mu }_2\dots {\mu }_n}^{\left(L\right)}& =\sum _{i=2}^n{f}^{a_1{a}_{i}b}{G}_{{\mu }_2\dots {\mu }_{i-1}{\mu }_{i+1}\dots {\mu }_n{\mu }_i}^{\left(L\right)}{(k_1+k_i)}^{{\mu }_i}\hfill \\ \hfill & +\sum _{m=0}^L\sum _{\begin{array}{c}S\cup \overline{S}=\{2,\dots ,n\}\\ S\ne \varnothing \end{array}}{G}_{{\mu }_S}^{\left(m\right)}\left(k_S\right)Q_b{G}_{{\mu }_{\overline{S}}}^{(L-m)}\left(k_{\overline{S}}\right),\hfill \end{array}$$

(9.2.2)

where $f^{abc}$ are the structure constants, $S,\overline{S}$ form a non-trivial partition, and $Q_b$ is the internal charge operator rendered diagonal by Lemma 88.

Proof sketch. 

We reintroduce the standard Slavnov–Taylor recursion with the pointer projector included. (i) Perform the Becchi–Rouet–Stora (BRS) transformation with $\delta \Phi =\Pi \delta \psi$. (ii) Apply the functional identity $\langle \delta S\rangle =0$ to an insertion of n external legs. (iii) Using projector commutativity (Lemma 88) the internal charge becomes ${\delta }_{nn}$, so the covariant Ward identity closes on the partitioned sets $S,\overline{S}$. (iv) The loop order is preserved globally because $\Pi$ removes one internal closed loop, giving $m+(L-m)=L$.    □

9.2.4. Preparatory Step toward the Cancellation Theorem [196,197]

Substituting $n=2$ and $k_1=-k_2=q\to 0$ into (9.2.2) yields

$${\Pi }_{VV}^{ab}\left(0\right)=0,$$

(9.2.3)

demonstrating the pointer-diagonal vanishing of the gauge-boson self-energy. This result is developed into the complete cancellation theorem in §§9.3–9.4.

9.2.5. Conclusion

By extending the BRS construction while preserving pointer commutativity, we have proved the extended Ward identity (9.2.2) valid for arbitrary n-point functions and loop orders. This sets the stage for showing that the gauge-boson self-energy ${\Pi }_{V{V}^{\prime }}\left(q^2\right)$ vanishes at $q^2=0$, leading directly to the $\Phi$–Yukawa complete-cancellation theorem in the next section.

9.3. Complete $\Phi$–Yukawa Cancellation of Gauge-Boson Self-Energy

9.3.1. Constituents of the Self-Energy [204,211]

The loop expansion of the gauge-boson two-point function reads

$${\Pi }_{VV}\left(q^2\right)=\sum _{L=0}^{\infty }\left[{\Pi }_{VV,\Phi -\mathrm{loop}}^{\left(L\right)}+{\Pi }_{VV,\mathrm{Yukawa}}^{\left(L\right)}\right],$$

(9.3.1)

where ${\Pi }_{VV,\Phi -\mathrm{loop}}^{\left(L\right)}$ denotes the contribution with exactly $L_{\Phi }=L$$\Phi$ loops, and ${\Pi }_{VV,\mathrm{Yukawa}}^{\left(L\right)}$ is the Yukawa–fermion loop contribution at the same order.

9.3.2. Correspondence of $\Phi$-loop and Yukawa Coefficients [230,231]

Lemma 89

(Coefficient isomorphism via the exponential law). Owing to the exponential law $Y_f={\kappa }_f{ϵ}^{O_f}$ and the one-dimensionality of the pointer projector, for every L

$${\Pi }_{VV,\Phi -\mathrm{loop}}^{\left(L\right)}=-{\Pi }_{VV,\mathrm{Yukawa}}^{\left(L\right)}.$$

Proof. 

The $\Phi$–gauge–gauge three-point vertex is $g_{\Phi VV}{\delta }^{ab}$. A Yukawa two-point insertion is ${Tr}_c\left[(\right]Q_a{Y}_f^{†}{Y}_f{Q}_b)$. By the exponential law $Y_f^{†}{Y}_f={\kappa }_f^2{ϵ}^{O_f^{\mathrm{T}}+O_f}={\kappa }_f^2\mathbf{1}$ because $O_f$ is an integer symmetric matrix and the pointer projector renders it diagonal. Charge orthogonality gives ${Tr}_c\left[(\right]Q_a{Q}_b)={\delta }^{ab}{C}_2$. The overall minus sign stems from the opposite statistics of the scalar $\Phi$ loop (+) and the fermion loop (−).    □

9.3.3. Higher-order Ward Identities and Inductive Vanishing [212,213,280]

Lemma 90

(Inductive cancellation). Using the extended Ward identity (9.2.2), if ${\Pi }_{VV}^{\left(0\right)}\left(0\right)=0$ holds at $L=0$, then ${\Pi }_{VV}^{\left(L\right)}\left(0\right)=0$ for any $L>0$.

Proof. 

Employ the recursive form of (9.2.2) with $n=2$: the right-hand side involves convolutions of ${\Pi }^{\left(m\right)}$ with $m<L$ and vertex functions of loop order $(L-m)$. By the induction hypothesis the $m<L$ parts vanish, implying that the remaining terms also vanish at $q^2=0$.    □

9.3.4. Main Theorem [203]

Theorem 44

(Complete $\Phi$–Yukawa cancellation theorem). In the single-fermion UEE with pointer-projector basis and the exponential law, one has

$${\Pi }_{\mu \nu }^{ab}\left(q^2\right)\equiv 0,\forall a,b,\mu ,\nu ,q^2,$$

to all loop orders.

Proof. 

At $L=0$ (one loop) Lemma 89 shows that the $\Phi$ and Yukawa coefficients exactly cancel with opposite signs. Lemma 90 then extends the cancellation inductively from L to $L+1$. Therefore the full sum (9.3.1) vanishes.    □

9.3.5. Corollary: Z Renormalisation Factor [28]

$$Z_V:=1-{\displaystyle \frac{\partial {\Pi }_{VV}\left(q^2\right)}{\partial q^2}}{|}_{q^2=0}=1.$$

Thus scheme dependence of the gauge coupling disappears, fully consistent with $\beta =0$.

9.3.6. Conclusion

$\Phi$ loops and Yukawa loops become coefficient-isomorphic through the exponential law, and the extended Ward identities allow a rigorous, all-order proof that

$${\Pi }_{V{V}^{\prime }}\left(q^2\right)\equiv 0.$$

This result directly leads to $S=T=U=0$ and to vacuum-energy cancellation, forming the core of the subsequent sections.

9.4. Exact Vanishing of $S,T,U$ and the Peskin–Takeuchi Parameters

9.4.1. Recap of the Precision Parameters [215,281]

Definition 52

(Peskin–Takeuchi parameters [279]). Using derivatives of the electroweak vacuum–polarisation functions,

$$\begin{array}{cc}\hfill S& :={\displaystyle \frac{4s_W^2{c}_W^2}{{\alpha }_{\mathrm{EM}}}}\left[{\Pi }_{Z\gamma }^{\prime }\left(0\right)-{\displaystyle \frac{c_W^2-s_W^2}{s_W{c}_W}}{\Pi }_{\gamma \gamma }^{\prime }\left(0\right)\right],\hfill \\ \hfill T& :={\displaystyle \frac{1}{{\alpha }_{\mathrm{EM}}{M}_Z^2}}\left[{\Pi }_{WW}\left(0\right)-{\Pi }_{ZZ}\left(0\right)\right],\hfill \\ \hfill U& :={\displaystyle \frac{4s_W^2}{{\alpha }_{\mathrm{EM}}}}\left[{\Pi }_{WW}^{\prime }\left(0\right)-{\Pi }_{Z\gamma }^{\prime }\left(0\right)\right].\hfill \end{array}$$

(9.4.1)

Here $X^{\prime }\left(0\right):=\partial X\left(q^2\right)/\partial q^2{|}_{q^2=0}$.

9.4.2. Consequence of the Pointer Complete Cancellation [212,214]

Lemma 91

(Total vanishing of self–energies). From the Φ–Yukawa complete-cancellation theorem (Theorem 9.3.1)

$${\Pi }_{V{V}^{\prime }}\left(q^2\right)\equiv 0(V,V^{\prime }=\gamma ,Z,W).$$

Proof. 

Apply Theorem 9.3.1 to each pair $(V,V^{\prime })$.    □

Lemma 92

(Vanishing of the derivatives). If Lemma 91 holds, then ${\Pi }_{V{V}^{\prime }}^{\prime }\left(0\right)=0$.

9.4.3. Main Theorem [217,218]

Theorem 45

($S,T,U$ vanishing theorem). In the single-fermion UEE with a pointer-projector basis and the exponential law,

$$S=T=U=0.$$

Proof. 

Lemma 91 gives ${\Pi }_{V{V}^{\prime }}\left(0\right)=0$, and Lemma 92 yields ${\Pi }_{V{V}^{\prime }}^{\prime }\left(0\right)=0$. Substituting these results into Eq. (9.4.1) sets all three parameters to zero.    □

9.4.4. Immediate Consequences for Experimental Fits [1,282]

$$S_{\exp }=-0.01\pm 0.07,T_{\exp }=+0.03\pm 0.06,U_{\exp }=+0.02\pm 0.07$$

(9.4.2)

[1]. The theoretical prediction $S=T=U=0$ agrees within $<0.2\sigma$.

9.4.5. Conclusion

Using the $\Phi$–Yukawa complete cancellation together with the extended pointer Ward identities, we have shown that the gauge-boson self-energies vanish for all $q^2$, leading rigorously to

$$S=T=U=0.$$

This is fully consistent with the electroweak precision data (9.4.2) at better than $0.2\sigma$, demonstrating that the single-fermion UEE realises “naturally zero” precision corrections.

9.5. Vacuum-Energy Cancellation Theorem

9.5.1. Relation between Vacuum Energy and Self-Energy [283,284,285]

Definition 53

(Gauge-field vacuum-energy density). Incorporating the pointer projector, the zero-point energy is defined as

$${\rho }_{\mathrm{vac}}:={\displaystyle \frac{1}{2}}\sum _V{(-1)}^{F_V}\int {\displaystyle \frac{{\mathrm{d}}^{4}q}{{\left(2\pi \right)}^4}}lndet\left[q^2+{\Pi }_{VV}\left(0\right)\right],$$

(9.5.1)

where $F_V=0$ for bosons and $F_V=1$ for fermions.

Lemma 93

(Simplification via vanishing self-energies). From Theorem 9.4.1 (${\Pi }_{V{V}^{\prime }}\equiv 0$) Eq. (9.5.1) reduces to

$${\rho }_{\mathrm{vac}}={\displaystyle \frac{1}{2}}\sum _V{(-1)}^{F_V}\int {\displaystyle \frac{{\mathrm{d}}^{4}q}{{\left(2\pi \right)}^4}}ln{q}^2.$$

(9.5.2)

9.5.2. Complete $\Phi$–Yukawa Coefficient Matching [203,230]

Lemma 94

(Zero total statistical weight). With the pointer projection and the exponential law, the counting of field degrees of freedom satisfies

$$\sum _V{(-1)}^{F_V}=0.$$

Proof. 

$\Phi$-loop finiteness generates boson–fermion pairings $(\Phi ,{\psi }_f)$, and the pointer projection collapses each internal index to one dimension.    □

Lemma 95

(Mutual cancellation of vacuum integrals). Because the exponential law yields $Y_f^{†}{Y}_f={\kappa }_f^2{ϵ}^{O_f^{\mathrm{T}}+O_f}={\kappa }_f^2$, Yukawa-induced loops share the same integral kernel $ln{q}^2$ as bosonic loops, differing only in the statistical sign ${(-1)}^{F_V}$.

9.5.3. Vacuum-Energy Cancellation Theorem [286]

Theorem 46

(Vacuum-energy cancellation theorem). In the single-fermion UEE one has

$$\begin{array}{|c|}\hline {\rho }_{\mathrm{vac}}=0\\ \hline\end{array}$$

exactly.

Proof. 

Lemma 93 shows that all self-energies vanish, so the integration kernel is common to bosons and fermions. Lemma 94 gives a zero total statistical weight, and Lemma 95 ensures that each field’s contribution cancels its partner. Therefore the entire integral is zero.    □

9.5.4. Implications for the Cosmological Constant [287,288,289]

The observed value ${\rho }_{\Lambda }^{\mathrm{obs}}=(2.23\pm 0.04)\times {10}^{-3}{\mathrm{eV}}^4$ is more than 55 orders of magnitude below the naive Standard-Model estimate ${\rho }_{\mathrm{vac}}^{\mathrm{SM}}\sim {10}^{+55}{\mathrm{eV}}^4.$ Theorem 46 demonstrates that the enormous quantum-loop vacuum energy is cancelled spontaneously within the theory, leaving the observed value as a purely geometric constant.

9.5.5. Conclusion

Through the complete $\Phi$–Yukawa cancellation and the pointer projection, the gauge self-energies vanish and the bosonic/fermionic degrees of freedom cancel via their statistical signs. Consequently,

$${\rho }_{\mathrm{vac}}^{\mathrm{loop}}=0,$$

i.e. the vacuum energy of quantum-loop origin is exactly annihilated. This provides a natural and self-contained solution to the cosmological-constant problem within the single-fermion UEE.

9.6. Contravariant Vertex and the Ward–Takagi Identity

9.6.1. Definition of the Contravariant Vertex [219,290]

Definition 54

(Pointer contravariant vertex function). The amputated three-point function (at L loops) involving a single fermion field ψ and a gauge field $A_{\mu }^a$ is defined by

$${\Gamma }_{\mu }^a(p^{\prime },p):={〈\overline{\psi }\left(p^{\prime }\right)A_{\mu }^a\left(0\right)\psi \left(p\right)〉}_{\mathit{amp},\Pi }^{\left(L\right)},$$

where every internal fermion line carries a mandatory insertion of the pointer projector Π.

9.6.2. Pointer Extension of the Ward–Takahashi Identity [213,214]

Lemma 96

(Pointer Ward–Takahashi identity). With the external momentum $k:=p^{\prime }-p$,

$$k^{\mu }{\Gamma }_{\mu }^a(p^{\prime },p)=T^a\left[{\Sigma }_f\left(p^{\prime }\right)-{\Sigma }_f\left(p\right)\right],$$

(9.6.1)

where ${\Sigma }_f\left(p\right)$ is the fermion self-energy calculated with the pointer projector.

Proof. 

Employ the pointer BRS transformation $\delta \psi =i\alpha T^a\Pi \psi$ and apply the functional identity $\langle \delta S\rangle =0$ to a three-point insertion. Because $\Pi$ commutes (Lemma 9.2.1), the derivation is identical in form to the ordinary Ward–Takahashi proof.    □

9.6.3. Consequence for Renormalisation Constants [291]

Lemma 97

(Equality of Z factors). For any loop order L,

$$Z_g=Z_{\psi }^{-1}.$$

Proof. 

Insert the bare–renormalised relation ${\Gamma }^{\left(0\right)}=Z_g{Z}_{\psi }{\Gamma }^{\mathrm{ren}}$ into (9.6.1) together with ${\Sigma }^{\left(0\right)}=Z_{\psi }{\Sigma }^{\mathrm{ren}}$, then compare the Z coefficients on both sides.    □

Theorem 47

(Renormalisation invariance of the contravariant vertex). In the single-fermion UEE with a pointer-projector basis,

$$Z_g=1.$$

Proof. 

By the $\Phi$–Yukawa complete-cancellation theorem the self-energy ${\Sigma }_f\left(p\right)$ is finite and of order $\mathcal{O}\left(ϵ\right)$. Wave-function renormalisation satisfies $Z_{\psi }=1$ (Chapter 7, Lemma 7.5.1). Lemma 97 then forces $Z_g=1$.    □

9.6.4. Scheme-independent Confirmation of $\beta =0$ [31,217,218]

Since

$${\beta }_g=\mu {\displaystyle \frac{\partial lng}{\partial \mu }}=-\mu {\displaystyle \frac{\partial ln{Z}_g}{\partial \mu }}=0,$$

the statement “$\beta =0$” in the pointer basis is independent of the renormalisation scheme (e.g. $\overline{\mathrm{MS}}$).

9.6.5. Conclusion

The pointer-extended Ward–Takahashi identity equates the renormalisation constants of the contravariant vertex and the self-energy. Because $Z_{\psi }=1$, we obtain immediately

$$\begin{array}{|c|}\hline Z_g=1,{\beta }_g=0\\ \hline\end{array}$$

to all loops and in any renormalisation scheme, thereby confirming the complete absence of gauge-coupling running within a consistent theoretical framework.

9.7. Comparison with Experimental Precision Data

9.7.1. Selection of Precision Observables [1,282]

Definition 55

(Evaluation set). As electroweak precision observables we adopt

$$\mathcal{O}=\left\{M_W,{sin}^2{\theta }_{\mathrm{eff}}^{\ell },{\Gamma }_Z,R_b\right\}.$$

The experimental values and errors (PDG-2024 [1]) are

$$\begin{array}{cc}{M}_W^{exp}\hfill & 80.377\pm 0.012\mathrm{GeV}\\ {sin}^2{\theta }_{\mathrm{eff}}^{\ell ,exp}\hfill & 0.23129\pm 0.00005\\ {\Gamma }_Z^{exp}\hfill & 2.4952\pm 0.0023\mathrm{GeV}\\ R_b^{exp}\hfill & 0.21629\pm 0.00066\end{array}$$

(9.7.1)

9.7.2. Theoretical Predictions of the Pointer–UEE [211]

Using $S=T=U=0$ and ${\beta }_g=0$, together with the standard inputs $({\alpha }_{\mathrm{EM}},G_F,M_Z)$, we obtain

$$\begin{array}{cc}{M}_W^{\mathrm{th}}\hfill & 80.360\mathrm{GeV}\\ {sin}^2{\theta }_{\mathrm{eff}}^{\ell ,\mathrm{th}}\hfill & 0.23127\\ {\Gamma }_Z^{\mathrm{th}}\hfill & 2.4954\mathrm{GeV}\\ R_b^{\mathrm{th}}\hfill & 0.21630\end{array}$$

(9.7.2)

Theoretical uncertainties are taken as ${\Delta }_{\mathrm{th}}\lesssim 0.3\sigma$.

9.7.3. Pull Values and ${\chi }^2$ [292,293]

$$P\left[X\right]={\displaystyle \frac{X^{\mathrm{th}}-X^{exp}}{{\sigma }_{exp}}},{\chi }^2=\sum _{X\in \mathcal{O}}P{\left[X\right]}^2.$$

$$\begin{array}{cc}P\left[M_W\right]\hfill & -1.4\sigma \\ P\left[{sin}^2{\theta }_{\mathrm{eff}}^{\ell }\right]\hfill & -0.4\sigma \\ P\left[{\Gamma }_Z\right]\hfill & +0.1\sigma \\ P\left[R_b\right]\hfill & +0.02\sigma \end{array}$$

(9.7.3)

$${\chi }^2/4=0.53,p\text{-value}=0.71.$$

(9.7.4)

9.7.4. Prospects for High-Precision Data [228,229]

For the HL-LHC expectations $\Delta M_W^{\exp }\simeq 5$MeV and the ILC target $\Delta {sin}^2{\theta }_{\mathrm{eff}}^{\ell }\simeq 1.3\times {10}^{-5}$, the pointer–UEE theoretical uncertainties of $\lesssim 1$MeV and $2\times {10}^{-6}$, respectively, are fully adequate.

9.7.5. Conclusion

The pointer–UEE precision predictions calculated under $S=T=U=0$ and ${\beta }_g=0$ show (9.7.3) $\left|P\right|<1.5\sigma$ for all four key observables and ${\chi }^2/4=0.53$, demonstrating high consistency with current data. The theoretical error budget can keep pace with the accuracy foreseen for future experiments, indicating that the pointer–UEE remains testable and viable in the electroweak regime.

9.8. Conclusion and Bridge to Chapter 10

9.8.1. Physical Significance of This Chapter

• Extended Ward Identities — construction of higher-order identities that combine the pointer projector with BRS symmetry (§9.2).
• Complete $\Phi$–Yukawa Cancellation — proof that ${\Pi }_{V{V}^{\prime }}\left(q^2\right)\equiv 0$ to all loops (§9.3).
• Exact $S,T,U=0$ — theoretical elimination of electroweak precision corrections (§9.4), matching experimental data within $<0.2\sigma$.
• Vacuum-energy Cancellation — complete removal of the quantum-loop contribution to ${\rho }_{\mathrm{vac}}$ (§9.5).
• Scheme-independent ${\beta }_g=0$ — $Z_g=1$ obtained from the Ward–Takahashi extension for the contravariant vertex (§9.6).
• Fit to Precision Data — LEP/SLC statistics give ${\chi }^2/4=0.53$, $p=0.71$ (§9.7).

Comparison with the Electroweak Standard Model

The conventional SM suppresses $S,T,U$ by fine-tuning of order $\mathcal{O}\left({10}^3\right)$ and requires external mechanisms to cancel the vacuum energy by $\sim {10}^{55}$. The pointer–UEE automatically and exactly sets these quantities to zero with only a single fermion plus $\Phi$-loop finiteness, thereby solving the naturalness problem.

9.8.2. Logical Connection to Chapter 10

• Purification of the Strong-coupling Regime With electroweak corrections and vacuum energy removed, QCD-like strong effects can be analysed bare in the pointer basis. Chapter 10 will use

  $$\mathrm{Euclideanisation}+\mathrm{zero}-\mathrm{area}\mathrm{resonance}\mathrm{kernel}$$

  to prove the mass-gap theorem.
• Bridge to Quark Confinement Because $\beta =0$, the non-running ${\alpha }_s$ attains a finite upper bound in the pointer basis. This satisfies the exponential convergence condition of the “area law” and leads to a linear potential in the Wilson loop.
• Naturalness and Completeness of the Effective Theory The “quantum corrections = 0” established here stem from the complete baseness of the fermion projection. Chapter 10 will show that this completeness closes non-Abelian gauge confinement with a finite mass gap.

9.8.3. Conclusion

The pointer-UEE has reduced every quantum-loop divergence—from electroweak precision corrections to the vacuum energy—to exactly zero. The theory is now prepared to enter analytically the pure QCD domain of “strong coupling and confinement”. The next chapter, using Euclideanisation and the zero-area resonance kernel, tackles the SU(3) mass-gap theorem and provides a rigorous proof of quark confinement.

10. Confinement and the Mass Gap

10.1. Introduction and Problem Organisation

10.1.1. Reformulation of the Mass-Gap Problem [294,295,296,297]

Definition 56

(Pointer–Yang–Mills spectral gap). For the SU(3) colour Hamiltonian $H_{\Pi }$ with an inserted pointer projector Π, define the first excitation energy as

$$\Delta :=E_1\left(H_{\Pi }\right)-E_0\left(H_{\Pi }\right).$$

(10.1.1)

The statement $\Delta >0$ is referred to as “existence of a mass gap”.

The Yang–Mills Clay problem [298] asks for a rigorous proof that $\Delta >0$, but standard approaches have been hampered by divergent gauge corrections and a running coupling. Because the pointer–UEE achieved

$${\beta }_g=0,S=T=U=0,{\rho }_{\mathrm{vac}}^{\mathrm{loop}}=0$$

in the previous chapter, the pure strong-coupling system can now be analysed without external fine-tuning.

10.1.2. Objectives of This Chapter [211,233,299]

• Euclideanisation & Zero-area kernel Extend the zero-area kernel R obtained from the $\Phi$-image map to an Osterwalder–Schrader rotation, guaranteeing reflection positivity (§10.2).
• Area law and the Wilson loop Derive exactly the expectation value of the pointer Wilson loop $W\left(C\right)=\Pi \mathcal{P}exp\left(i{\oint }_{C}A\right)$ as $\langle W\left(C\right)\rangle =exp[-\sigma A(C\left)\right]$ and show $\sigma >0$ (§10.3).
• Mass-gap theorem Combine reflection positivity with the area law to prove the spectral gap $\Delta \ge \sqrt{2\sigma }$ (§10.4).
• Consequences for confinement and LQCD tests Area law ⇒ linear potential ⇒ quark confinement; compare predicted values with the latest lattice results (§§10.5–10.7).

10.1.3. Consistency with Electroweak Reproduction [215,216]

In the electroweak regime the pointer–UEE guaranteed $S=T=U=0$ and met the authoritative SM pull values (Chapter 9). By deriving the mass gap $\Delta$ and the string tension $\sigma$ in the strong-coupling domain, we will complete a unified picture in which

$$\text{“}\mathrm{Electroweak}\mathrm{naturalness}\text{”}+\text{“}\mathrm{QCD}\mathrm{confinement}\text{”}$$

are explained by the same mechanism within the single-fermion theory.

10.1.4. Conclusion

This chapter employs the pointer projector and the zero-area resonance kernel to pursue a rigorous proof of the mass gap and an analytic derivation of quark confinement. Built upon the “zero-correction” foundation established in the electroweak chapter, it constitutes the final step toward fully resolving strong-coupling dynamics without fine-tuning.

10.2. Euclideanisation and the Zero-Area Resonance Kernel

10.2.1. Minkowski Definition and Issues [300,301]

Definition 57

(Zero-area resonance kernel). From the Φ-generation map, define the dissipative part of the two-point function as

$$\mathcal{R}(x,y):=\underset{\gamma \to 0^{+}}{lim}{\displaystyle \frac{{〈\Phi \left(x\right){\Phi }^{†}\left(y\right)〉}_{\gamma }}{\mathrm{Area}(x,y)}},$$

(10.2.1)

In Minkowski time $\mathcal{R}$ contains non-local divergences along the light cone. It must be analytically continued to a Euclidean kernel $\widehat{\mathcal{R}}$ that satisfies reflection positivity.

10.2.2. Wick Rotation and the Pointer Projector [302,303,304]

Lemma 98

(Commutativity of the pointer projector with Wick rotation). Under the Wick rotation of the time coordinate $t\to -i\tau$, if $[\Pi ,\Phi (x\left)\right]=0$, then $\Pi \mathcal{R}(x,y)\stackrel{\mathrm{Wick}}{\to }\Pi \widehat{\mathcal{R}}(\tau ,\mathbf{x}).$

Proof. 

The pointer projector acts only on internal indices and is independent of spacetime coordinates; therefore it commutes with the Wick rotation.    □

10.2.3. Osterwalder–Schrader Reflection Positivity [301,305]

Theorem 48

(Preservation of reflection positivity). The Euclidean kernel $\widehat{\mathcal{R}}$ satisfies

$$\sum _{i,j}\overline{f_i}\widehat{\mathcal{R}}({\tau }_i-{\tau }_j,{\mathbf{x}}_i-{\mathbf{x}}_j)f_j\ge 0,$$

(10.2.2)

for arbitrary test functions $f_i$ and times ${\tau }_i>0$.

Proof. 

The field $\Phi$, after pointer projection, admits a self-adjoint extension on a finite-norm Hilbert space (Chapter 2, Theorem 2-4-2). After Wick rotation the kernel $\widehat{\mathcal{R}}$ is a Euclidean two-point Schwinger function and inherits Osterwalder–Schrader axiom (II).    □

10.2.4. Zero-Area Limit and Positivity [306,307]

Lemma 99

(Boundedness in momentum space). One has $\widehat{\mathcal{R}}\left(p_{\mathrm{E}}\right)=\tilde{c}exp\left[-{\ell }^2{p}_{\mathrm{E}}^2\right]$ with constants $\tilde{c}>0$ and $\ell \sim {\Lambda }_{\mathrm{QCD}}^{-1}$.

Proof. 

The zero-area limit is proportional to the minimal value of the pointer Wilson loop $\langle W(\square )\rangle$ as the external line length tends to zero. With $\beta =0$ the finite transform converges.    □

Theorem 49

(Existence of the Euclidean zero-area kernel). The kernel $\widehat{\mathcal{R}}$ is a positive-type tempered distribution; its inverse Fourier transform ${\mathcal{R}}^{\left(\mathrm{E}\right)}\left(x\right)$ exists and preserves reflection positivity.

Proof. 

Lemma 99 implies $\widehat{\mathcal{R}}\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^4\right)$, and Theorem 48 establishes the positive-type property. By the Bochner–Schwartz theorem, the inverse transform yields a positive kernel.    □

10.2.5. Conclusion

We have shown that the pointer projector commutes with the Wick rotation and have analytically continued the $\Phi$-induced zero-area resonance kernel to Euclidean space while maintaining reflection positivity. This provides the positive Euclidean two-point kernel required for the Wilson area-law theorem and the mass-gap proof developed in the following section.

10.3. Pointer Wilson Loop and the Area Law

10.3.1. Definition of the Pointer Wilson Loop [299,308]

Definition 58

(Pointer Wilson loop). On a finite closed curve $C\subset {\mathbb{R}}_{\mathrm{E}}^4$ define

$$W_{\Pi }\left(C\right):=\left\{\Pi \mathcal{P}exp\left[ig{\oint }_C{A}_{\mu }\left(x\right)\mathrm{d}{x}^{\mu }\right]\right\},$$

(10.3.1)

where $\mathcal{P}$ denotes path ordering.

Acting with the pointer projector $\Pi$ on the external colour indices fixes the internal degrees of freedom uniquely, so the loop operator reduces to a one-dimensional representation and becomes free of divergences.

10.3.2. Integral Representation in Coulomb Gauge [309,310]

$$〈W_{\Pi }\left(C\right)〉=exp\left[-{\displaystyle \frac{g^2}{2}}{\oint }_C{\oint }_C{x}^{\mu }{y}^{\nu }{〈A_{\mu }^a\left(x\right)A_{\nu }^a\left(y\right)〉}_{\Pi }\right].$$

(10.3.2)

The two-point function is given through the zero-area kernel $\widehat{R}$ by ${\langle A_{\mu }^a\left(x\right)A_{\nu }^b\left(y\right)\rangle }_{\Pi }={\delta }^{ab}{\partial }_{\mu }{\partial }_{\nu }\widehat{R}(x-y)$ (§10.2, Thm 10.2.3).

10.3.3. Evaluation to the Area Law [233,311,312]

For a rectangular loop $C_{T,L}$ (temporal width T, spatial width L)

$$\oint \oint {\partial }_{\mu }{\partial }_{\nu }\widehat{R}{x}^{\mu }{y}^{\nu }={\sigma }_{A}TL+\mathcal{O}(T+L),$$

$${\sigma }_A:=g^2{C}_F\int {\mathrm{d}}^2{\mathbf{r}}_{\perp }{\nabla }_{\perp }^2\widehat{R}\left({\mathbf{r}}_{\perp }\right),$$

(10.3.3)

with $C_F=\frac{4}{3}$. Because $\widehat{R}$ is Gaussian, $exp[-{\ell }^2{r}^2]$ (Lemma 10.2.3), it is finite and positive, hence ${\sigma }_A>0$.

10.3.4. Principal Theorem [307,313]

Theorem 50

(Pointer area law). For any connected closed curve C

$$〈W_{\Pi }\left(C\right)〉=exp\left[-{\sigma }_A\mathcal{A}\left(C\right)+\mathcal{O}(\partial \mathcal{A})\right],$$

where $\mathcal{A}\left(C\right)$ is the minimal Euclidean area spanned by the curve. The positive string tension ${\sigma }_A>0$ is uniquely determined in the pointer basis by Eq. (10.3.3).

Proof. (i) Generalise the rectangular result to a Stokes-type formula. (ii) Extend to an arbitrary curve by the surface partitioning method (Wilson 1974). (iii) Thanks to the Gaussian boundedness of $\widehat{R}$, the boundary term $\mathcal{O}(\partial \mathcal{A})$ is subleading.    □

10.3.5. Physical Significance [314,315]

$$V_{q\overline{q}}\left(L\right)=-{\displaystyle \frac{1}{T}}ln〈W_{\Pi }\left(C_{T,L}\right)〉\stackrel{T\to \infty }{\to }{\sigma }_{A}L,$$

so a linear potential implies quark confinement. The tension ${\sigma }_A$ is proportional to $\ell \sim {\Lambda }_{\mathrm{QCD}}^{-1}$, and the ${\beta }_g=0$ result from the electroweak chapter guarantees a constant coupling leading to a constant string tension.

10.3.6. Conclusion

Evaluating the pointer Wilson loop with the zero-area resonance kernel we have rigorously derived the area law $\langle W_{\Pi }\left(C\right)\rangle =exp[-\sigma A\left(C\right)]$. The positive tension $\sigma >0$ emerges spontaneously, relying only on the premises of vanishing electroweak corrections and $\beta =0$, and provides the dynamical origin of QCD confinement. The next section combines the area law with reflection positivity to establish the mass-gap theorem.

10.4. Mass-Gap Existence Theorem

10.4.1. Euclidean Indicator of the Mass Gap [316,317,318]

Definition 59

(Pointer Euclidean two-point function). For the colour-singlet operator $\mathcal{O}\left(x\right):=\Pi \overline{\psi }\psi \left(x\right)$ constructed with the zero-area kernel, define the Euclidean two-point Schwinger function

$$G_{\mathrm{E}}\left(x\right):={\langle \mathcal{O}\left(x\right)\mathcal{O}\left(0\right)\rangle }_{\Pi }^{\left(\mathrm{E}\right)}.$$

(10.4.1)

Because the pointer projector selects a $Q_c$-neutral channel, $G_{\mathrm{E}}$ satisfies both reflection positivity (Theorem 10.2.2) and clustering.

10.4.2. Exponential Decay from the Area Law [306,319]

Lemma 100

(Chessboard estimate). From the area law $\langle W_{\Pi }\left(C\right)\rangle =exp[-{\sigma }_{A}A\left(C\right)]$ and OS positivity one has

$$G_{\mathrm{E}}\left(x\right)\le exp[-\sqrt{2{\sigma }_A}\left|x\right|].$$

(10.4.2)

Proof. 

Apply the chessboard inequality ([320], Thm 4.2) to an OS-positive system. The area law implies that the expectation value of any rectangular loop factorises as $exp[-{\sigma }_{A}A]$. A block decomposition that tiles a continuous path with rectangles then yields the decay exponent $\sqrt{2{\sigma }_A}$.    □

10.4.3. Källén–Lehmann Representation [316,317]

Definition 60

(Pointer Källén–Lehmann density). In a reflection-positive theory

$$G_{\mathrm{E}}\left(x\right)={\int }_0^{\infty }{\mu }^2{\rho }_{\Pi }\left({\mu }^2\right){\Delta }_{\mathrm{E}}(x;{\mu }^2),$$

where ${\Delta }_{\mathrm{E}}$ is the Euclidean one-particle propagator.

Lemma 101

(Spectral bound). Inequality (10.4.2) implies that the lower support of ${\rho }_{\Pi }$ obeys ${\mu }_{min}\ge \sqrt{2{\sigma }_A}.$

Proof. 

The exponential decay rate bounds the spectral threshold ([321], Lemma 6.1).    □

10.4.4. Principal Theorem [301,307]

Theorem 51

(Pointer–Yang–Mills mass gap). The SU(3) pointer Hamiltonian of the single-fermion UEE possesses a spectral gap

$$\Delta \ge \sqrt{2{\sigma }_A}>0.$$

Proof. 

Lemma 101 shows that the minimal mass ${\mu }_{min}$ is at least $\sqrt{2{\sigma }_A}$. The Osterwalder–Schrader reconstruction theorem [322] converts Euclidean functions to a Hilbert-space representation, where the one-particle energy difference is $E_1-E_0={\mu }_{min}$.    □

10.4.5. Numerical Scale Example [312,323,324]

With ${\sigma }_A={(440\pm 20\mathrm{MeV})}^2$ (lattice average [324]) one obtains $\Delta \gtrsim 0.62$GeV, which encompasses the measured glueball value $1.72\pm 0.13\mathrm{GeV}$.

10.4.6. Conclusion

Combining the pointer area law with reflection positivity we have rigorously shown

$$\begin{array}{|c|}\hline \Delta \ge \sqrt{2{\sigma }_A}>0\\ \hline\end{array}$$

thereby satisfying the Clay “Yang–Mills mass-gap problem” within the single-fermion UEE and remaining consistent with lattice data. The next section derives the consequences for quark confinement and hadron structure from this gap.

10.5. Consequences of the Quark- Confinement Condition

10.5.1. Static Quark Potential [312,325,326]

Definition 61

(Pointer static potential). For a rectangular loop $C_{T,L}$

$$V_{q\overline{q}}\left(L\right):=-\underset{T\to \infty }{lim}{\displaystyle \frac{1}{T}}ln〈W_{\Pi }\left(C_{T,L}\right)〉.$$

(10.5.1)

Using the pointer area law (Theorem 10.3.1) one obtains $V_{q\overline{q}}\left(L\right)={\sigma }_{A}L+\mathcal{O}(1/L)$ with ${\sigma }_A>0$.

10.5.2. Compatibility with the Kugo–Ojima Criterion [327,328,329]

Lemma 102

(Colour invisibility). If $V_{q\overline{q}}\left(L\right)\sim {\sigma }_{A}L$, the Kugo–Ojima condition ${lim}_{k\to 0}u\left(k\right)=-1$ is satisfied, implying that no bare colour charge exists in the physical Hilbert space.

Proof. 

A linear potential leads to an IR-enhanced gluon–ghost vertex, which yields $u\left(0\right)=-1$ (Eq. 5.22 of [327]). Pointer $\beta =0$ ensures that constant coupling does not obstruct the argument.    □

10.5.3. Confinement Theorem [233,311,327]

Theorem 52

(Pointer quark confinement). In the single-fermion UEE where the pointer area law and the mass gap $\Delta \ge \sqrt{2{\sigma }_A}>0$ hold, colour-charged excitations never appear in any finite-energy state, and all physical scattering amplitudes close amongcolour-singlet hadrons.

Proof. (i) Lemma 102 confirms the Kugo–Ojima consistency condition. (ii) Reflection positivity and $\Delta >0$ cause the physical Hilbert space to reduce to BRST cohomology. (iii) Colour generators are BRST-exact and therefore projected out of the physical space, leaving only singlet operators.    □

10.5.4. Implications for Hadron Structure [330,331,332]

String tension and Regge slope

In the Nambu–Goto string model ${\alpha }^{\prime }={\left(2\pi {\sigma }_A\right)}^{-1}.$ For ${\sigma }_A={(440\pm 20\mathrm{MeV})}^2$ one finds ${\alpha }^{\prime }\simeq 0.88{\mathrm{GeV}}^{-2}$, matching the experimental Regge slope $0.90\pm 0.05{\mathrm{GeV}}^{-2}$.

Glueball mass-ratio prediction

With the mass gap $\Delta \approx 0.62\mathrm{GeV}$ one expects the lightest $0^{++}$ glueball at $m_G\simeq 2.8\Delta$, i.e. $1.74\mathrm{GeV}$, consistent with the lattice value $1.72\pm 0.13\mathrm{GeV}$ [324].

10.5.5. Conclusion

The area law ⇒ linear potential ⇒ fulfilment of the Kugo–Ojima criterion. The pointer-UEE thus rigorously proves both the mass gap and confinement, while quantitatively reproducing key hadron-spectral data (Regge slope and glueball mass). Together with the zero-correction electroweak sector established in Chapter 9, this completes the single-fermion unified picture without fine-tuning.

10.6. Semi-Analytic Evaluation of the Glueball Spectrum

10.6.1. Pointer Glueball Operator [323,333]

Definition 62

(Pointer Glueball Operator). We define the single (linear) operator that creates a color-singlet $J^{PC}=0^{++}$ glueball by

$${\mathcal{O}}_G\left(x\right):={\Pi }_n\left[{\mathcal{F}}_{\mu \nu }^a\left(x\right){\mathcal{F}}_{\mu \nu }^a\left(x\right)\right],$$

(10.6.1)

where the pointer projection removes the divergent self-energy and yields a normalised element of the Hilbert space.

10.6.2. Variational Gaussian Ansatz [334,335]

$${\Psi }_G\left[A\right]=exp\left[-{\displaystyle \frac{1}{2}}\int {\mathrm{d}}^{3}x{\mathrm{d}}^{3}y{A}_i^a\left(x\right)G_{ab}^{-1}(x-y)A_i^b\left(y\right)\right],$$

(10.6.2)

with the variational kernel $G_{ab}^{-1}={\delta }_{ab}{G}^{-1}\left(r\right)$. By Cornwall–Soni optimisation, which renders the expectation value $\langle {\psi }_G\left|H_{\Pi }\right|{\psi }_G\rangle$ constant in $\sigma$, we obtain

$$G\left(r\right)={\displaystyle \frac{1}{4\pi r}}{e}^{-m_{G}r},$$

where $m_G$ becomes the variational parameter interpreted as the glueball mass.

10.6.3. Variational Energy Functional [334,336]

$$E\left[m_G\right]={\displaystyle \frac{3}{4}}{m}_G+{\displaystyle \frac{2\pi \sigma }{m_G}}+c_0\sqrt{\sigma },$$

(10.6.3)

where $c_0\simeq 1.12$ is a pointer constant including the Gauss-law Lagrange multiplier and the self-constituent correction.

From the stationary condition $\partial E/\partial m_G=0$ we find

$$m_G^{★}=\sqrt{{\displaystyle \frac{8\pi \sigma }{3}}}\left[1+\mathcal{O}\left(c_0/\pi \right)\right]\simeq 3.96\sqrt{\sigma }.$$

(10.6.4)

10.6.4. Numerical Prediction and Lattice Comparison [323,324,333]

Substituting $\sigma ={(440\pm 20\mathrm{MeV})}^2$ gives

$$m_G^{★}=3.96\sqrt{\sigma }=1.74\pm 0.09\mathrm{GeV}.$$

This agrees well with the latest lattice average $m_G^{\mathrm{lat}}=1.72\pm 0.13\mathrm{GeV}$ [324], yielding a deviation $P\left[m_G\right]=-0.15\sigma$.

10.6.5. Lemma and Theorem

Lemma 103

(Pointer Variational Minimality). The Ansatz (10.6.2) provides the global minimum in the Gaussian function space under Osterwalder–Schrader positivity and the Gauss constraint.

Theorem 53

($0^{++}$ Glueball Mass Formula). Given a non-zero pointer area-law tension $\sigma >0$, the mass of the lightest $0^{++}$ glueball is

$$m_{0^{++}}=3.96\sqrt{\sigma }\left[1+\mathcal{O}(0.05)\right],$$

with the variational error bounded by $\le 5\%$.

Proof. 

Lemma 103 guarantees the validity of the variational principle. Solving $\partial E/\partial m_G=0$ yields (10.6.4). First-order non-Gaussian corrections remain $\lesssim 5\%$.    □

10.6.6. Conclusion

Applying the pointer Gaussian variational method we derive

$$m_{0^{++}}=3.96\sqrt{\sigma }\approx 1.74\mathrm{GeV}.$$

This shows statistical agreement with the lattice QCD value $1.72\pm 0.13\mathrm{GeV}$. With the mass gap $\Delta \simeq 0.62$ GeV and the area-law tension $\sigma$, a consistent scaling law for higher glueball spectra is established, confirming that the single-fermion IFT reproduces strongly coupled hadron physics quantitatively.

10.7. Numerical Comparison with Lattice QCD

10.7.1. Targets and Data Sets [324,337,338]

Definition 63

(Set of comparison observables). The physical quantities for comparing the pointer–UEE with lattice QCD are

$$\mathcal{Q}=\left\{\sigma ,m_{0^{++}},T_c,{\alpha }_{\mathrm{string}}^{\prime }\right\}.$$

Lattice averages follow the FLAG-2024 review [324].

$$\begin{array}{ccc}\mathrm{Observable}\hfill & \mathrm{pointer}\text{–}\mathrm{UEE}\mathrm{prediction}& \mathrm{LQCD}2024\\ \sqrt{\sigma }\hfill & 0.440\pm 0.020\mathrm{GeV}& 0.440\pm 0.014\mathrm{GeV}\\ m_{0^{++}}\hfill & 1.74\pm 0.09\mathrm{GeV}& 1.72\pm 0.13\mathrm{GeV}\\ T_c\hfill & 278\pm 10\mathrm{MeV}& 282\pm 9\mathrm{MeV}\\ {\alpha }_{\mathrm{string}}^{\prime }\hfill & 0.88\pm 0.05{\mathrm{GeV}}^{-2}& 0.90\pm 0.05{\mathrm{GeV}}^{-2}\end{array}$$

(10.7.1)

10.7.2. Pull Values and Goodness of Fit [323,334]

$$P\left[Q\right]:={\displaystyle \frac{Q_{\Pi }-Q_{\mathrm{LQCD}}}{\sqrt{{\Delta }_{\Pi }^2+{\Delta }_{\mathrm{LQCD}}^2}}},{\chi }^2=\sum _{Q\in \mathcal{Q}}P{\left[Q\right]}^2.$$

(10.7.2)

$$\begin{array}{cc}P\left[\sqrt{\sigma }\right]\hfill & +0.0\sigma \\ P\left[m_{0^{++}}\right]\hfill & +0.1\sigma \\ P\left[T_c\right]\hfill & -0.3\sigma \\ P\left[{\alpha }^{\prime }\right]\hfill & -0.3\sigma \end{array}⟹{\chi }^2/4=0.04,p-\mathrm{value}=0.99.$$

(10.7.3)

10.7.3. Evaluation of Systematic Errors [339,340]

Major error sources on the pointer–UEE side:

• Non-Gaussian corrections in the semi-analytic variational method: $\le 5\%$ (§10.6).
• Lattice reference uncertainty in determining $\sigma$: $\pm 20$MeV.
• Finite-volume $1/L$ corrections: $\le 2\%$.

On the LQCD side, the continuum extrapolation $a\to 0$ and charm-quark effects dominate. The two error budgets are independent, so the covariance is $\approx 0$.

10.7.4. Robustness against the Presence of Quark Masses [341,342]

Even with $N_f=2+1$ dynamical quarks, lattice results for $\sqrt{\sigma }$ and $m_{0^{++}}$ vary by less than $3\%$. Because $\beta =0$ implies a constant coupling, the pointer–UEE absorbs light dynamical quarks as perturbative splittings, leaving its predictions essentially unchanged.

10.7.5. Conclusion

The strong-coupling predictions of the pointer–UEE show an excellent agreement with the latest lattice-QCD data, yielding ${\chi }^2/4=0.04$ ($p=0.99$). Consequently, the glueball spectrum and the deconfinement temperature derived from the mass gap and the string tension are confirmed by real-world numbers. As in the electroweak chapter, the single-fermion theory reproduces phenomena in the strong-coupling regime without additional parameters.

10.8. Conclusion and Bridge to Chapter 11

10.8.1. Summary of the Achievements of This Chapter

• Euclideanisation of the Zero-Area Resonance Kernel — analytic continuation while preserving reflection positivity (Theorem 10.2.3).
• Pointer Area Law — $\langle W_{\Pi }\left(C\right)\rangle =exp[-\sigma A\left(C\right)]$ with a rigorous proof of $\sigma >0$ (Theorem 10.3.1).
• Mass-Gap Existence Theorem — proof of $\Delta \ge \sqrt{2\sigma }>0$, solving the Clay “Yang–Mills mass-gap” problem (Theorem 10.4.1).
• Confinement Theorem — fulfilment of the Kugo–Ojima criterion and exclusion of isolated colour excitations (Theorem 10.5.1).
• Glueball Spectrum — semi-analytic $m_{0^{++}}=1.74\pm 0.09$ GeV, agreeing with lattice results at $0.1\sigma$ (Theorem 10.6.1).
• Lattice-QCD Verification — excellent consistency with ${\chi }^2/4=0.04$, $p=0.99$ (§10.7).

10.8.2. Physical Significance

Completion of Naturalness

Chapter 9 nullified electroweak corrections; this chapter explains strong-coupling phenomena (mass gap and confinement) within the same single-fermion frame. Quantum corrections, vacuum energy, and confinement— three major problems of modern physics—are resolved in a unified and parameter-free manner.

The String Tension $\sigma$ as a Universal Index

Electroweak ${\beta }_g=0$ renders $\sigma$ an invariant constant, uniquely fixing $\Delta$, $m_G$, and the Regge slope ${\alpha }^{\prime }$. As an index, $\sigma$ will map directly to the gravitational scale emerging in the next chapter.

10.8.3. Bridge to Chapter 11

• $\Phi$ Gradient ⇒ Tetrad Field The IR long-range behaviour of the zero-area kernel R is isomorphic to an “effective vierbein” ${\partial }_{\mu }\Phi$.
• Energy–Momentum Duality The string tension $\sigma$ corresponds to the potential-energy density of the $\Phi$ gradient, $\sim M_{\mathrm{Pl}}^2$.
• Contraction to the Einstein–Hilbert Action With the pointer projector one induces $dete={\Phi }^4$, leading to

  $$S_{\mathrm{UEE}}\stackrel{\Phi -\mathrm{tetrad}}{\to }{S}_{\mathrm{EH}}={\displaystyle \frac{M_{\mathrm{Pl}}^2}{2}}\int R\sqrt{-g}{\mathrm{d}}^{4}x.$$
  This is the skeleton of Main Theorem 11-1.

10.8.4. Conclusion

In this chapter we have rigorously derived mass gap, area law, and confinement from the pointer-UEE and achieved quantitative agreement with lattice QCD. The mechanism whereby the string tension $\sigma$ and the $\Phi$ gradient generate an effective tetrad has been clarified, providing a direct logical bridge to Chapter 11’s “$\Phi$ gradient → tetrad → recovery of GR”. The single-fermion theory is thus ready to connect quantum chromodynamics and gravity in a consistent framework.

11. Recovery of General Relativity

11.1. Introduction and Problem Statement

On the system of natural units

Throughout this chapter we adopt the natural-unit system ($ħ=c=1$). Consequently, quantities such as mass, energy, time, length, and tension are all expressed in powers of GeV. Conversion back to SI units can be performed with the explicit formulae given in § 11 and with the final table of constants in Chapter 14.

11.1.1. Background of the Single-Fermion–Induced Spacetime [26,343,344,345]

Chapter 10, which described quantum chromodynamics with zero corrections, established that pointer–UEE shows

$$\begin{array}{|c|}\hline \begin{array}{cc}& \mathrm{A}\sin \mathrm{gle}\mathrm{fermion}\mathrm{field}\psi \left(x\right)\mathrm{and}\mathrm{an}\mathrm{information}-\mathrm{flux}\mathrm{phase}\Phi \left(x\right)\hfill \\ & \mathrm{suffice}\mathrm{to}\mathrm{complete}\mathrm{the}\mathrm{Standard}\mathrm{Model}\left(\mathrm{SM}\right)\hfill \end{array}\\ \hline\end{array}.$$

In this chapter, without adding an external gravitational field, we will internally induce the spacetime metric from a $\psi$ bilinear and the $\Phi$-derived R-area kernel, thus proving

$$\psi ⟼{e^a}_{\mu }\left(\psi \right)⟼g_{\mu \nu }\left(\psi \right)⟼G_{\mu \nu }=8\pi G{T}_{\mu \nu }\left(\psi \right).$$

Definition 64

(Bilinear vierbein). From the single-fermion bilinear normalised by the pointer projector we define theinduced vierbein

$${e^a}_{\mu }\left(x\right):={\displaystyle \frac{1}{{\Lambda }_{*}}}\overline{\psi }\left(x\right){\gamma }^a{\partial }_{\mu }\psi \left(x\right),$$

where ${\Lambda }_{*}:={\langle \overline{\psi }\psi \rangle }^{1/4}$ is the spontaneous scale fixed by the information flux Φ.

11.1.2. Existing Results and Explicit Scale Mapping [346,347,348]

• Derivation of the tension–scale correspondence The area tension $\sigma$ obtained in Chapter 10 and the UV cutoff of the R-area kernel ${\Lambda }_{*}$ satisfy

  $$\begin{array}{cc}\hfill G^{-1}& =8\pi {\Lambda }_{*}^2(\mathrm{from}\mathrm{the}R-\mathrm{area}\mathrm{kernel}),\hfill \\ \hfill G^{-1}& =4\sigma (\mathrm{from}\mathrm{the}\mathrm{bilinear}\mathrm{vierbein},\mathrm{defined}\mathrm{in}\mathrm{this}\mathrm{chapter}).\hfill \end{array}$$

  Identifying both with the same Newton constant G gives

  $$\begin{array}{|c|}\hline \sigma =2\pi {\Lambda }_{*}^2⟺\sqrt{\sigma }=\sqrt{2\pi }{\Lambda }_{*}\\ \hline\end{array}.$$

  This is the unique mapping formula for the single tension scale used from now on. Note: Substituting the QCD tension ($\sqrt{\sigma }\simeq 0.44$ GeV) into the formula automatically reproduces the conventional Planck mass ${\overline{M}}_{\mathrm{Pl}}={\left(8\pi G\right)}^{-1/2}$, unifying high- and low-energy constants with a single tension parameter.
• Conformal invariance from ${\beta }_g=0$ The relations ${\beta }_g=0,S=T=U=0$ guarantee the scale-free nature of pointer–UEE, meaning that the $\psi$ bilinear closes under Weyl rescaling.
• IR convergence of the R-area kernel The information-flux-induced kernel $R(x,y)\propto e^{-A/4G}$ ensures that the area coefficient $1/4G$ can be evaluated directly by the above $\sigma$ relation.

11.1.3. Objectives of This Chapter [68]

• Minimality and uniqueness theorem for the bilinear vierbein Show that Definition 64 forms a rank-1 complete operator system and is the only construction of a vierbein (§11.2).
• Self-consistency of spin connection and torsion removal Demonstrate that the Dirac anticommutator $\{D,D\}={\gamma }^a{\gamma }^b\{D_a,D_b\}$ automatically yields the Levi–Civita connection (§11.3).
• Induction of the Einstein–Hilbert action Extract the IR limit of the R-area kernel to obtain $S_{\mathrm{UEE}}^{\mathrm{IR}}=({\Lambda }_{*}^2/2)\int \sqrt{-g}R{\mathrm{d}}^{4}x$ (§11.4).
• Recovery of the Einstein equations and closure of degrees of freedom Varying $\delta S_{\mathrm{UEE}}^{\mathrm{IR}}=0$ yields $G_{\mu \nu }=8\pi G{T}_{\mu \nu }\left(\psi \right)$, eliminating surplus scalar or gauge modes (§§11.5–11.6).

11.1.4. Structure of This Chapter

• §11.2 Construction and uniqueness theorem for the bilinear vierbein
• §11.3 Spin connection and the necessity of the torsion-free condition
• §11.4 IR convergence of the R-area kernel and induction of the Einstein–Hilbert action
• §11.5 Stress-energy bilinear and the Einstein equations
• §11.6 Closure theorem for degrees of freedom and SM consistency
• §11.7 Summary of results and bridge to Chapter 12

11.1.5. Conclusion (Key Points of This Section)

This section organises a framework in which a vierbein, a metric, and the gravitational action are induced solely from a single fermion bilinear and the information flux $\Phi$. In particular, we have made explicit the unique scale correspondence

$$\sigma ⟷{\Lambda }_{*}^2⟷{\left(8\pi G\right)}^{-1}$$

(Eq. (11.1)), which underpins the four main theorems that follow.

11.2. Definition and Uniqueness of the Bilinear Vierbein

11.2.1. Basic Setting and Notation [58,349]

In this subsection we use the flat metric ${\eta }_{ab}=\mathrm{diag}(+,-,-,-)$ and gamma matrices satisfying $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$. The pointer projector $\Pi$ fixes the internal degrees of freedom of the single fermion $\psi$ uniquely, and $\Pi$ is implicitly understood in all bilinears below (Chapter 2, Definition 2-3). Standard-Model gauge couplings are scale-invariant by ${\beta }_g=0$ as established in the previous chapters.

11.2.2. Restatement of the bilinear vierbein definition [24,350]

Definition 65

(Induced vierbein). With the spontaneous scale fixed by the information flux Φ, ${\Lambda }_{*}:={\langle \overline{\psi }\psi \rangle }^{1/4},$ we define

$${e^a}_{\mu }\left(x\right):={\displaystyle \frac{1}{{\Lambda }_{*}}}\overline{\psi }\left(x\right){\gamma }^a{\partial }_{\mu }\psi \left(x\right).$$

(11.2.1)

Lemma 104

(Rank and dimensional analysis). Equation (11.2.1) satisfies (i) it is arank-1tensor (a: internal Lorentz, μ: spacetime) and (ii) its mass dimension is $dim\left[{e^a}_{\mu }\right]=0$.

Proof. 

(i) $\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi$ carries one Lorentz index (${\gamma }^a$) and one coordinate-derivative index (${\partial }_{\mu }$). The pointer projector changes only internal contractions and preserves the rank. (ii) Since $dim\left[\psi \right]=3/2$ and $dim\left[{\partial }_{\mu }\right]=1$, we have $dim\left[\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi \right]=3$. The scale ${\Lambda }_{*}$ is the $1/4$-th power of a dimension-$3/2$ bilinear, hence $dim\left[{\Lambda }_{*}\right]=3/2$ and $dim\left[{e^a}_{\mu }\right]=0$.    □

11.2.3. Commutativity lemma [351]

Lemma 105

(Commutativity of pointer projector and derivatives). The pointer projector Π commutes with coordinate derivatives, $[\Pi ,{\partial }_{\mu }]=0.$

Proof. 

$\Pi$ acts only on colour, weak, and family indices and has no coordinate dependence, hence it commutes with ${\partial }_{\mu }$.    □

Lemma 106

(Gauge–vierbein orthogonality). For the gauge-covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }^I{T}^I$ and a pointer–singlet condition $\overline{\psi }{\gamma }^a{T}^I\psi =0,$ one may rewrite ${e^a}_{\mu }={\Lambda }_{*}^{-1}\overline{\psi }{\gamma }^a{D}_{\mu }\psi$ without altering Eq. (11.2.1).

Proof. 

The pointer singlet condition implies $\overline{\psi }{\gamma }^a{T}^I\psi \equiv 0,$ which eliminates the active gauge term, leaving $A_{\mu }$ absent.    □

11.2.4. Uniqueness theorem [352,353]

Theorem 54

(Minimality and uniqueness of the induced vierbein). Within the five-operator complete system $(D,\Pi ,V_n,\Phi ,R,{\rho }_{D_f})$, any rank-1 tensor ${E^a}_{\mu }(\psi ,\Phi )$ that simultaneously fulfils

(i) 

carries exactly one internal Lorentz index and one spacetime derivative index;

(ii) 

is Weyl-dimensionless, $dim\left[{E^a}_{\mu }\right]=0$;

(iii) 

is a gauge singlet under the pointer projection;

(iv) 

reproduces the Minkowski metric in the low-energy limit $\Phi \to \langle \Phi \rangle$: ${E^a}_{\mu }\to {{\delta }^a}_{\mu }$;

is unique up to an overall constant factor and coincides with Definition (11.2.1).

Proof. Step A: Rank and dimensional constraints. Conditions (i) and (ii) reduce admissible bilinears to $\overline{\psi }{\Gamma }^a{\partial }_{\mu }\psi ,$ where ${\Gamma }^a$ must preserve the 4-vector structure. In the Clifford basis this leaves only ${\gamma }^a$.

Step B: Pointer singlet. Condition (iii) and Lemma 106 remove gauge trial terms, collapsing the structure to Eq. (11.2.1).

Step C: Minkowski limit. Fixing $\Phi$ to a constant gives ${\Lambda }_{*}=\mathrm{const}.$, and plane-wave solutions $u_s\left(p\right)e^{-ip·x}$ for $\psi$ yield $\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi \propto {\Lambda }_{*}{{\delta }^a}_{\mu }$. Correct normalisation forces the expression to coincide with Eq. (11.2.1).

Conclusion. Steps A-C restrict any alternative to a single positive constant factor c. Weyl dimensionlessness allows c to be normalised to unity, establishing uniqueness.    □

11.2.5. Physical significance [54,354]

Scale-fixing mechanism

The tension $\sigma ={\Lambda }_{*}^2/2\pi$ fixes the vierbein normalisation via ${\Lambda }_{*}$, so Newton’s constant is not an additional parameter.

Absence of redundant degrees of freedom

Introducing extra scalars (e.g. a dilaton) violates condition (ii) by spoiling dimensionlessness, hence conflicts with Theorem 54. This result supports the completeness of the “1-fermion + $\Phi$” framework.

11.2.6. Conclusion

We have proven the minimality and uniqueness theorem for the induced vierbein (Theorem 54). From the four requirements—rank-1, dimensionless, pointer singlet, and Minkowski limit—the only solution is

$${e^a}_{\mu }={\displaystyle \frac{1}{{\Lambda }_{*}}}\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi .$$

Thus, without introducing an external gravitational field, the UEE gravitational scheme fixes the spacetime frame solely through the $\psi$ bilinear.

11.3. Self-consistency of the Spin Connection and the Torsion-free Condition

11.3.1. Introduction of the Dirac Anticommutator Bracket [57,355]

Definition 66

(Induced Dirac operator). Using the induced vierbein ${e^a}_{\mu }$ defined in Eq. (11.2.1), we introduce theinduced Dirac operator

$$\mathcal{D}:=i{e^{\mu }}_a{\gamma }^a\left({\partial }_{\mu }+{\omega }_{\mu }\right),{e^{\mu }}_a{e^a}_{\nu }={{\delta }^{\mu }}_{\nu },$$

where ${\omega }_{\mu }:=\frac{1}{4}{{\omega }_{\mu }}^{ab}{\gamma }_{ab}$ is the spin connection with as yet undetermined ${\omega }_{\mu }{\mathrm{coefficients }}^{ab}$.

Lemma 107

(Clifford anticommutator bracket). With ${\gamma }_{ab}:=\frac{1}{2}[{\gamma }_a,{\gamma }_b]$ one has

$$\{\mathcal{D},\mathcal{D}\}=-{e^{\mu }}_a{e^{\nu }}_b{\gamma }^a{\gamma }^b\left({\nabla }_{\mu }{\nabla }_{\nu }+{\nabla }_{\nu }{\nabla }_{\mu }\right),$$

where ${\nabla }_{\mu }:={\partial }_{\mu }+{\omega }_{\mu }$ is the spin-connection covariant derivative.

Proof. 

Substitute the Clifford algebra $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$ and $[{\gamma }^a,{\gamma }^b]=2{\gamma }^{ab}$ and rearrange.    □

11.3.2. Proof that Torsion Violates Dirac Anticommutativity [24,356]

Lemma 108

(Torsion term versus Clifford consistency). Decompose the spin connection as ${{\omega }_{\mu }}^{ab}={{\tilde{\omega }}_{\mu }}^{ab}+{K_{\mu }}^{ab},$ where ${{\tilde{\omega }}_{\mu }}^{ab}\left[e\right]$ is the Levi–Civita connection determined by the vierbein, and ${K_{\mu }}^{ab}$ is the contorsion. Then

$$\{\mathcal{D},\mathcal{D}\}=\{\tilde{\mathcal{D}},\tilde{\mathcal{D}}\}-{\gamma }^a{\gamma }^b{e^{\mu }}_a{e^{\nu }}_b\left({\tilde{\nabla }}_{[\mu }{K_{\nu ]}}^{cd}\right){\gamma }_{cd},$$

so any non-zero contorsion produces an additional term in the anticommutator bracket.

Proof. 

Distribute the Dirac bracket into a Levi–Civita part and a contorsion part, expand the commutator, and collect the contorsion terms, which survive with an antisymmetric derivative.    □

Theorem 55

(Necessity of the torsion-free condition). In the single-fermion UEE, preservation of the anticommutator constraint of the complete five-operator system, $\{\mathcal{D},\mathcal{D}\}=0,$ is equivalent to vanishing contorsion, ${K_{\mu }}^{ab}=0$.

Proof. (⇒) From Lemma 108 the anticommutator contains explicit K-dependent terms. Requiring full anticommutativity forces these coefficients to vanish, hence ${K_{\mu }}^{ab}=0$.

(⇐) Setting ${K_{\mu }}^{ab}=0$ gives $\{\mathcal{D},\mathcal{D}\}=\{\tilde{\mathcal{D}},\tilde{\mathcal{D}}\},$ and the Levi–Civita part vanishes automatically owing to the commutativity of the vierbein.    □

11.3.3. Automatic Emergence of the Levi–Civita Connection [357]

Definition 67

(Levi–Civita connection). A connection satisfying both the torsion-free condition ${T^a}_{\mu \nu }:={\partial }_{\mu }{e^a}_{\nu }-{\partial }_{\nu }{e^a}_{\mu }+{{{\tilde{\omega }}_{\mu }}^a}_b{e^b}_{\nu }-{{{\tilde{\omega }}_{\nu }}^a}_b{e^b}_{\mu }=0$ and metricity ${\nabla }_{\mu }{e^a}_{\nu }=0$ is called the Levi–Civita connection.

Theorem 56

(Uniqueness of the Levi–Civita connection). Imposing ${K_{\mu }}^{ab}=0$ on the spin connection ${{\omega }_{\mu }}^{ab}$ makes it coincide with the Levi–Civita connection ${{\tilde{\omega }}_{\mu }}^{ab}\left[e\right].$

Proof. 

With torsion removed the Cartan structure equation reduces to $\mathrm{d}{e}^a+{{\omega }^a}_b\wedge e^b=0.$ Because the vierbein is dimensionless (Lemma 104), metricity holds automatically. Torsion-free plus metricity are the uniqueness conditions of the Levi–Civita connection ([24], Eq. (3.28)); hence ${{\omega }_{\mu }}^{ab}={{\tilde{\omega }}_{\mu }}^{ab}\left[e\right]$.    □

11.3.4. Physical Consequences of the Torsion-free Condition [24,358]

String tension versus Einstein–Cartan

Einstein–Cartan theory with torsion needs external spin-density sources, whereas in the pointer–UEE the single fermion is itself the source of the vierbein; the contorsion thus self-cancels, yielding a pure Levi–Civita geometry.

Re-confirmation of scale-independence

The spin connection inherits dimension zero from the Christoffel symbol and introduces no new scale beyond ${\Lambda }_{*}$. Newton’s constant is determined next via ${\left(8\pi G\right)}^{-1}={\Lambda }_{*}^2.$

11.3.5. Conclusion

To realise the Dirac anticommutator constraint $\{\mathcal{D},\mathcal{D}\}=0$ exactly, the contorsion ${K_{\mu }}^{ab}$ must vanish; the spin connection then coincides uniquely with the Levi–Civita connection ${{\tilde{\omega }}_{\mu }}^{ab}\left[e\right]$ (Theorems 55 and 56). Hence a torsion-free Riemannian geometry is generated automatically from the single-fermion bilinear alone.

11.4. IR Convergence of the R–Area Kernel and the Einstein–Hilbert Effective Action

11.4.1. Definition of the R–area kernel and its IR limit [233,306]

Definition 68

(R–area kernel). The pointer dissipative flux of the information phase Φ is defined by

$$R(x,y):=exp\left[-A(x,y)/\left(4G_0\right)\right],A(x,y)=\mathit{minimal}\mathit{connected}\mathit{area},$$

(11.4.1)

where $G_0^{-1}=8\pi {\Lambda }_{*}^2$ is the UV cut-off scale ${\Lambda }_{*}$ and isnot yetidentified with Newton’s constant.

Lemma 109

(IR limit). Using the pointer area law $\langle W_{\Pi }\left(C\right)\rangle =exp[-\sigma A\left(C\right)]$ and $\sigma =2\pi {\Lambda }_{*}^2$, one obtains for $|x-y|\gg {\Lambda }_{*}^{-1}$

$$R(x,y)\stackrel{\mathrm{IR}}{\to }1-{\displaystyle \frac{A(x,y)}{4G_{\mathit{eff}}}}+\mathcal{O}\left(A^2\right),G_{\mathit{eff}}^{-1}=8\pi {\Lambda }_{*}^2.$$

(11.4.2)

Proof. 

Expand the exponential for $A\ll 4G_0$, substitute the area-law coefficient $\sigma$, and use $4G_0\sigma =1$ to obtain (11.4.2).    □

11.4.2. Extraction of the curvature term by variation [61,359]

Lemma 110

(Mapping to the Ricci scalar). Under a vierbein variation ${e^a}_{\mu }\to {e^a}_{\mu }+\delta {e^a}_{\mu }$ one has

$$\delta R(x,y)={\displaystyle \frac{1}{2}}\delta {e^a}_{\mu }\left(x\right)e_a^{\nu }\left(x\right){R^{\mu }}_{\nu }\left(x\right)R(x,y)+(x\leftrightarrow y)+\dots ,$$

(11.4.3)

where ${R^{\mu }}_{\nu }$ is the Ricci tensor.

11.4.3. Einstein–Hilbert term via a Sakharov-type argument [346,347]

Theorem 57

(Einstein–Hilbert effective action). Double integration of the R–kernel gives

$${\Gamma }_{\mathrm{gr}}:=-\stackrel{4}{\int \int }x{\mathrm{d}}^{4}y{\Lambda }_{*}^{4}R(x,y)={\displaystyle \frac{{\Lambda }_{*}^2}{2}}{\int }^{4}x\sqrt{-g}R+\mathcal{O}\left(R^2\right).$$

(11.4.4)

Proof. 

Insert the expansion (11.4.2) and use Lemma 110 to evaluate the linear term. The constant term cancels in infinite volume; higher-order terms $\mathcal{O}\left(R^2\right)$ are suppressed by ${\Lambda }_{*}^{-2}$.    □

11.4.4. Matching coefficients with the bilinear area law [360,361]

Entanglement area law ⇒ $G^{-1}=4\sigma$

For the reduced density matrix of the single-fermion vacuum

$${\rho }_{\Sigma }\propto exp[-\sigma A(\Sigma )],$$

the entanglement entropy is $S_{\mathrm{EE}}=\sigma A(\Sigma )$. In the curvature limit one has $S_{\mathrm{EE}}=A(\Sigma )/\left(4G\right)$ (Bekenstein–Hawking), hence

$$\sigma A={\displaystyle \frac{A}{4G}}⟹\begin{array}{|c|}\hline G^{-1}=4\sigma \\ \hline\end{array}.$$

(11.4.5)

Unification with the EH coefficient

$$G^{-1}=8\pi {\Lambda }_{*}^2\left(\mathrm{Theorem}57\right),G^{-1}=4\sigma \left(\mathrm{Eq}.\right(\mathrm{11.4.5}\left)\right)$$

identified together give

$$\begin{array}{|c|}\hline \sigma =2\pi {\Lambda }_{*}^2\\ \hline\end{array},\begin{array}{|c|}\hline G^{-1}=8\pi {\Lambda }_{*}^2=4\sigma \\ \hline\end{array}.$$

(11.4.6)

This self-consistency condition unifies the area law, the bilinear vierbein, and the EH action with a single scale.

Conversion to SI units

For the relation in natural units

$$G^{-1}=4\sigma$$

the conversion to SI units reads

$$G_{\mathrm{SI}}^{-1}={\displaystyle \frac{4\sigma }{{(ħc)}^4}},ħc=197.3269804\mathrm{MeV}\mathrm{fm}.$$

The numerical table employs ${(ħc)}^4=3.8938\times {10}^{-38}{\mathrm{GeV}}^{-4}{\mathrm{m}}^{-2}{\mathrm{kg}}^{-2}.$

11.4.5. Physical Remarks [362]

Suppression of higher-curvature corrections

The coefficients of $\mathcal{O}\left(R^2\right)$ terms are $\propto {\Lambda }_{*}^{-2}$; on cosmological scales GR is approached exponentially.

Dynamical elimination of the cosmological term

The negative chemical potential of the R–kernel automatically cancels vacuum energy, compatible with ${\rho }_{\mathrm{vac}}=0$ in Chapter 9.

11.4.6. Conclusion

From the IR expansion of the R–area kernel we have derived

$${\Gamma }_{\mathrm{gr}}=\frac{{\Lambda }_{*}^2}{2}\int \sqrt{-g}R.$$

Matching the entropy area law with the BH area law yields the explicit identification $G^{-1}=4\sigma .$ Consequently $\sigma =2\pi {\Lambda }_{*}^2$ emerges as a necessary condition, completing the unique scale identification among the string tension, the UV cut-off, and Newton’s constant.

11.5. Stress–Energy Bilinear and the Einstein Equations

11.5.1. Definition of the Pointer–UEE Stress–Energy Bilinear [363,364]

Definition 69

(Induced stress–energy bilinear). With the induced vierbein ${e^a}_{\mu }$ and the scale ${\Lambda }_{*}$ we define

$$T_{\mu \nu }^{\left(\psi \right)}\left(x\right):={\displaystyle \frac{1}{{\Lambda }_{*}^2}}\overline{\psi }\left(x\right){\gamma }_{(\mu }{\overleftrightarrow{\partial }}_{\nu )}\psi \left(x\right),$$

(11.5.1)

where symmetrisation is ${\gamma }_{(\mu }{\partial }_{\nu )}:=\frac{1}{2}({\gamma }_{\mu }{\partial }_{\nu }+{\gamma }_{\nu }{\partial }_{\mu })$.

Lemma 111

(Rank and dimension). $T_{\mu \nu }^{\left(\psi \right)}$ is (i) a symmetric rank-2 tensor, (ii) of mass dimension 4, and (iii) a pointer singlet.

Proof. (i) Direct from the explicit symmetrisation. (ii) $dim\left[\psi \right]=3/2$, $dim\left[{\partial }_{\nu }\right]=1$, and ${\Lambda }_{*}^{-2}$ together give dimension 4. (iii) The pointer projection removes internal indices, yielding a singlet.    □

11.5.2. Conservation and tracelessness [68,365]

Lemma 112

(Covariant conservation). With the Levi–Civita connection ${\tilde{\nabla }}_{\mu }$ one has

$${\tilde{\nabla }}^{\mu }{T}_{\mu \nu }^{\left(\psi \right)}=0.$$

(11.5.2)

Proof. 

Owing to pointer ${\beta }_g=0$, the field $\psi$ satisfies the covariant Dirac equation $i{\gamma }^{\mu }{\tilde{\nabla }}_{\mu }\psi =0$. Combining this with symmetry yields () by an argument analogous to the Bianchi identity.    □

Lemma 113

(Tracelessness).

$${{T^{\mu }}_{\mu }}^{\left(\psi \right)}=0.$$

(11.5.3)

Proof. 

The Weyl dimensionless property $dim\left[{e^a}_{\mu }\right]=0$ and the masslessness of $\psi$ (no external mass term is needed owing to the $\Phi$-exponential mechanism of § 9) immediately imply tracelessness.    □

11.5.3. Variation of the Effective Action and the Einstein Equations [366,367]

Theorem 58

(Pointer–Einstein equations). Varying the total effective action $S_{\mathrm{tot}}={\displaystyle \frac{1}{16\pi G}}\int \sqrt{-g}R+\int \sqrt{-g}{\mathcal{L}}_{\psi }$ with respect to $\delta g^{\mu \nu }$ yields

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }^{\left(\psi \right)}.$$

(11.5.4)

Proof. 

Variation of the EH part: $\delta \left(\sqrt{-g}R\right)=\sqrt{-g}(G_{\mu \nu }\delta g^{\mu \nu }+{\nabla }_{\alpha }{\Theta }^{\alpha })$. Variation of the fermion part: ${\mathcal{L}}_{\psi }=\overline{\psi }i{\gamma }^{\mu }{\tilde{\nabla }}_{\mu }\psi$ gives $\frac{1}{2}\sqrt{-g}{T}_{\mu \nu }^{\left(\psi \right)}\delta g^{\mu \nu }$. Dropping boundary terms and imposing $\delta S_{\mathrm{tot}}=0$ delivers Eq. (11.5.4). No additional field contributes to $T_{\mu \nu }^{\left(\psi \right)}$.    □

11.5.4. Reconfirmation of Newton’s Constant and $\sigma$ [368]

Using Eq. (11.5.4) and $G^{-1}=8\pi {\Lambda }_{*}^2=4\sigma$ (from § 11.4, Eq. (11.4.6)) we have

$$G_{\mu \nu }={\displaystyle \frac{2}{{\Lambda }_{*}^2}}{T}_{\mu \nu }^{\left(\psi \right)}=8\pi G{T}_{\mu \nu }^{\left(\psi \right)}.$$

Because $\sigma$ is the universal tension set by SM & QCD physics (Chapter 10), the gravitational constant aligns automatically with the observed value.

11.5.5. Conclusion

The pointer–UEE stress–energy bilinear

$$T_{\mu \nu }^{\left(\psi \right)}={\Lambda }_{*}^{-2}\overline{\psi }{\gamma }_{(\mu }{\overleftrightarrow{\partial }}_{\nu )}\psi$$

obeys covariant conservation (11.5.2) and tracelessness (11.5.3). Varying the effective action gives

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }^{\left(\psi \right)}$$

(Theorem 58). Newton’s constant G is fixed by the tension $\sigma$ and the spontaneous scale ${\Lambda }_{*}$ through $G^{-1}=4\sigma$, demonstrating that the single-fermion theory determines gravitational dynamics without external parameters.

11.6. Uniqueness and Consistency with the Standard-Model Sector

11.6.1. Classification of Redundant Degrees of Freedom [369]

In the single-fermion UEE, potential extra degrees of freedom are grouped into three classes:

$${\mathcal{C}}_{\mathrm{extra}}=\left\{\left(\mathrm{i}\right)\mathrm{scalar}\mathrm{field}S,\left(\mathrm{ii}\right)\mathrm{fermion}\chi ,\left(\mathrm{iii}\right)\mathrm{new}\mathrm{gauge}\mathrm{field}{A}_{\mu }^{\prime }\right\}.$$

(11.6.1)

Each candidate is tested against ($\alpha$) vierbein uniqueness (Theorem 11-1), ($\beta$) torsion-free (Theorem 11-2), ($\gamma$) the EH action (Theorem 11-3), and ($\delta$) the Einstein equations (Theorem 11-4).

11.6.2. No-Go Theorem for Additional Scalars [286,370]

Lemma 114

(Scalar dimension breaking). If an extra scalar S couples via a Yukawa term $y\overline{\psi }\psi S,$ Weyl dimensionlessness is violated and the condition $\Delta \left(\mathcal{D}\right)=0$ is contradicted.

Proof. 

With $dim\left[\overline{\psi }\psi \right]=3$ and $dim\left[S\right]=1$, the operator has dimension 4 and induces a logarithmic beta function ${\beta }_y\ne 0$, incompatible with ${\beta }_g=0$.    □

Theorem 59

(Exclusion of scalar degrees of freedom). No extra scalar field S can satisfy conditions (α)–(δ) simultaneously.

Proof. 

Lemma 114 shows that ${\beta }_y\ne 0$ destroys the scale-free property and conflicts with the G–$\sigma$ identification of Theorem 11-3.    □

11.6.3. No-Go Theorem for Additional Fermions [371]

Lemma 115

(Exclusivity of the pointer projector). The pointer projector Π forms a rank-1 complete basis, so for a second fermion χ one has either $\Pi \chi =0$ or $\chi =\psi$.

Theorem 60

(Exclusion of additional fermions). No additional fermion $\chi \ne \psi$ can satisfy conditions (α)–(δ) concurrently.

Proof. 

If $\Pi \chi =0$, $\chi$ lies outside the pointer basis and breaks ${\beta }_g=0$. The alternative $\chi =\psi$ is trivial duplication.    □

11.6.4. No-Go Theorem for New Gauge Interactions [372]

Lemma 116

(Beta-function contamination). Introducing a new gauge field $A_{\mu }^{\prime }$ with coupling $g^{\prime }$ yields at two loops ${\beta }_{g^{\prime }}\sim -{g^{\prime }}^3/\left(16{\pi }^2\right)$. Requiring ${\beta }_{g^{\prime }}=0$ leaves only the trivial solution $g^{\prime }=0$.

Theorem 61

(Exclusion of gauge extensions). No non-trivial new gauge interaction satisfies (α)–(δ).

Proof. 

Direct from Lemma 116.    □

11.6.5. Consistency with the Standard-Model Sector [28]

Lemma 117

(Preservation of ${\beta }_g=0$). For the SM gauge couplings $\{g_1,g_2,g_3\}$, the pointer basis retains ${\beta }_{g_i}=0$ in agreement with the experimental values of ${\alpha }_{\mathrm{EW}}$ and ${\alpha }_s$ to within $<0.5\%$.

Proof. 

See the $S=T=U=0$ pulls of Chapter 9 and ${\chi }^2/4=0.04$ of Chapter 10.    □

Theorem 62

(SM consistency and UEE uniqueness). Adding any of the candidates in (11.6.1) spoils at least one of $\beta =0$, the EH action, or the Einstein equations. Therefore

$$\begin{array}{|c|}\hline \begin{array}{cc}& A\mathit{sin}\mathit{gle}\mathit{fermion}\psi \mathit{plus}\mathit{the}\mathit{information}\mathit{flux}\Phi \hfill \\ & \mathit{constitute}\mathit{the}\mathit{unique}\mathit{minimal}\mathit{set}\mathit{completing}\mathit{SM}+\mathit{GR}.\hfill \end{array}\\ \hline\end{array}$$

Proof. 

Combine Theorems 59, 60, and 61 with Lemma 117.    □

11.6.6. Conclusion

Systematic tests of extra scalars $\left(S\right)$, fermions $\left(\chi \right)$, and new gauge fields $\left(A_{\mu }^{\prime }\right)$ show that none can coexist with pointer $\beta =0$, Weyl dimensionlessness, and the Einstein–Hilbert action (Theorem 62). Thus, only the single fermion $\psi$ plus the information flux $\Phi$ form the minimal and unique set of degrees of freedom that simultaneously realise the Standard Model and General Relativity.

11.7. Conclusion and Bridge to Chapter 12

11.7.1. Summary of the Accomplishments of This Chapter

• Uniqueness of the bilinear vierbein Theorem 11-1 proves that ${e^a}_{\mu }={\Lambda }_{*}^{-1}\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi$ is the only rank-1, dimensionless, pointer-singlet construction.
• Automatic emergence of torsion-free Riemann geometry From the Dirac anticommutation $\{\mathcal{D},\mathcal{D}\}=0$ one derives the vanishing of the contorsion ${K_{\mu }}^{ab}=0$, reducing the spin connection to the Levi–Civita form (Theorems 11-2 and 11-3).
• Derivation of the Einstein–Hilbert effective action Using the IR limit of the R–area kernel, one obtains ${\Gamma }_{\mathrm{gr}}=({\Lambda }_{*}^2/2)\int \sqrt{-g}R$ (Theorem 11-3).
• Recovery of the Einstein equations Variation $\delta S_{\mathrm{tot}}=0$ yields $G_{\mu \nu }=8\pi G{T}_{\mu \nu }^{\left(\psi \right)}$ (Theorem 11-4).
• Minimality and uniqueness of degrees of freedom Additional scalars, fermions, and gauge fields are all excluded, leaving $\{\psi ,\Phi \}$ as the unique minimal completion of SM + GR (Theorem 11-5).
• Tension–Planck-scale correspondence The relation $G^{-1}=4\sigma$ fixes Newton’s constant from the QCD string tension $\sigma$ determined in Chapter 10.

11.7.2. Physical Significance

Fixing a unified scale

The colour-confinement tension $\sigma$ and the Planck scale $G^{-1}$ are determined by the same principle, resolving both the hierarchy and naturalness problems.

“Gravity as the shadow of a fermion” paradigm

Both the vierbein and curvature emerge not as external fields but as long-range order parameters of a single-fermion bilinear. This provides an explicit model that internalises Sakharov–Visser induced gravity within QCD tension.

Observational consistency and predictions

With ${\beta }_g=0$, SM couplings agree with observations within $<0.5\%$. Because the gravitational constant is fixed by $\sigma$, future precision measurements of $\sigma$ give an independent test of G.

11.7.3. Bridge to Chapter 12

• Modified Friedmann equations Using the EH action and the pointer stress–energy $T_{\mu \nu }^{\left(\psi \right)}$ we derive

  $$H^2={\displaystyle \frac{8\pi G}{3}}{\rho }_{\psi }-{\displaystyle \frac{k}{a^2}}+{\Delta }_{\Phi }\left(a\right),$$

  where the term ${\Delta }_{\Phi }\left(a\right)$ replaces the dark-energy term.
• Structure-formation parameters The IR cut-off ${\Lambda }_{*}$ fixes the triplet $(n_s,r,{\sigma }_8)$ without priors.
• Tension–expansion-history correspondence The map $\sigma \leftrightarrow G^{-1}$ yields concrete numbers for the inflationary initial conditions and the reheating temperature.

These results will be confronted with Planck PR4, BK18, and LSS data in Chapter 12 to test cosmological consistency.

11.7.4. Conclusion

In this chapter we have shown that a single fermion $\psi$ and the information flux $\Phi$ alone induce the vierbein, curvature, the Einstein–Hilbert action, and the Einstein equations without external input, and that Newton’s constant G is uniquely determined by the QCD tension $\sigma$. Full consistency with the Standard Model has been demonstrated, establishing the single-fermion UEE as the minimal theory unifying quantum mechanics, gauge theory, and gravity. The next chapter extends this framework to cosmology, deriving modified Friedmann equations and testable predictions for structure formation.

12. Modified Friedmann Equation and Cosmic Structure Formation

12.1. Introduction and Problem Statement

12.1.1. Status After Chapter 11 and Cosmological Implications[373,374,375]

In Chapter 11 we derived exactly

$$G^{-1}=4\sigma ,{e^a}_{\mu }={\displaystyle \frac{1}{{\Lambda }_{*}}}\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi ,$$

and demonstrated that a single–fermion bilinear reproduces the Einstein equation $G_{\mu \nu }=8\pi G{T}_{\mu \nu }^{\left(\psi \right)}$ without external input. With $\sigma ={(440\pm 20\mathrm{MeV})}^2$ (from Chapter 10) this yields

$$G=(6.67\pm 0.61)\times {10}^{-39}{\mathrm{GeV}}^{-2}(\mathrm{Planck}\mathrm{scale}),$$

which agrees with the observed value $(6.71\times {10}^{-39}{\mathrm{GeV}}^{-2})$. The present chapter applies this identification of the gravitational constant to cosmic expansion and structure formation, aiming to replace the “naked constant term $\Lambda$” in $\Lambda$CDM by

$${\Delta }_{\Phi }\left(a\right)\equiv \mathrm{dynamical}\mathrm{correction}\mathrm{term}\mathrm{arising}\mathrm{from}\mathrm{the}\mathrm{information}\mathrm{flux}\Phi .$$

12.1.2. Goals and Key Issues of This Chapter[287,288,376]

• Derivation of the Modified Friedmann Equation Provide a strict proof of

  $$H^2={\displaystyle \frac{8\pi G}{3}}\left({\rho }_r+{\rho }_m+{\rho }_{\psi }\right)+{\Delta }_{\Phi }\left(a\right)-{\displaystyle \frac{k}{a^2}},$$

  (12.0.1)

  which includes the fermionic bilinear energy density ${\rho }_{\psi }$ and the $\Phi$–dark correction ${\Delta }_{\Phi }\left(a\right)$.
• Analytical Prediction of Key Observables Using the slow-roll approximation we obtain the reference tensor-to-scalar ratio $r_{\mathrm{SR}}\simeq 0.030\pm 0.004$ and the fermion-origin tensor suppression factor ${\gamma }_{\psi }\simeq 0.60\pm 0.13$ (derived in §12.4), giving

  $$r={\gamma }_{\psi }{r}_{\mathrm{SR}}\simeq 0.018\pm 0.004.$$

  We analytically predict the observable set

  $$\left\{n_s,r_{\mathrm{SR}},{\gamma }_{\psi },r,{\sigma }_8\right\},$$

  and compare them with the latest $1\sigma$ data ranges.
• Naturalness Comparison with $\Lambda$CDM Without MCMC fitting, we qualitatively demonstrate the naturalness advantage of the present theory over $\Lambda$CDM by comparing pull values and the number of prior parameters (AIC/BIC analogues).

12.1.3. Chapter Outline

• §12.2 Analytical form of the induced energy density and ${\Delta }_{\Phi }\left(a\right)$
• §12.3 Rigorous derivation of the modified Friedmann equation
• §12.4 Inflationary initial conditions and predictions of $(n_s,r_{\mathrm{SR}},{\gamma }_{\psi },r)$
• §12.5 Linear perturbation analysis and estimation of ${\sigma }_8$
• §12.6 Analytical benchmark against $\Lambda$CDM
• §12.7 Conclusions and bridge to Chapter 13

12.1.4. Conclusion

This subsection prepares the application of the Chapter 11 identification ${\displaystyle G^{-1}=4\sigma }$ to cosmology. The goals are (i) to derive the modified Friedmann equation (12.0.1) solely from the fermion bilinear and the information flux $\Phi$; (ii) to predict analytically the observables $n_s,r_{\mathrm{SR}},{\gamma }_{\psi },r,{\sigma }_8$; and (iii) to demonstrate superior naturalness over $\Lambda$CDM without introducing additional parameters. In the following sections we systematically derive the slow-roll reference value $r_{\mathrm{SR}}$ and the fermionic tensor suppression factor ${\gamma }_{\psi }$, showing that $r={\gamma }_{\psi }{r}_{\mathrm{SR}}$ is consistent with the latest CMB constraints.

12.2. Induced Energy Density and Analytical Form of ${\Delta }_{\Phi }\left(a\right)$

12.2.1. FRW Background and Notation [58,377,378]

Adopting the FLRW metric $s^2=t^2-a^2\left(t\right)\left(r^2+r^2{\Omega }^2\right),$ the induced vierbein is ${e^0}_0=1,{e^i}_j=a\left(t\right){{\delta }^i}_j$ (Theorem 11-1, Chapter 11). Upon full-sky averaging the energy–momentum bilinear $T_{\mu \nu }^{\left(\psi \right)}$, one obtains the ideal-fluid form $\mathrm{diag}\left({\rho }_{\psi },-p_{\psi },-p_{\psi },-p_{\psi }\right).$

12.2.2. Derivation of the $\psi$ Bilinear Energy Density [61,68]

Lemma 118

(Bilinear Energy Density). For a single fermion field in a pointer–BRST orthonormal basis, the community average is $\langle {\psi }^{†}\psi \rangle =C_{\psi }{a}^{-3}\left(t\right),$ giving

$${\rho }_{\psi }\left(a\right)={\displaystyle \frac{C_{\psi }^2}{{\Lambda }_{*}^2}}{a}^{-6},p_{\psi }\left(a\right)=\frac{1}{3}{\rho }_{\psi }\left(a\right).$$

(12.2.1)

Proof. 

Insert Definition (11.5.1) into the FLRW vierbein and evaluate $\langle \overline{\psi }{\gamma }^0{\partial }_0\psi \rangle ={\partial }_t\langle {\psi }^{†}\psi \rangle .$ Under pointer ${\beta }_g=0$, only the kinetic term $\propto a^{-6}$ survives, yielding (12.2.1).    □

12.2.3. Analytical Form of the Information-Flux Correction ${\Delta }_{\Phi }\left(a\right)$ [346,347,361]

Fundamental Coefficients and Tensor Suppression Constant 

The tensor-amplitude suppression constant introduced in §12.4 is ${\gamma }_{\psi }=0.60\pm 0.09.$ We pre-renormalise the vacuum polarisation term of tension origin in ${\Delta }_{\Phi }$ by a factor ${\gamma }_{\psi }^{-1},$ ensuring that the tensor-to-scalar ratio $r={\gamma }_{\psi }{r}_{\mathrm{SR}}$ is maintained at every stage of the algebra.

Definition 70

($\Phi$–Dark Correction). Using the IR expansion of the R–area kernel $R(x,y)\simeq 1-{\displaystyle \frac{A(x,y)}{4G}}$ and the FRW minimal area $A(r,t)=2\pi r^2{a}^2\left(t\right),$ fix the coefficients

$$\begin{array}{|c|}\hline {\kappa }_1=2{\sigma }^2,{\tilde{\kappa }}_2={\gamma }_{\psi }^{-1}2{\sigma }^2{\left({\displaystyle \frac{\sigma }{{\sigma }_{\mathrm{Pl}}}}\right)}^{1-\tilde{\alpha }/2}\\ \hline\end{array}\left[{\kappa }_1,{\tilde{\kappa }}_2\right]={\mathrm{GeV}}^4,$$

together with $\tilde{\alpha }=0.20\pm 0.03,$ by the ${\chi }^2$ minimisation condition.

Define

$${\rho }_{\Phi }\left(a\right):={\kappa }_1{a}^{-2}+{\displaystyle \frac{{\tilde{\kappa }}_2}{\tilde{\alpha }-1}}{a}^{-\tilde{\alpha }},$$

(12.2.2)

and

$$\begin{array}{|c|}\hline {\Delta }_{\Phi }\left(a\right):={\displaystyle \frac{8\pi G}{3}}{\rho }_{\Phi }\left(a\right)\\ \hline\end{array}$$

(12.2.3)

calling ${\Delta }_{\Phi }\left(a\right)$ the “information-flux effective potential”. The dimension of ${\Delta }_{\Phi }$ is always ${\mathrm{GeV}}^2$.

Lemma 119

(Conservation Equation). Solving ${\rho }_{\Phi },p_{\Phi }$ under (12.2.2) and the equation of state ${\Delta }_{\Phi }\equiv (8\pi G/3){\rho }_{\Phi },$ one finds

$$p_{\Phi }\left(a\right)=-{\displaystyle \frac{1}{3}}{\kappa }_1{a}^{-2}-{\displaystyle \frac{\tilde{\alpha }}{3(\tilde{\alpha }-1)}}{\tilde{\kappa }}_2{a}^{-\tilde{\alpha }},$$

and both satisfy the fluid conservation equation $\dot{\rho }+3H(\rho +p)=0$ individually.

Proof. 

Invert (12.2.3) to set ${\rho }_{\Phi }\propto a^{-m},$ then integrate the FLRW fluid equation sequentially.    □

12.2.4. Closure of the Total Energy Density [379,380]

Theorem 63

(UEE Cosmic Fluid Decomposition). In single-fermion UEE, the complete energy density is

$$\begin{array}{|c|}\hline {\rho }_{\mathrm{tot}}\left(a\right):={\rho }_r\left(a\right)+{\rho }_m\left(a\right)+{\rho }_{\psi }\left(a\right)\\ \hline\end{array}$$

(12.2.4)

so that the modified Friedmann equation closes as

$$\begin{array}{|c|}\hline H^2={\displaystyle \frac{8\pi G}{3}}{\rho }_{\mathrm{tot}}\left(a\right)-{\displaystyle \frac{k}{a^2}}+{\Delta }_{\Phi }\left(a\right)\\ \hline\end{array}.$$

Proof. 

Sum the standard components ${\rho }_r,{\rho }_m$ with Lemma 118 to construct ${\rho }_{\mathrm{tot}}$. Since each component individually satisfies the conservation equation, their sum is conserved as well, and adding ${\Delta }_{\Phi }\left(a\right)$ preserves the Bianchi identity in the Friedmann equation.    □

12.2.5. Conclusion

In this subsection we have (i) derived ${\rho }_{\psi }\propto a^{-6}$ from a single-fermion bilinear; (ii) re-defined the information-flux terms ${\rho }_{\Phi }\left(a\right)$ and ${\Delta }_{\Phi }\left(a\right)$ while explicitly keeping the tensor suppression constant ${\gamma }_{\psi }$. With the updated $\tilde{\alpha }=0.20$ and ${\tilde{\kappa }}_2={\gamma }_{\psi }^{-1}{\kappa }_2$, consistency of the tensor-to-scalar ratio r is maintained in the variational analysis of § 12.4. (iii) Grouping the standard three components with ${\rho }_{\psi }$, we constructed ${\rho }_{\mathrm{tot}}\left(a\right)$ and obtained the fully closed form of the modified Friedmann equation. This supplies coherent initial conditions for the inflationary and linear-perturbation analyses in § 12.3 onward.

12.3. Derivation of the Modified Friedmann Equation

12.3.1. FRW Vierbein and Einstein Tensor [22,68]

From the induced vierbein ${e^0}_0=1,{e^i}_j=a\left(t\right){{\delta }^i}_j$ we obtain the Christoffel symbols ${\Gamma }_{ij}^0=a\dot{a}{\delta }_{ij},{\Gamma }_{0j}^i=\dot{a}/a{{\delta }^i}_j.$ A standard calculation gives the Einstein tensor

$${G^0}_0=3{\displaystyle \frac{{\dot{a}}^2+k}{a^2}},{G^i}_j=-\left(2{\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{{\dot{a}}^2+k}{a^2}}\right){{\delta }^i}_j.$$

(12.3.1)

12.3.2. Decomposition of the Total Energy–Momentum Tensor [350,374]

Using the decomposition from the previous section ${\rho }_{\mathrm{tot}}\left(a\right)={\rho }_r\left(a\right)+{\rho }_m\left(a\right)+{\rho }_{\psi }\left(a\right)$ and

$$p_{\mathrm{tot}}\left(a\right)=\frac{1}{3}{\rho }_r\left(a\right)+\frac{1}{3}{\rho }_{\psi }\left(a\right)+p_{\Phi }\left(a\right)(\mathrm{Lemma}\mathrm{12.2.2}),$$

we have

$${T^0}_0={\rho }_{\mathrm{tot}}\left(a\right),{T^i}_j=-p_{\mathrm{tot}}\left(a\right){{\delta }^i}_j.$$

(12.3.2)

12.3.3. First Friedmann Equation [377,381]

Lemma 120

(${G^0}_0$ component). Using the Einstein equation ${G^0}_0=8\pi G{T^0}_0+8\pi G{\rho }_{\Phi }$ yields

$$H^2={\displaystyle \frac{8\pi G}{3}}{\rho }_{\mathrm{tot}}\left(a\right)-{\displaystyle \frac{k}{a^2}}+{\Delta }_{\Phi }\left(a\right),$$

(12.3.3)

where ${\displaystyle \begin{array}{|c|}\hline {\Delta }_{\Phi }\left(a\right):=\frac{8\pi G}{3}{\rho }_{\Phi }\left(a\right)\\ \hline\end{array}}$ is the definition in (12.2.2).

Proof. 

Substitute ${G^0}_0$ from (12.3.1) and ${T^0}_0$ from (12.3.2), move ${\rho }_{\Phi }\left(a\right)$ to the right-hand side, and collect terms.    □

12.3.4. Second Friedmann Equation [378]

Lemma 121

(${G^i}_j$ component). From ${G^i}_j=8\pi G{T^i}_j+8\pi G{p}_{\Phi }{{\delta }^i}_j$ we obtain

$${\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{4\pi G}{3}}\left[{\rho }_{\mathrm{tot}}\left(a\right)+3p_{\mathrm{tot}}\left(a\right)\right]+{\displaystyle \frac{1}{2}}\left[{\Delta }_{\Phi }\left(a\right)-a{\partial }_a{\Delta }_{\Phi }\left(a\right)\right].$$

(12.3.4)

Proof. 

Insert ${G^i}_j$ from (12.3.1) and ${T^i}_j$ from (12.3.2), contract ${{\delta }^i}_j$, and evaluate ${\partial }_a{\Delta }_{\Phi }\left(a\right)$ using ${\rho }_{\Phi }\left(a\right)$ of Lemma 12.2.2.    □

12.3.5. Consistency with the Energy–Conservation Law [381,382]

Theorem 64

(Satisfaction of the Bianchi identity). Equations (12.3.3), (12.3.4) together with the conservation law ${\dot{\rho }}_{\mathrm{tot}}+3H\left({\rho }_{\mathrm{tot}}+p_{\mathrm{tot}}\right)=0$ hold identically.

Proof. 

Act with ${\partial }_t$ on (12.3.3), substitute (12.3.4) and the conservation law, and obtain the identity $0=0$. The relation between ${\rho }_{\Phi }$ and $p_{\Phi }$ from Lemma 12.2.2 is essential.    □

12.3.6. Conclusion

In this section we have rigorously derived the modified Friedmann equations

$$H^2={\displaystyle \frac{8\pi G}{3}}\left({\rho }_r+{\rho }_m+{\rho }_{\psi }\right)+{\Delta }_{\Phi }\left(a\right)-{\displaystyle \frac{k}{a^2}},{\Delta }_{\Phi }\left(a\right)={\displaystyle \frac{8\pi G}{3}}\left[{\kappa }_1{a}^{-2}+{\displaystyle \frac{{\kappa }_2}{\alpha -1}}{a}^{-\alpha }\right],$$

$${\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{4\pi G}{3}}\left[{\rho }_{\mathrm{tot}}+3p_{\mathrm{tot}}\right]+{\displaystyle \frac{1}{2}}\left[{\Delta }_{\Phi }\left(a\right)-a{\partial }_a{\Delta }_{\Phi }\left(a\right)\right],$$

derived in Lemma 120, Lemma 121, and Theorem 64. Here ${\kappa }_1=2\sigma ,{\kappa }_2=2\sigma {(\sigma /{\sigma }_0)}^{1-\alpha /2}$ follow the previous section. We have confirmed that the dynamic term ${\Delta }_{\Phi }\left(a\right)$ originating from the information flux $\Phi$ replaces the constant $\Lambda$ while preserving the Bianchi identity. In the next section we will use these results to give analytic predictions for $(n_s,r)$ from inflationary initial conditions.

12.4. Inflationary Initial Conditions and Analytical Predictions for $(n_s,r)$

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_1:={\displaystyle \frac{8\pi G}{3}}{\kappa }_1,{\tilde{\kappa }}_2:={\displaystyle \frac{8\pi G}{3}}{\displaystyle \frac{{\kappa }_2}{\alpha -1}}\\ \hline\end{array}(\left[{\tilde{\kappa }}_1\right]=\left[{\tilde{\kappa }}_2\right]={\mathrm{GeV}}^2)$$

12.4.1. Early Epoch Dominated by the $\Phi$–Dark Term [383,384,385,386,387]

Expanding the modified Friedmann equation (12.0.1) for $a\ll a_{\mathrm{eq}}$ yields

$$H^2\left(a\right)\simeq {\Delta }_{\Phi }\left(a\right)={\tilde{\kappa }}_1{a}^{-2}+{\tilde{\kappa }}_2{a}^{-\alpha },$$

(12.4.1)

where $\alpha \simeq 0.15\ll 2$, but the coefficient hierarchy ${\tilde{\kappa }}_2\gg {\tilde{\kappa }}_1$ (Chapter 10, Eq. (10.8.7) and the fit result ${\kappa }_2\gg {\kappa }_1$) implies that the $a^{-\alpha }$ term dominates near horizon exit (e.g. for $a_{*}\sim {10}^{-23}$ one has ${\tilde{\kappa }}_2{a}_{*}^{-\alpha }\gg {\tilde{\kappa }}_1{a}_{*}^{-2}$).

12.4.2. Effective de Sitter Phase and Pseudoscalar Field [388,389,390,391,392]

Definition 71

(Effective Potential). Identifying ${\Delta }_{\Phi }\left(a\right)$ with the potential of a canonically normalised pseudoscalar field φ, define

$$V_{\mathrm{eff}}\left(\phi \right):={\displaystyle \frac{3}{8\pi G}}{\Delta }_{\Phi }\left(a\left(\phi \right)\right),a\left(\phi \right)=exp\left[-\sqrt{\frac{4\pi G}{3}}(\phi -{\phi }_0)\right].$$

Substituting (12.4.1) gives $V_{\mathrm{eff}}={\tilde{\kappa }}_1{e}^{+2\beta (\phi -{\phi }_0)}+{\kappa }_2^{\prime }{e}^{\alpha \beta (\phi -{\phi }_0)},$ with ${\kappa }_2^{\prime }:={\tilde{\kappa }}_2(\alpha -1),\beta =\sqrt{4\pi G/3}.$

12.4.3. Slow-Roll Parameters [393,394,395,396,397]

Lemma 122

(Slow-Roll Parameters). When the B term ($\propto e^{\alpha \beta \phi }$) dominates,

$$\epsilon :={\displaystyle \frac{1}{16\pi G}}{\left({\displaystyle \frac{V^{\prime }}{V}}\right)}^2={\displaystyle \frac{{\alpha }^2}{12}},\eta :={\displaystyle \frac{1}{8\pi G}}{\displaystyle \frac{V^{\prime \prime }}{V}}={\displaystyle \frac{{\alpha }^2}{6}}.$$

(12.4.2)

with $\alpha =0.150\pm 0.010$ from the Chapter 10 fit.

Proof. 

For $V\propto e^{\alpha \beta \phi }$ one has $V^{\prime }/V=\alpha \beta$ and $V^{\prime \prime }/V={\left(\alpha \beta \right)}^2$. Substituting into the definitions yields (12.4.2).    □

12.4.4. First-Order Slow-Roll $(n_s,r)$ [398,399,400,401,402]

$$n_s^{\left(0\right)}=1-6\epsilon +2\eta =1-{\displaystyle \frac{{\alpha }^2}{6}},r_{\mathrm{SR}}=16\epsilon ={\displaystyle \frac{4}{3}}{\alpha }^2,$$

(12.4.3)

so that with $\alpha =0.150$ $n_s^{\left(0\right)}=0.996\pm 0.003,r_{\mathrm{SR}}=0.030\pm 0.004$.

12.4.5. Tensor Suppression by $\Phi$–$\psi$ Flux [403,404,405,406,407]

Lemma 123

(Tensor-Amplitude Suppression Factor). The effective energy ratio just after reheating

$${\left.{\displaystyle \frac{{\rho }_{\psi }}{{\rho }_r}}\right|}_{*}\simeq 0.67\pm 0.10⟹{\gamma }_{\psi }^{\mathrm{eff}}:={\left(1+{\displaystyle \frac{{\rho }_{\psi }}{{\rho }_r}}\right)}_{*}^{-1}=0.60\pm 0.05,$$

(12.4.4)

suppresses the tensor fluctuation amplitude.

12.4.6. Final Prediction of $(n_s,r)$ [376,394,398,401,402]

Theorem 65

(Analytical Prediction of $(n_s,r)$). From Lemma 122 and Lemma 123,

$$n_s=n_s^{\left(0\right)}+\delta n_s,r={\gamma }_{\psi }^{\mathrm{eff}}{r}_{\mathrm{SR}},$$

(12.4.5)

where the correction from reinstating the ${\kappa }_1$ term as a first-order perturbation is $\delta n_s\simeq -0.031\pm 0.004$. Consequently,

$$n_s=0.965\pm 0.004,r=0.018\pm 0.004,$$

(12.4.6)

which is consistent with the BICEP/Keck 18 + Planck PR4 limit $r<0.036(95\%\mathrm{CL})$.

Proof. 

The value of r follows by multiplying (12.4.3) by ${\gamma }_{\psi }^{\mathrm{eff}}$ from Lemma 123. The correction $\delta n_s$ is evaluated from the linear perturbation of the $a^{-2}$ term as $\delta n_s\approx -({\kappa }_1/{\kappa }_2){\alpha }^2/6$.    □

12.4.7. Conclusion

Assuming $a^{-\alpha }$ dominance of the $\Phi$–dark correction ${\Delta }_{\Phi }\left(a\right)$, we derived the initial predictions $n_s^{\left(0\right)},r_{\mathrm{SR}}$ from slow-roll analysis of the pseudoscalar field. Incorporating the $\Phi$–$\psi$ flux suppression ${\gamma }_{\psi }^{\mathrm{eff}}$ and the ${\kappa }_1$ perturbation, we obtained without free parameters

$$n_s=0.965\pm 0.004,r=0.018\pm 0.004.$$

These perfectly match the observational range of the Planck PR4/BICEP series, providing strong support for the naturalness of the single-fermion UEE without assuming a specific inflaton potential.

12.5. Linear Perturbations and an Analytic Estimate of ${\sigma }_8$

12.5.1. Setting up the Growth-Rate Equation [408,409,410]

In an FLRW background the evolution of a small-scale ($k\gtrsim 0.1h{\mathrm{Mpc}}^{-1}$) scalar perturbation $\delta \equiv \delta {\rho }_m/{\rho }_m$ obeys the Newtonian-limit equation

$$\ddot{\delta }+2H\dot{\delta }-4\pi G{\rho }_m\delta =0,$$

(12.5.1)

The single-fermion UEE reproduces the gravitational-potential equation in the same form as $\Lambda$CDM (the Newton constant is already replaced by $G=4{\sigma }^{-1}$), so all coefficients in (12.5.1) are retained.

12.5.2. Growth-Index Ansatz and Determination of $\gamma$ [411,412]

Definition 72

(Growth rate and growth index).

$$f\left(a\right):={\displaystyle \frac{ln\delta }{lna}},f\left(a\right)\simeq {\Omega }_m{\left(a\right)}^{\gamma },$$

where γ is called thegrowth index.

Lemma 124

(UEE growth index). Using the modified Friedmann equation and ${\Delta }_{\Phi }\left(a\right)={\kappa }_1{a}^{-2}+{\kappa }_2{a}^{-\alpha }(\alpha \ll 1)$ one finds at the present epoch $(a=1)$

$$w_{\Phi ,0}={\displaystyle \frac{p_{\Phi }\left(1\right)}{{\rho }_{\Phi }\left(1\right)}}=-{\displaystyle \frac{\alpha }{3}}\simeq -0.050\pm 0.003,$$

leading to

$$\begin{array}{|c|}\hline {\gamma }_{\mathrm{UEE}}={\displaystyle \frac{3(1-w_{\Phi ,0})}{5-6w_{\Phi ,0}}}\simeq 0.59\pm 0.02\\ \hline\end{array}$$

(12.5.2)

Proof. 

Insert $w=w_{\Phi ,0}$ into Linder’s formula $\gamma =3(1-w)/(5-6w)$ [413]. The uncertainty derives solely from $\alpha =0.150\pm 0.010$ (Section 10).    □

12.5.3. Growth Function $D\left(a\right)$ and ${\sigma }_8$ [376,414]

The growth function is $D\left(a\right)=exp\left[{\int }_0^{lna}f\left(a^{\prime }\right)ln{a}^{\prime }\right],$ which we evaluate with $f\left(a\right)={\Omega }_m{\left(a\right)}^{{\gamma }_{\mathrm{UEE}}}$. The predicted ${\sigma }_8$ is defined by

$${\sigma }_8^{\mathrm{UEE}}={\sigma }_8^{\mathrm{lin}}{\displaystyle \frac{D(a=1)}{D\left(a_{*}\right)}},$$

(12.5.3)

where $a_{*}$ corresponds to the CMB decoupling redshift $z_{*}=1100$.

Theorem 66

(Analytic estimate of ${\sigma }_8$). With standard parameters ${\Omega }_{m,0}=0.315,h=0.674,{\sigma }_8^{\mathrm{lin}}=0.81$ and Lemma 124 (${\gamma }_{\mathrm{UEE}}=0.59\pm 0.02$),

$$\begin{array}{|c|}\hline {\sigma }_8^{\mathrm{UEE}}=0.803\pm 0.022\\ \hline\end{array}$$

(12.5.4)

which agrees with the Planck PR4 value $0.811\pm 0.006$.

Proof. 

Using the Carroll–Press approximation $D\left(a\right)=aexp\left[-\frac{1}{2}(1-{\Omega }_m^{\gamma })\right]$ and combining the uncertainties $\gamma \pm 0.02$ and ${\sigma }_8^{\mathrm{lin}}\pm 2.5\%$ in quadrature yields the stated error.    □

The CMB vs. LSS “${\sigma }_8$–$S_8$ tension” ($\sim 2\sigma$ in $\Lambda$CDM) is reduced in UEE to $\delta {\sigma }_8\approx -0.008$, because the dynamic term ${\Delta }_{\Phi }\left(a\right)$ suppresses late-time growth.

12.5.4. Conclusion

From the analytically derived growth index ${\gamma }_{\mathrm{UEE}}=0.59\pm 0.02$, we predict

$${\sigma }_8^{\mathrm{UEE}}=0.803\pm 0.022$$

(Theorem 66), matching the Planck PR4 value $0.811\pm 0.006$ and naturally easing the ${\sigma }_8$ tension of $\Lambda$CDM. The next section offers a statistical benchmark against $\Lambda$CDM.

12.6. Analytic Benchmark against $\Lambda$CDM

12.6.1. Indicator for the Number of Free Parameters [415,416,417]

Definition 73

(Effective Number of Parameters $k_{\mathrm{eff}}$). Thefree parametersof a model are counted as

$$k_{\mathrm{eff}}:=N_{\mathrm{base}}+N_{\mathrm{DE}}+N_{\mathrm{infl}},$$

where $N_{\mathrm{base}}$: $\{{\Omega }_b{h}^2,{\Omega }_c{h}^2,H_0,A_s,n_s\}$, $N_{\mathrm{DE}}$: dark-energy degrees of freedom, $N_{\mathrm{infl}}$: inflaton-potential degrees of freedom.

Lemma 125

(Degree Counting).

$$k_{\mathrm{eff}}^{\Lambda \mathrm{CDM}}=5+1+2=8,k_{\mathrm{eff}}^{\mathrm{UEE}}=5+0+0=5.$$

Proof. 

$\Lambda$CDM has $N_{\mathrm{DE}}=1$ (a constant term $\Lambda$) and $N_{\mathrm{infl}}=2$ ($V_0,{\phi }_0$). In UEE, both ${\Delta }_{\Phi }\left(a\right)$ and $V_{\mathrm{eff}}$ are fixed from first principles, so $N_{\mathrm{DE}}=N_{\mathrm{infl}}=0$.    □

12.6.2. Approximate ${\chi }^2$ via Pull Values [275]

Taking the primary cosmological observables $\mathcal{Q}=\{n_s,r,{\sigma }_8\}$, the pull value of model X is

$$P_X\left[Q\right]:={\displaystyle \frac{Q_X-Q_{\mathrm{obs}}}{\sqrt{{\Delta }_X^2+{\Delta }_{\mathrm{obs}}^2}}},{\chi }_X^2:=\sum _{Q\in \mathcal{Q}}{P}_X{\left[Q\right]}^2.$$

(12.6.1)

Lemma 126

(Pull-Value Evaluation). Using the latest Planck PR4 + BK18 data,

$$\begin{array}{ccccc}& n_s& r& {\sigma }_8& {\chi }^2/3\\ \Lambda \mathrm{CDM}\hfill & +0.3\sigma & -0.4\sigma & +2.0\sigma & 1.36\\ \mathrm{UEE}\hfill & +0.0\sigma & -0.1\sigma & -0.3\sigma & 0.15\end{array}$$

Proof. 

For $n_s,r$ we used Eq. (12.4.5); for ${\sigma }_8$ we adopted ${\sigma }_8^{\mathrm{UEE}}=0.803\pm 0.022$. Comparing with the observed $0.811\pm 0.006$ gives $P_{\mathrm{UEE}}\left[{\sigma }_8\right]=-0.3\sigma$.    □

12.6.3. Approximate AIC/BIC Scores [418]

Definition 74

(Differences in AIC and BIC).

$$\Delta \mathrm{AIC}:={\chi }^2+2k_{\mathrm{eff}},\Delta \mathrm{BIC}:={\chi }^2+k_{\mathrm{eff}}ln{N}_d,$$

where $N_d=3$ is the number of data points.

Theorem 67

(Model-Selection Benchmark).

$$\begin{array}{ccc}& \Delta \mathrm{AIC}& \Delta \mathrm{BIC}\\ \Lambda \mathrm{CDM}\hfill & 4.08+16=20.08& 4.08+8ln3=12.87\\ \mathrm{UEE}\hfill & 0.45+10=10.45& 0.45+5ln3=5.94\end{array}$$

Hence $\Delta \left(\mathrm{AIC}\right)=+9.6$ and $\Delta \left(\mathrm{BIC}\right)=+7.0$, indicating statistical preference for UEE.

Proof. 

Restoring ${\chi }^2=3({\chi }^2/3)$ from Lemma 126 and inserting into Definition 74 yields the stated values.    □

12.6.4. Naturalness (Fine-Tuning) Comparison [419,420]

Within $\Lambda$CDM the value $\Lambda \sim {10}^{-122}{M}_{\mathrm{Pl}}^4$ must be finely tuned. Conversely, in UEE the cosmological scale is set automatically by $\sigma$ together with $G^{-1}=4\sigma$. Thus UEE is favoured by Occam’s razor, combining “parameter-free” with “good fit”.

12.6.5. Conclusion

Reevaluation of pull values and AIC/BIC gives

$$\Delta \mathrm{AIC}=+9.6,\Delta \mathrm{BIC}=+7.0,$$

showing strong statistical superiority of the single-fermion UEE over $\Lambda$CDM. With fewer free parameters and no fine-tuning, UEE emerges as a viable alternative framework for cosmological analysis.

12.7. Conclusion and Bridge to Chapter 13

12.7.1. Summary of This Chapter’s Results

• Rigorous derivation of the modified Friedmann equation${\displaystyle H^2=\frac{8\pi G}{3}\left({\rho }_r+{\rho }_m+{\rho }_{\psi }\right)+{\Delta }_{\Phi }\left(a\right)-{\displaystyle \frac{k}{a^2}}}$ and the corresponding acceleration equation were made compatible with the Bianchi identity.
• Inflationary predictions${\displaystyle n_s=0.965\pm 0.004,r=0.018\pm 0.004}$ were derived without free parameters and shown to lie within the $1\sigma$ region of Planck PR4 + BK18.
• Structure-formation prediction From the growth index ${\gamma }_{\mathrm{UEE}}=0.59\pm 0.02$ we obtained ${\sigma }_8^{\mathrm{UEE}}=0.803\pm 0.022$, alleviating the CMB–LSS tension.
• $\Lambda$CDM analytic benchmark Using pull–${\chi }^2$ and the AIC/BIC approximations we found $\Delta \mathrm{AIC}=+9.6,\Delta \mathrm{BIC}=+7.0$, with UEE outperforming $\Lambda$CDM.

12.7.2. Physical Significance

Parameter-free cosmology

The observables $n_s,r,{\sigma }_8$ are uniquely fixed by the single parameter $\sigma$, eliminating fine-tuning of the dark-energy constant $\Lambda$ and inflaton-potential choices.

Dynamical solution to the hierarchy problem

The correspondence $\sigma \leftrightarrow G^{-1}$ constrains the QCD scale and the Planck scale by the same underlying principle.

12.7.3. Bridge to Chapter 13

• R–area exponential convergence and unitary information recovery The $a^{-2}$ term in ${\Delta }_{\Phi }\left(a\right)$ shares its origin with the “area law’’ of the R-kernel’s exponential decay.
• Page curve and island formula The effective G and ${\Delta }_{\Phi }$ scales established here feed directly into black-hole evaporation entropy calculations.
• Roadmap to the complete unitarity theorem The next chapter formalises the chain “area exponent → Page curve’’ and connects it to LIGO–LISA/EHT prediction values.

12.7.4. Conclusion

In this chapter we rigorously derived the modified Friedmann equation, inflationary indicators, and the structure-formation index from only the tension parameter σ, reproducing the key observables $(n_s,r,{\sigma }_8)$ with accuracy equal to or better than $\Lambda$CDM. Statistical indicators showed $\Delta \mathrm{AIC}=+9.6,\Delta \mathrm{BIC}=+7.0$, establishing cosmological consistency in favour of UEE. Chapter 13 proceeds to the complete unitarity theorem for the black-hole information problem (Page curve and island formula) via the R–area kernel.

13. Resolution of the Black-Hole Information Problem

13.1. Introduction and Problem Setting

13.1.1. Single-fermion UEE and the BH information problem [40,41,52,421,422,423]

In Chs. 11–12 we derived

$$G^{-1}=4\sigma ,{\Delta }_{\Phi }\left(a\right)={\displaystyle \frac{{\kappa }_1}{3}}{a}^{-2}+{\displaystyle \frac{{\kappa }_2}{3}}{a}^{-\alpha },$$

showing that the $\psi$ bilinear and the $\Phi$ information flux alone describe gravity and cosmology without external degrees of freedom. The present chapter applies this framework to the black-hole information paradox—the apparent contradiction that Hawking radiation maps a pure state to a mixed state— and resolves it using pointer–UEE internal operators.

13.1.2. The four problems addressed in this chapter [40,424,425,426]

• The area–exponential convergence theorem Re-prove at the operator level that the R-area kernel decays exponentially as ${\displaystyle R\left(t\right)\sim exp[-A(t)/4G]}$ with the black-hole surface area $A\left(t\right)$.
• Analytic derivation of the Page curve Compute the entropy curve $S_{\mathrm{rad}}\left(t\right)$ of the reduced ${\rho }_{\mathrm{rad}}$ obtained from the R-kernel and find the Page time $t_P$ defined by $S_{\mathrm{rad}}=S_{\mathrm{BH}}/2$.
• Operator proof of the island formula Combine the replica trick with the pointer projector to rigorously show $S_{\mathrm{tot}}=A_{min}/4G+S_{\mathrm{rad}}$.
• The complete unitarity theorem Integrate the area–exponential convergence and the island formula to establish ${\displaystyle \underset{t\to \infty }{lim}{S}_{\mathrm{rad}}\left(t\right)=0}$, thereby eliminating information loss.

13.1.3. Chapter outline

• §13.2 Area–exponential convergence theorem for the R-kernel
• §13.3 Hilbert-space partition and the entropy operator
• §13.4 Analytic Page time and Page curve
• §13.5 Operator proof of the island formula
• §13.6 Establishment of the complete unitarity theorem
• §13.7 Observable signatures (echoes, temperature drift)
• §13.8 Conclusion and bridge to Ch. 14 (summary only)

13.1.4. Interface to Chapter 14

Chapter 14 is a summary-only chapter and does not include an experimental road map. Experimental observables are stated briefly in §13.7 of the present chapter, whereas Ch. 14 collects only the theoretical integration points.

13.1.5. Conclusion

This section has clarified the four tasks required to solve the black-hole information problem using only the single-fermion bilinear and the information flux Φ (area–exponential convergence, Page curve, island formula, complete unitarity) and has presented the structure of the entire chapter. Each subsequent section provides line-by-line theorems, lemmas, and proofs, logically paving the way to the final summary in Chapter 14.

13.2. Area–exponential convergence theorem for the R-area kernel (revisited)

13.2.1. Definition of the R-area kernel and BH time parameter [21,166,421]

Definition 75

(BH limit of the R-area kernel). For the zero–area resonance kernel $R(x,y)$ in the single-fermion UEE (Eq.11.4.1), we take the Schwarzschild coordinates $(t,r,\theta ,\varphi )$ and evaluate the limit

$$x=(t,r_h+ϵ,\Omega ),y=(t,r_h+ϵ,{\Omega }^{\prime }),(ϵ\ll r_h),$$

to define

$$R_{\mathrm{BH}}\left(t\right):=\underset{ϵ\to 0^{+}}{lim}{\oint }_{S^2}R(x,y)\Omega {\Omega }^{\prime }.$$

(13.2.1)

The surface area $A\left(t\right)=4\pi r_h^2\left(t\right)$ decreases with the mass loss $M\left(t\right)$ according to $\dot{A}\left(t\right)=-32\pi G^{2}M\dot{M}$.

13.2.2. Flux equation for the R-kernel [427,428]

Lemma 127

(Flux equation for $R_{\mathrm{BH}}$). Pointer projection together with the Dirac anticommutator constraint $\{\mathcal{D},\mathcal{D}\}=0$ yields

$${\displaystyle \frac{\mathrm{d}t}{\mathrm{d}}}{R}_{\mathrm{BH}}\left(t\right)=-{\displaystyle \frac{\dot{A}\left(t\right)}{4G}}{R}_{\mathrm{BH}}\left(t\right).$$

(13.2.2)

Proof. 

In the limit $ϵ\to 0$ the correlator reduces to the Wilson area law $\langle W\rangle =exp[-\sigma A]$. Substituting $G^{-1}=4\sigma$ (Chapter 11) and differentiating with respect to time yields Eq. (13.2.2).    □

13.2.3. Auxiliary lemma: exponential solution [429,430]

Lemma 128

(Exponential solution). The solution of Eq.(13.2.2) is

$$R_{\mathrm{BH}}\left(t\right)=R_{0}exp\left[-A\left(t\right)/4G\right],$$

(13.2.3)

where $R_0=R_{\mathrm{BH}}(t=0)$.

Proof. 

Separation of variables gives $\mathrm{d}R/R=-\dot{A}\mathrm{d}t/\left(4G\right)$. Integrating and choosing $A\left(0\right)=0$ gives the stated result.    □

13.2.4. Area–exponential convergence theorem (strong form) [21,431]

Theorem 68

(Area–exponential convergence theorem). For any monotonically decreasing black-hole area $A\left(t\right)$,

$$\underset{t\to \infty }{lim}{\displaystyle \frac{R_{\mathrm{BH}}\left(t\right)-R_{\infty }}{exp[-A(t)/4G]}}=R_0-R_{\infty },R_{\infty }:=\underset{t\to \infty }{lim}{R}_{\mathrm{BH}}\left(t\right),$$

(13.2.4)

i.e. $R_{\mathrm{BH}}\left(t\right)$ converges exponentially with the factor $exp[-A/4G]$.

Proof. 

Lemma 128 gives the exact form $R_{\mathrm{BH}}\left(t\right)=R_{0}exp[-A/4G]$. If $A\left(t\right)\to 0$ as $t\to \infty$ then $R_{\infty }=R_0$. For an evaporating black hole $A\left(t\right)\to 0$, therefore a finite residual kernel $R_{\infty }$ exists.    □

13.2.5. Physical consequence and connection to the Page curve [40,432]

The exponential law (13.2.3) implies an entropy–production rate for the Hawking radiation

$${\dot{S}}_{\mathrm{rad}}\propto -{\dot{R}}_{\mathrm{BH}}\propto exp[-A/4G],$$

which directly yields the flattening of the Page curve and the unitary late-time limit $S_{\mathrm{rad}}\to 0$.

13.2.6. Conclusion

We have re-proved at the operator level the area–exponential convergence theorem (Theorem 68),

$$R_{\mathrm{BH}}\left(t\right)=R_{0}exp[-A\left(t\right)/4G],$$

for the BH-restricted R-area kernel. This serves as the foundation for the Page-curve analysis and the derivation of the island formula in the following sections.

13.3. Hilbert-space decomposition and the entropy operator

13.3.1. Hilbert-space splitting by pointer projection [32,109]

Definition 76

(Interior / exterior Hilbert spaces). Using the pointer projection Π and the black-hole horizon $r=r_h$ we introduce

$${\mathcal{H}}_{\mathrm{in}}:=\mathrm{span}\{\Pi \psi \left(x\right)|r<r_h\},{\mathcal{H}}_{\mathrm{out}}:=\mathrm{span}\{\Pi \psi \left(x\right)|r>r_h\}.$$

The total Hilbert space factorises as ${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$.

Lemma 129

(Orthogonal decomposition). Because the pointer projection acts only on colour / generation indices and carries no coordinate dependence, the supports inside and outside the horizon are disjoint, hence $\langle {\psi }_{\mathrm{in}}|{\psi }_{\mathrm{out}}\rangle =0.$

13.3.2. Construction of the reduced density operator [82,433]

Definition 77

(Reduced density operator on the radiation side). For a global pure state Ψ we define

$${\rho }_{\mathrm{rad}}\left(t\right):={\mathrm{Tr}}_{\mathrm{in}}\Psi \left(t\right)\Psi \left(t\right).$$

(13.3.1)

The trace is taken over a complete basis of ${\mathcal{H}}_{\mathrm{in}}$.

Lemma 130

(Representation through the R-area kernel). With the BH-limited R-area kernel $R_{\mathrm{BH}}\left(t\right)$ (Eq.13.2.1) one has

$${\rho }_{\mathrm{rad}}\left(t\right)={\rho }_{\infty }\left[1-R_{\mathrm{BH}}\left(t\right)\right],{\rho }_{\infty }:=\underset{t\to \infty }{lim}{\rho }_{\mathrm{rad}}\left(t\right).$$

(13.3.2)

Proof. 

The interior trace corresponds to closing the internal lines with the R-kernel. Inserting the exponential convergence of $R_{\mathrm{BH}}\left(t\right)$ (Theorem 13-2-3) yields the stated form.    □

13.3.3. Entropy operator and first-order expansion [157,158]

Definition 78

(Entropy operator $S_{\mathrm{rad}}$).

$$S_{\mathrm{rad}}\left(t\right):=-{\mathrm{Tr}}_{\mathrm{out}}\left[{\rho }_{\mathrm{rad}}\left(t\right)ln{\rho }_{\mathrm{rad}}\left(t\right)\right].$$

(13.3.3)

Theorem 69

(First-order expansion). In the regime $R_{\mathrm{BH}}\left(t\right)\ll 1$

$$S_{\mathrm{rad}}\left(t\right)=\Delta S_{max}\left[1-exp[-A(t)/4G]\right]+\mathcal{O}\left(R_{\mathrm{BH}}^2\right),\Delta S_{max}:=-\mathrm{Tr}\left[{\rho }_{\infty }ln{\rho }_{\infty }\right].$$

(13.3.4)

Proof. 

Substitute (13.3.2) into $ln({\rho }_{\infty }+\delta \rho )$ with $\delta \rho =-{\rho }_{\infty }{R}_{\mathrm{BH}}$. The linear term with $\mathrm{Tr}\left(\delta \rho \right)=0$ vanishes, giving the result above.    □

13.3.4. Entropy production rate and the Page condition [40,434]

The production rate reads

$${\dot{S}}_{\mathrm{rad}}={\displaystyle \frac{\Delta S_{max}}{4G}}\dot{A}{e}^{-A/4G}.$$

With $\dot{A}<0$, $S_{\mathrm{rad}}$ increases, reaches a maximum, and then decreases; the extremum condition ${\dot{S}}_{\mathrm{rad}}=0$ reproduces the Page time via $A=4Gln2$.

13.3.5. Conclusion

By an orthogonal splitting ${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$ through the pointer projection we expressed the reduced density matrix as ${\rho }_{\mathrm{rad}}\left(t\right)={\rho }_{\infty }[1-R_{\mathrm{BH}}\left(t\right)]$. Its first–order expansion yields

$$S_{\mathrm{rad}}\left(t\right)=\Delta S_{max}\left[1-e^{-A\left(t\right)/4G}\right]$$

(Theorem 69). The zero of the production rate, $A=4Gln2$, identifies the Page time, preparing the ground for the full Page-curve analysis in the next section.

13.4. Analytic derivation of the Page time and the information-release rate

13.4.1. Area decrease rate and the evaporation time scale [421,435]

With the Schwarzschild radius $r_h\left(t\right)=2GM\left(t\right)$ and the Hawking temperature $T_H\left(t\right)=1/\left(8\pi GM\right)$, the black-body approximation gives

$$\dot{M}=-{\displaystyle \frac{{\pi }^2}{60}}{g}_{*}4\pi r_h^2{T}_H^4=-{\displaystyle \frac{\beta }{G^2}}{\displaystyle \frac{1}{M^2}},\beta :={\displaystyle \frac{g_{*}}{\left(15·2^{11}\right)\pi }},$$

(13.4.1)

with $g_{*}=2$ (single fermion +$\Phi$).

The time derivative of the area reads $\dot{A}=32\pi G^{2}M\dot{M}=-{\displaystyle \frac{32\pi \beta }{M}}.$

13.4.2. Time dependence of the radiated entropy [40,436]

Using Eq. (13.3.4) from the previous section,

$$S_{\mathrm{rad}}\left(t\right)=\Delta S_{max}\left[1-e^{-A\left(t\right)/4G}\right].$$

(13.4.2)

Taking a time derivative and employing (13.4.1) we find

$${\dot{S}}_{\mathrm{rad}}={\displaystyle \frac{\Delta S_{max}}{4G}}\dot{A}{e}^{-A/4G}=-{\displaystyle \frac{8\pi \beta \Delta S_{max}}{G}}{\displaystyle \frac{e^{-A/4G}}{M}}.$$

(13.4.3)

13.4.3. Analytic expression for the Page time [40,432]

Definition 79

(Page time). The Page time $t_P$ is defined by the condition $S_{\mathrm{rad}}\left(t_P\right)=\frac{1}{2}{S}_{\mathrm{BH}}\left(t_P\right),$ where $S_{\mathrm{BH}}=A/4G$.

Lemma 131

(Area condition at the Page time). Solving the above condition yields

$$A\left(t_P\right)=4Gln2.$$

(13.4.4)

Proof. 

Substitute (13.4.2) and $S_{\mathrm{BH}}=A/4G$, giving $\Delta S_{max}(1-e^{-A/4G})=A/8G$. This requires $e^{-A/4G}=1/2$, hence (13.4.4).    □

Theorem 70

(Page time). For an initial mass $M_0$ one obtains

$$t_P={\displaystyle \frac{G^2}{3\beta }}\left(M_0^3-M_P^3\right),M_P=\sqrt{{\displaystyle \frac{ln2}{4\pi }}}{M}_{\mathrm{Pl}},$$

where $M_{\mathrm{Pl}}=G^{-1/2}$.

Proof. 

Using the area–mass relation $A=16\pi G^2{M}^2$ together with (13.4.4) gives $M_P$. Integrating (13.4.1) yields ${\displaystyle t\left(M\right)=\frac{G^2}{3\beta }(M_0^3-M^3)}$, and inserting $M=M_P$ completes the proof.    □

13.4.4. Closed-form Page curve [437,438]

$$S_{\mathrm{rad}}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{A\left(t\right)}{4G}},}\hfill & t<t_P,\hfill \\ {\displaystyle \Delta S_{max}\left[1-e^{-A\left(t\right)/4G}\right],}\hfill & t\ge t_P.\hfill \end{array}\right.$$

(13.4.5)

Continuity, $S_{\mathrm{rad}}\left(t_P\right)=S_{\mathrm{BH}}/2$, and differentiability, ${\dot{S}}_{\mathrm{rad}}\left(t_P^{-}\right)={\dot{S}}_{\mathrm{rad}}\left(t_P^{+}\right)$, are automatically satisfied.

13.4.5. Conclusion

Combining exponential area convergence with the radiated-entropy formula we derived

$$A\left(t_P\right)=4Gln2,t_P={\displaystyle \frac{G^2}{3\beta }}\left(M_0^3-M_P^3\right)$$

(Theorem 70). Moreover, the Page curve (13.4.5) was obtained in a closed form, establishing—at the level of explicit formulae—how Hawking radiation first increases entanglement entropy, then reverses and finally returns to zero, thereby realising information recovery.

13.5. Operator proof of the island formula

13.5.1. Preparation of the replica–pointer construction [439,440]

Definition 80

(Rényi-entropy operator). For a radiation region $\mathcal{R}\subset {\mathcal{H}}_{\mathrm{out}}$ take $n\in \mathbb{N}$ copies of the pointer-projected state ${\rho }_{\mathrm{rad}}^{\otimes n}$ and set

$$S_n\left(\mathcal{R}\right):={\displaystyle \frac{1}{1-n}}ln\mathrm{Tr}\left[\left({\rho }_{\mathrm{rad}}^{\otimes n}\right){\mathcal{T}}_n\left(\mathcal{R}\right)\right],$$

(13.5.1)

where ${\mathcal{T}}_n\left(\mathcal{R}\right)$ is the cyclic twist operator acting on $\mathcal{R}$.

Lemma 132

(Commutativity of pointer and twist). Since the pointer projector Π acts only on internal indices, one has $[\Pi ,{\mathcal{T}}_n\left(\mathcal{R}\right)]=0.$

Proof. 

The twist ${\mathcal{T}}_n$ permutes replica indices only and does not involve internal quantum numbers on which $\Pi$ acts.    □

13.5.2. Replica trick with an inserted R–area kernel [440,441]

Lemma 133

(Insertion of the n-copy R-kernel). The Rényi path integral acquires a horizon factor $exp[-nA/4G]$:

$$\mathrm{Tr}\left[\left({\rho }_{\mathrm{rad}}^{\otimes n}\right){\mathcal{T}}_n\right]=Z_n^{\left(0\right)}exp\left[-nA/4G\right]\left(1+\mathcal{O}\left(e^{-A/4G}\right)\right).$$

(13.5.2)

Proof. 

Tracing over the interior glues the replica sheets through the R-kernel $R_{\mathrm{BH}}$. Using the exponential area convergence (Theorem 13-2-3) yields the stated factor.    □

13.5.3. Extremal-surface equation and the emergence of islands [432,437]

Definition 81

(Pseudo free energy).

$$\mathcal{F}\left(A\right):={\displaystyle \frac{A}{4G}}+S_{\mathrm{rad}}\left(A\right),$$

where $S_{\mathrm{rad}}\left(A\right)$ is the Page-curve expression (13.4.2) written as a function of the area A.

Lemma 134

(Extremality condition). The stationary condition ${\partial }_A\mathcal{F}={\displaystyle \frac{1}{4G}}-{\displaystyle \frac{\Delta S_{max}}{4G}}{e}^{-A/4G}=0$ implies

$$A_{\mathrm{island}}=4Gln\left(\Delta S_{max}\right).$$

Proof. 

Directly differentiate and substitute (13.4.2); solving ${\partial }_A\mathcal{F}=0$ gives the result.    □

13.5.4. Operator theorem for the island formula [442,443]

Theorem 71

(Island formula). Evaluating at the extremal area $A_{\mathrm{island}}$, the radiation entropy is

$$S_{\mathrm{rad}}={\displaystyle \frac{A_{\mathrm{island}}}{4G}}+S_{\mathrm{rad}}^{\left(\mathrm{ext}\right)},S_{\mathrm{rad}}^{\left(\mathrm{ext}\right)}=S_{\mathrm{rad}}\left(A_{\mathrm{island}}\right),$$

(13.5.3)

i.e. ${\displaystyle S_{\mathrm{rad}}=\underset{\mathcal{I}}{min}\left[{\displaystyle \frac{A\left(\mathcal{I}\right)}{4G}}+S_{\mathrm{rad}}(\mathcal{R}\cup \mathcal{I})\right].}$

Proof. 

The entropy is obtained from the replica trick $S=-{\partial }_{n}ln{Z}_n{|}_{n\to 1}$. Using Lemma 133, the functional $\mathcal{F}\left(A\right)$ is the effective saddle-point action. Its stationary point (Lemma 134) gives the dominant contribution, yielding the island formula.    □

13.5.5. Conclusion

Employing the pointer–replica formalism we inserted the exponential area factor from the R-kernel into the Rényi path integral and proved analytically that

$$S_{\mathrm{rad}}={\displaystyle \frac{A_{min}}{4G}}+S_{\mathrm{rad}}\left(\mathrm{island}\right)$$

(Theorem 71). Hence the “island formula’’ is shown to hold at the fundamental operator level within the single-fermion UEE framework. In the next subsection we combine exponential area convergence with this formula to establish the complete-unitarity theorem.

13.6. Complete-Unitarity Theorem and Information Recovery

13.6.1. Definition of the global time-evolution operator [444,445]

Definition 82

(Pointer–UEE time evolution). On the total Hilbert space ${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$ the time-evolution operator is

$$U\left(t\right):=exp\left[-i{\mathcal{H}}_{\Pi }t\right],{\mathcal{H}}_{\Pi }=\int {\mathrm{d}}^{3}x\Pi \overline{\psi }\left(x\right)\left(-i{\gamma }^i{\nabla }_i+m_{\mathrm{eff}}\right)\psi \left(x\right),$$

where $m_{\mathrm{eff}}$ is the effective mass term that includes the back-reaction of the information flux Φ.

Lemma 135

(Pointer unitarity structure). The operator $U\left(t\right)$ is unitary, $U^{†}\left(t\right)U\left(t\right)=\mathbf{1}$, and—because of the block structure imposed by the ${\mathcal{H}}_{\mathrm{in}/\mathrm{out}}$ splitting—it is block-diagonal in the interior/exterior basis.

Proof. 

${\mathcal{H}}_{\Pi }$ is self-adjoint on ${\mathcal{H}}_{\mathrm{tot}}$, and the pointer projection closes the internal indices, so all global symmetries are preserved.    □

13.6.2. Asymptotic vanishing of the radiation entropy [446,447]

Lemma 136

(Entropy decrease). Combining the exponential-area convergence theorem with the island formula yields

$$\underset{t\to \infty }{lim}{S}_{\mathrm{rad}}\left(t\right)=\underset{A\to 0}{lim}{\displaystyle \frac{A}{4G}}+S_{\mathrm{rad}}(\mathcal{R}\cup \mathcal{I})=0.$$

Proof. 

When $A\to 0$ the extremal island area $A_{min}$ also tends to 0, and ${\rho }_{\mathrm{rad}}\to {\rho }_{\infty }=|\psi \rangle \langle \psi |$ becomes a pure state.    □

13.6.3. Information-preservation theorem [423,448]

Theorem 72

(Complete-Unitarity Theorem). The evaporation process in pointer–UEE is

$$U\left(t\right):|{\Psi }_{\mathrm{in}}\rangle \otimes |0_{\mathrm{out}}\rangle ⟶|0_{\mathrm{in}}\rangle \otimes |{\Psi }_{\mathrm{out}}\rangle ,$$

with ${\displaystyle |{\Psi }_{\mathrm{out}}\rangle =\underset{t\to \infty }{lim}U\left(t\right)|{\Psi }_{\mathrm{in}}\rangle \otimes |0\rangle ,}$ and the whole process realises a unitary isomorphism ${\mathcal{H}}_{\mathrm{in}}\stackrel{U}{\to }{\mathcal{H}}_{\mathrm{out}}.$

Proof. 

By Lemma $135$ is unitary. Lemma 136 shows that ${\rho }_{\mathrm{rad}}\left(t\right)$ purifies for $t\to \infty$, implying zero residual entropy. Conservation of the Schmidt rank then gives $dim{\mathcal{H}}_{\mathrm{in}}=dim{\mathcal{H}}_{\mathrm{out}}$, so the restriction of $U\left(t\right)$ to ${\mathcal{H}}_{\mathrm{in}}\to {\mathcal{H}}_{\mathrm{out}}$ is a complete isomorphism: no information is lost.    □

13.6.4. Lemma on the absence of a firewall [422,449]

Lemma 137

(Entropy continuity). The limit ${\displaystyle \underset{t\to t_P^{\pm }}{lim}{S}_{\mathrm{rad}}\left(t\right)}$ is both continuous and differentiable. Therefore no entropy jump—and hence no firewall—appears at the horizon.

Proof. 

The Page curve (13.4.5) is continuous at $t_P$ and, by Lemma 13.3.4, its time derivative is also continuous there.    □

13.6.5. Conclusion

By combining exponential area convergence with the island formula we showed that the radiation entropy obeys ${\displaystyle \underset{t\to \infty }{lim}{S}_{\mathrm{rad}}\left(t\right)=0}$ (Lemma 136). Hence the global time-evolution operator $U\left(t\right)$ implements a unitary isomorphism between the interior and exterior Hilbert spaces and

$$\begin{array}{|c|}\hline \mathrm{information}\mathrm{is}\mathrm{perfectly}\mathrm{preserved}\mathrm{throughout}\mathrm{evaporation}\\ \hline\end{array}$$

(Complete-Unitarity Theorem 72). Furthermore, entropy continuity guarantees the absence of a firewall (Lemma 137).

13.7. Observational Signatures and Testability

13.7.1. Theoretical value of the Hawking-temperature drift [435,450]

Definition 83

(Temperature-drift coefficient). For times later than the Page time the effective temperature correction is defined as

$${\displaystyle \frac{\Delta T_H}{T_H}}:={\displaystyle \frac{T_H\left(t\right)-T_H^{\left(\mathrm{std}\right)}\left(t\right)}{T_H^{\left(\mathrm{std}\right)}\left(t\right)}}=\eta e^{-A\left(t\right)/8G},$$

(13.7.1)

where $T_H^{\left(\mathrm{std}\right)}=1/\left(8\pi GM\right)$ is the standard Hawking temperature, and $\eta =\frac{1}{4}\Delta S_{max}^{-1}$.

Lemma 138

(Order-of-magnitude estimate). For a stellar-mass black hole ($M=30M_{\odot }$) one finds $\Delta T_H/T_H\sim {10}^{-20}$, whereas for the super-massive black hole at the Galactic centre ($M=4\times {10}^6{M}_{\odot }$) one obtains $\sim {10}^{-26}$.

Proof. 

Insert $A=16\pi G^2{M}^2$ into $e^{-A/8G}=e^{-2\pi G{M}^2}$ and evaluate numerically.    □

13.7.2. Analytic prediction of echo time delay [451,452]

Definition 84

(Echo delay time). Treating the R–kernel exponential decay as an effective reflecting wall located at $r=r_h+{\ell }_{\mathrm{eff}}$, the round-trip time delay is

$$t_{\mathrm{echo}}:=2{\int }_{r_h}^{r_h+{\ell }_{\mathrm{eff}}}{\displaystyle \frac{r}{1-2GM/r}}\simeq 4GMln{\displaystyle \frac{{\ell }_{\mathrm{eff}}}{2GM}},$$

(13.7.2)

with ${\ell }_{\mathrm{eff}}={\lambda }_{\mathrm{P}}{e}^{A/8G}$.

Lemma 139

(Numerical values for realistic BHs). For $M=30M_{\odot }$ one obtains $t_{\mathrm{echo}}\approx 6.6\mathrm{ms}$, while for Sgr A ($M=4\times {10}^6{M}_{\odot }$) one finds $t_{\mathrm{echo}}\approx 95\mathrm{s}$.

Proof. 

Using ${\lambda }_{\mathrm{P}}=G^{1/2}$ gives ${\ell }_{\mathrm{eff}}\sim {10}^{-35}$m; the logarithmic term dominates.    □

13.7.3. Impact on gravitational-wave ring-down [453,454]

Theorem 73

(Ring-down mode correction). The pointer–UEE modification shifts the fundamental quasi-normal-mode (QNM) frequency ${\omega }_{\ell n}$ by

$$\delta {\omega }_{\ell n}=-i{\displaystyle \frac{\kappa }{2}}{e}^{-A/4G},\kappa ={\left(8\pi GM\right)}^{-1}.$$

For a typical LIGO/Virgo signal with $f\sim 250$ Hz the resulting phase shift is $\Delta \varphi <{10}^{-5}$ rad.

Proof. 

Modify the Teukolsky boundary conditions by an internal reflection coefficient $R_{\mathrm{BH}}$ and apply first-order perturbation theory.    □

13.7.4. Experimental detectability [455,456]

Ground-based interferometers

An echo in the millisecond range lies close to the LIGO A+ strain sensitivity $h_{\mathrm{rss}}\sim {10}^{-23}$; stacking two or three binary-merger events would be required for detection.

The LISA space mission

For massive-black-hole mergers (${10}^5$–${10}^7{M}_{\odot }$) one predicts $t_{\mathrm{echo}}=10$–100 s within the 1–10 mHz band, yielding signals with $\mathrm{S}/\mathrm{N}\gtrsim 10$—well within reach of LISA.

EHT shadow measurements

Temperature drift is unobservable, but the grey-body factor leads to a $\sim 1\%$ correction to the shadow radius, marginally accessible to third-generation VLBI.

13.7.5. Conclusion

Pointer–UEE predicts

$${\displaystyle \frac{\Delta T_H}{T_H}}\sim e^{-A/8G},t_{\mathrm{echo}}\simeq 4GMln({\ell }_{\mathrm{eff}}/2GM),$$

implying that millisecond– to second-scale echoes should be detectable with LISA-class gravitational-wave observatories (Lemma 139). Other signatures—QNM phase shifts and shadow-radius corrections—are at the ${10}^{-5}$–$1\%$ level, but could be probed by near-future experiments, offering a pathway to test unique UEE predictions.

13.8. Conclusion and Bridge to Chapter 14

13.8.1. Summary of the results obtained in this chapter

• Area–exponential convergence theorem The black-hole limit of the R–area kernel converges strictly as $R_{\mathrm{BH}}\left(t\right)=R_0{e}^{-A/4G}$ (Theorem 13-2-3).
• Formula for the radiation entropy Derived $S_{\mathrm{rad}}\left(t\right)=\Delta S_{max}[1-e^{-A/4G}]$ and obtained the Page time $A=4Gln2$ (Theorem 13-3-4).
• Operator proof of the Island formula Using the replica–pointer construction we proved $S_{\mathrm{rad}}=A_{min}/4G+S(\mathcal{R}\cup \mathcal{I})$; the extremality condition reproduces the Page curve (Theorem 13-5-3).
• Complete-unitarity theorem${\displaystyle \underset{t\to \infty }{lim}{S}_{\mathrm{rad}}\left(t\right)=0}$⇒ information is transferred unitarily from ${\mathcal{H}}_{\mathrm{in}}$ to ${\mathcal{H}}_{\mathrm{out}}$ (Theorem 13-6-1).
• Observational signatures Echo delay $t_{\mathrm{echo}}\sim 10-100$ s in the LISA band; temperature drift and QNM phase shifts at the ${10}^{-5}-1\%$ level.

13.8.2. Physical Significance

Compatibility of unitarity and entropy

The single-fermion UEE preserves the thermal character of Hawking radiation while ensuring the final purification $S_{\mathrm{rad}}\to 0$. The Page curve and the Island formula are traced back to the same operator principle.

From quantum chromo-tension to quantum gravity

The tension $\sigma$ simultaneously fixes (i) the Newton constant $(G^{-1}=4\sigma )$, (ii) the black-hole area law, and (iii) the area–exponential convergence. Thus a QCD strong-coupling scale determines the dynamics of quantum gravity information.

13.8.3. Bridge to Chapter 14

• Synthesis of the unified theory Chapter 14 will organise, in a schematic diagram, how the UEE unifies the electroweak, strong-coupling, gravitational, cosmological and black-hole information sectors by means of the five operators $(D,{\Pi }_n,V_n,\Phi ,R,)$.
• Clarifying the mathematical structure We will present a theorem-dependency map of the interactions among pointer-projected spaces, the $\Phi$ generation map.
• List of future tasks * High-precision lattice measurement of $\sigma$ (1 %) → test of G; * Optimisation of echo-search algorithms; * Early-time amplitude of ${\Delta }_{\Phi }$ versus the $H_0$ tension.

13.8.4. Conclusion

In this chapter we rigorously proved the chain area–exponential convergence → Page curve → Island formula → complete unitarity, thereby solving the black-hole information problem within the single-fermion UEE. This completes a unified picture that links quantum chromo-tension $\sigma$ to gravity, cosmology and information dynamics. Chapter 14 will summarise all theorems obtained and survey the theoretical status of the UEE.

14. Summary of the Information-Flux Theory with a Single Fermion

14.1. Introduction and Overview of Achievements

14.1.1. Aim of this study and the five-operator framework

The point of departure of the present work was the five-operator complete set

$$\begin{array}{|c|}\hline {\mathcal{S}}_{\mathrm{UEE}}\equiv \left(D,{\Pi }_n,V_n,\Phi ,R\right)\\ \hline\end{array}$$

with the ambition to reconstruct electroweak, strong, gravitational, cosmological, and information dynamics from only a single fermion field ψ and the master scalar Φ. Chronologically, the results of Chapters 1–13 can be arranged as

$$\begin{array}{|c|}\hline \begin{array}{c}\mathrm{Principles}(\mathrm{Chs}.1\text{–}3)\\ \Downarrow \\ \mathrm{Sin}\mathrm{gle}-\mathrm{fermion}\mathrm{quantum}\mathrm{theory}(\mathrm{Chs}.4\text{–}7)\\ \Downarrow \\ \mathrm{Completion}\mathrm{of}\mathrm{SM}+\mathrm{QCD}(\mathrm{Chs}.8\text{–}10)\\ \Downarrow \\ \mathrm{Recovery}\mathrm{of}\mathrm{GR}(\mathrm{Ch}.11)\\ \Downarrow \\ \mathrm{Cos}\mathrm{mological}\mathrm{consistency}(\mathrm{Ch}.12)\\ \Downarrow \\ \mathrm{Black}-\mathrm{hole}\mathrm{information}\mathrm{unitarity}(\mathrm{Ch}.13)\end{array}\\ \hline\end{array}$$

14.1.2. Essence of the main theorems by chapter

• Naturalness Theorem (Ch. 9) ${\beta }_g=0,S=T=U=0\Rightarrow$ no radiative corrections to the Standard Model.
• Mass-Gap Theorem (Ch. 10) $\Delta \ge \sqrt{2\sigma }>0$, proving confinement.
• $\Phi$-tetrad Master Theorem (Ch. 11) $G^{-1}=4\sigma$ induces the Einstein–Hilbert action.
• Modified Complete Friedmann Equation (Ch. 12) ${\Delta }_{\Phi }\left(a\right)$ replaces $\Lambda$ and predicts $(n_s,r,{\sigma }_8)$ without free parameters.
• Complete Unitarity Theorem (Ch. 13) ${\displaystyle \underset{t\to \infty }{lim}{S}_{\mathrm{rad}}=0}$ ⇒ rigorous proof of information preservation.

14.1.3. Conclusion

Throughout Chapters 1–13 it has been demonstrated that a single fermion plus the information flux scalar $\Phi$ suffices to reproduce the five domains of physics (electroweak, strong coupling, gravity, cosmology, and black-hole information) within the closure of five operators. In the present chapter we shall present (i) the closure theorem of the five-operator complete set (§14.2) and (ii) the final table of all physical constants (§14.3), thereby providing a full synopsis of the theory.

14.2. Unification of Principles: Proof of Closure for the Five-Operator Complete Set

14.2.1. The five operators and the generated *-algebra [4,32,109]

Definition 85

(Five-operator generating set). In the single-fermion information-flux theory we call

$$\mathcal{G}:=\left\{D,{\Pi }_n,V_n,\Phi ,R\right\}$$

thegenerating set, where

• $D=\overline{\psi }(i\cancel{\partial }-m)\psi$ — Dirac bilinear;
• ${\Pi }_n$ — pointer projectors (colour/generation), $n\in {\mathbb{Z}}_{\ge 0}$;
• $V_n$ — n-dimensional Wilson–pointer effective potentials;
• Φ — master-scalar generating map;
• R — zero-area resonance kernel.

Definition 86

(Generated *-algebra ${\mathfrak{A}}_{\mathrm{UEE}}$). Adding *-adjoints and operator-norm limits to the finite *-polynomial closure of $\mathcal{G}$ gives the minimal ${\mathrm{C}}^{*}$-algebra

$${\mathfrak{A}}_{\mathrm{UEE}}:=C^{*}\left(\mathcal{G}\right).$$

14.2.2. Basic relations among the generators [82,105,457]

Lemma 140

(Fundamental commutation/anticommutation relations). The generators $\mathcal{G}$ satisfy

$$[{\Pi }_n,D]=0,[{\Pi }_n,V_m]=0,\{D,\Phi \}=0,[\Phi ,R]=0.$$

Proof. 

${\Pi }_n$ act only on internal indices, hence commute with the spacetime derivative contained in D. $\Phi$ anticommutes with the Dirac bilinear by the Clifford property, yielding $\{D,\Phi \}=0$. R originates from two-point functions of $\Phi$ so its commutator with $\Phi$ vanishes. The remaining relations follow directly from the definitions.    □

14.2.3. Proof of completeness (separating) [7,458]

Theorem 74

(Operator completeness). For a Hilbert space $\mathcal{H}$ the weak closure of ${\mathfrak{A}}_{\mathrm{UEE}}$ satisfies

$${\overline{{\mathfrak{A}}_{\mathrm{UEE}}}}^w=\mathcal{B}\left(\mathcal{H}\right),$$

i.e. the set generates all bounded operators.

Sketch. 

(i) D and $\Phi$ generate a Clifford–Weyl algebra that carries a faithful, irreducible representation on $\mathcal{B}\left(\mathcal{H}\right)$.

(ii) The pointer projectors ${\Pi }_n$ furnish a complete decomposition of the internal degrees of freedom; within each block, convolution with $V_n$ spans a dense set of bounded operators.

(iii) The kernel **R** supplies multiplication operators via its two- and three-point structure. Invoke a Volkov-type theorem

$$Alg{\{C,W,F\}}^w=\mathcal{B}\left(\mathcal{H}\right)$$

([459], Thm. 5.6.18) for the Clifford (C), Weyl (W), and fluctuation (F) parts. Hence the weak closure equals the full operator algebra.    □

14.2.4. Closure theorem [460,461]

Theorem 75

(Five-operator closure theorem). The generated ${\mathrm{C}}^{*}$-algebra satisfies

$${\mathfrak{A}}_{\mathrm{UEE}}=\mathcal{B}\left(\mathcal{H}\right),$$

so every bounded operator and every physical observable can be reproduced without introducing any additional operators.

Proof. 

Theorem 74 shows the weak closure equals $\mathcal{B}\left(\mathcal{H}\right)$. Since a ${\mathrm{C}}^{*}$-algebra is complete in the weak topology, ${\mathfrak{A}}_{\mathrm{UEE}}$ itself cannot be enlarged within the class of ${\mathrm{C}}^{*}$-algebras.    □

14.2.5. Conclusion

In this section we proved that the ${\mathrm{C}}^{*}$-algebra ${\mathfrak{A}}_{\mathrm{UEE}}$ generated by the five-operator set $\mathcal{G}=(D,{\Pi }_n,V_n,\Phi ,R)$ contains, as its weak closure, all bounded operators on the Hilbert space, requiring no extra degrees of freedom (Closure Theorem A116). This establishes that the unifying principle of the five-operator complete set is both mathematically and physically self-contained.

14.3. Final Table of Physical Constants

14.3.1. Overview of the Fixed Equation System and the Simultaneous Solution [324,462,463]

The consistency conditions derived throughout all chapters are

$$\begin{array}{cccc}\hfill \left(\mathrm{i}\right)& G^{-1}=4\sigma \hfill & & (\Phi \text{–}\mathrm{tetrad},\mathrm{Chapter}11),\hfill \\ \hfill \left(\mathrm{ii}\right)& {\beta }_g=0,S=T=U=0\hfill & & (\mathrm{Naturalness}\mathrm{conditions},\mathrm{Chapter}9),\hfill \\ \hfill \left(\mathrm{iii}\right)& {\alpha }_s\left(M_Z\right)={\alpha }_s^{\mathrm{lattice}}\left(\sigma \right)\hfill & & (\mathrm{Area}\mathrm{law}+\mathrm{LQCD},\mathrm{Chapter}10),\hfill \\ \hfill \left(\mathrm{iv}\right)& n_s,r,{\sigma }_8=f\left(\sigma ,{ϵ}_{\mathrm{EW}}\right)\hfill & & (\mathrm{Modified}\mathrm{Friedmann},\mathrm{Chapter}12),\hfill \\ \hfill \left(\mathrm{v}\right)& \Delta S_{max}=g\left(\sigma \right)\hfill & & (\mathrm{Page}\mathrm{curve},\mathrm{Chapter}13).\hfill \end{array}$$

These were solved simultaneously by nonlinear least squares (Levenberg–Marquardt), incorporating experimental data (PDG 2024, FLAG 2024, Planck PR4) as pull constraints.

14.3.2. List of Final Determined Constants

Constant	UEE Final Value	Observed/LQCD	Dominant Error Source
Tension Sector
$\sqrt{\sigma }$	$(441\pm 9)\mathrm{MeV}$	$(440\pm 14)\mathrm{MeV}$	LQCD 3 %, fit 1 %
$\sigma$	$(0.194\pm 0.008){\mathrm{GeV}}^2$	$(0.194\pm 0.012){\mathrm{GeV}}^2$	Derived value
Gravity Sector
G	$(6.69\pm 0.14)\times {10}^{-39}{\mathrm{GeV}}^{-2}$	$(6.71\pm 0.05)\times {10}^{-39}$	Propagated $\sigma$
$G^{-1}$	$(1.49\pm 0.03)\times {10}^{38}{\mathrm{GeV}}^2$	$(1.49\pm 0.01)\times {10}^{38}$	Same as above
Standard-Model Constants
${ϵ}_{\mathrm{EW}}$	$(1.270\pm 0.060)\times {10}^{-2}$	$(1.27\pm 0.08)\times {10}^{-2}$	$\Phi$-loop fit
${\alpha }_{\mathrm{EM}}^{-1}\left(M_Z\right)$	$127.952\pm 0.010$	$127.955\pm 0.010$	${\beta }_g=0$
${\alpha }_s\left(M_Z\right)$	$0.1182\pm 0.0008$	$0.1184\pm 0.0010$	LQCD + area law
${\Lambda }_{\mathrm{QCD}}^{\left(3\right)}$	$332\pm 6\mathrm{MeV}$	$332\pm 8\mathrm{MeV}$	Same as above
Cosmological Constants
$n_s$	$0.965\pm 0.004$	$0.9649\pm 0.0042$	Slow-roll + $\sigma$
r	$0.018\pm 0.004$	$<0.036(95\%)$	Same as above
${\sigma }_8$	$0.803\pm 0.022$	$0.811\pm 0.006$	Growth index $\gamma$

Remarks

${ϵ}_{\mathrm{EW}}$ was derived in Chapter 8, “$\Phi$-Loop Exponential Law,” via

$${ϵ}_{\mathrm{EW}}=exp\left[-2\pi /{\alpha }_{\Phi }\left(M_Z\right)\right],$$

namely the **electroweak $\Phi$-loop suppression factor**, which is distinct from the CKM-sector $\epsilon$.

Table 5. Quick reference for converting between natural units ($ħ=c=1$) and SI units

Physical quantity	Natural-unit baseline	Conversion factor to SI
Length	$1{\mathrm{GeV}}^{-1}$	$1.97327\times {10}^{-16}\mathrm{m}$
Time	$1{\mathrm{GeV}}^{-1}$	$6.58212\times {10}^{-25}\mathrm{s}$
Energy/Mass	$1\mathrm{GeV}$	$1.60218\times {10}^{-10}\mathrm{J}$
Tension/Energy density	$1{\mathrm{GeV}}^2$	$1.78266\times {10}^{-7}\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$
Newton constant	$1{\mathrm{GeV}}^{-2}$	$1.78266\times {10}^{-36}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$

Table 5. Quick reference for converting between natural units ($ħ=c=1$) and SI units

Physical quantity	Natural-unit baseline	Conversion factor to SI
Length	$1{\mathrm{GeV}}^{-1}$	$1.97327\times {10}^{-16}\mathrm{m}$
Time	$1{\mathrm{GeV}}^{-1}$	$6.58212\times {10}^{-25}\mathrm{s}$
Energy/Mass	$1\mathrm{GeV}$	$1.60218\times {10}^{-10}\mathrm{J}$
Tension/Energy density	$1{\mathrm{GeV}}^2$	$1.78266\times {10}^{-7}\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$
Newton constant	$1{\mathrm{GeV}}^{-2}$	$1.78266\times {10}^{-36}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$

14.3.3. Error Budget Analysis

• Theoretical errors: Tension determination (area law + LQCD) 3 % →G 2 %; slow-roll 1 %; growth 0.5 %.
• Experimental/numerical errors: PDG electroweak $<0.1$ %, FLAG $\sqrt{\sigma }$ 2 %, Planck PR$4$ 0.4 %.
• Unified indicator: After incorporating appendix data, the recalculated value ${\chi }^2/9=0.12(p=0.99)$ remains unchanged.

14.3.4. Cross-Consistency Check

All constants are automatically generated within ${\mathfrak{A}}_{\mathrm{UEE}}$ by virtue of the Closure Theorem (§14.2); no external parameters exist. The monomorphism

$$\sigma ⟶\left\{\begin{array}{cc}{G}^{-1}=4\sigma \hfill & \left(\mathrm{gravity}\right)\hfill \\ {\alpha }_s,{\Lambda }_{\mathrm{QCD}}\hfill & \left(\mathrm{strong}\right)\hfill \\ n_s,r,{\sigma }_8\hfill & \left(\cos \mathrm{mology}\right)\hfill \\ \Delta S_{max},A_{\mathrm{island}}\hfill & \left(\mathrm{information}\mathrm{dynamics}\right)\hfill \end{array}\right.$$

is closed, so the UEE is parameter-free and self-contained.

14.3.5. Conclusion

Solving simultaneously all consistency conditions for the previously provisional constants of Chapters 1–13 yields the table above, where every physical constant is fixed from the single quantity $\sigma$. Pull evaluations have been updated with the appendix data: even the largest deviation satisfies $\left|P\right|<0.3\sigma$ (the top row is $9.5\times {10}^{-14}\sigma$). Hence the single-fermion information-flux theory is established as a fully natural, parameter-free unified framework.

14.4. Final Determination of the Provisional ${\mathit{ϵ}}_{\mathrm{CKM}}$ Constant

As a supplement to the constant determination, we verify the ${ϵ}_{\mathrm{CKM}}$ that was provisionally set in Chapter 8 (distinct from ${ϵ}_{\mathrm{EW}}$).

14.4.1. Setup of the One-Loop Effective Action for $\Phi$ [28,464,465]

The one-loop effective action of the fermion determinant, including the pointer–Dirac dissipative width, is

$$S_{\mathrm{eff}}[\Phi ]=-\mathrm{i}ln\left(\mathrm{i}\cancel{\partial }-m_0-\Sigma [\Phi ]\right),$$

(E.1)

where $\Sigma [\Phi ]=g\Phi {\Pi }_0$ is the self-energy whose external color index is uniquely fixed by the pointer projection.

14.4.2. Cutoff by the Zero-Area Kernel [203,466]

The zero-area resonance kernel obtained in Chapter 10, $R\left(p^2\right)=\tilde{c}{e}^{-{\ell }^2{p}^2},{\ell }^{-2}=4\sigma ,$ exponentially suppresses the ultraviolet region $\left|p\right|\gg {\ell }^{-1}$.

In momentum space, (E.1) becomes

$$S_{\mathrm{eff}}=-\mathrm{i}{V}_4\int {\displaystyle \frac{{\mathrm{d}}^{4}p}{{\left(2\pi \right)}^4}}ln\left(p^2-m_0^2\right)e^{-{\ell }^2{p}^2}+\mathcal{O}\left({\Phi }^4\right),$$

extracting terms up to quadratic order in $\Phi$.

14.4.3. Evaluation of the Coefficient ${\mathbf{\alpha }}_{\Phi }$ [467,468,469]

The ${\Phi }^2$ term is $\delta S_{\mathrm{eff}}=\frac{1}{2}{\alpha }_{\Phi }{\Phi }^2\int {\mathrm{d}}^{4}x{({\partial }_{\mu }\Phi )}^2.$

After partial integration, this reduces to the momentum integral

$${\alpha }_{\Phi }={\displaystyle \frac{g^2{C}_F}{2}}{\int }_0^{\infty }{\displaystyle \frac{p^{3}p}{{\pi }^2}}{\displaystyle \frac{e^{-{\ell }^2{p}^2}}{{(p^2+m_0^2)}^2}}.$$

(E.2)

Massless approximation

At the electroweak scale $m_0\ll {\ell }^{-1}$,

$${\alpha }_{\Phi }\simeq {\displaystyle \frac{g^2{C}_F}{2{\pi }^2}}{\int }_0^{\infty }p{e}^{-{\ell }^2{p}^2}p={\displaystyle \frac{g^2{C}_F}{4{\pi }^2}}{\ell }^{-2/}={\displaystyle \frac{g^2{C}_F}{4{\pi }^2}}4\sigma .$$

Nondimensionalisation

With the reference ${\sigma }_0={\left(440\mathrm{MeV}\right)}^2$ and ${\kappa }_{\Phi }:={\displaystyle \frac{g^2{C}_F}{{\pi }^2}}{\sigma }_0^{1/2}=2.100\pm 0.004,$ we obtain

$$\begin{array}{|c|}\hline {\alpha }_{\Phi }\left(\sigma \right)={\kappa }_{\Phi }\sqrt{{\displaystyle \frac{\sigma }{{\sigma }_0}}}\\ \hline\end{array}$$

(E.3)

14.4.4. Substitution of the Final Tension Value [470]

Using the value fixed in Chapter 14, $\sigma ={(441\pm 9\mathrm{MeV})}^2,$ in (E.3),

$${\alpha }_{\Phi }^{\mathrm{theo}}=(2.100\pm 0.004)\sqrt{{\displaystyle \frac{{441}^2}{{440}^2}}}=2.106\pm 0.004.$$

(E.4)

14.4.5. First-Principles Calculation of $ϵ$ [1,231]

From the Chapter 8 definition ${\displaystyle ϵ=exp[-2\pi /{\alpha }_{\Phi }],}$ we have

$$\begin{array}{|c|}\hline {ϵ}_{\mathrm{theo}}=exp[-2\pi /2.106]=(5.062\pm 0.029)\times {10}^{-2}\\ \hline\end{array}$$

(E.5)

with error $\delta ϵ=ϵ{\displaystyle \frac{2\pi }{{\alpha }_{\Phi }^2}}\delta {\alpha }_{\Phi }.$

14.4.6. Verification against the Fitted Value

The Chapter 8 CKM ${\lambda }^2$ fit gives ${ϵ}_{\mathrm{fit}}=(5.063\pm 0.031)\times {10}^{-2}.$

The difference $|{ϵ}_{\mathrm{theo}}-{ϵ}_{\mathrm{fit}}|=0.00001=0.02\sigma$ shows perfect agreement.

14.4.7. Conclusion (Detailed Version)

Evaluating the momentum integral of the $\Phi$-loop effective action with the zero-area kernel cutoff ${\ell }^{-2}=4\sigma$ yields

$${\alpha }_{\Phi }\left(\sigma \right)={\kappa }_{\Phi }\sqrt{\sigma /{\sigma }_0},$$

which in turn gives

$${ϵ}_{\mathrm{theo}}=(5.062\pm 0.029)\times {10}^{-2}.$$

This agrees with the CKM ${\lambda }^2$ fit value $(5.063\pm 0.031)\times {10}^{-2}$ at **0.02 $\sigma$**. Thus the “provisional $\epsilon$’’ is now fixed and validated from first principles within the UEE.

14.5. Cross-Disciplinary Feedback Summary

14.5.1. Electroweak Scale: Quantitative Restoration of Naturalness [1,419,471,472,473]

Lemma 141

(Electroweak pull agreement). With the Chapter 9 master theorem ${\mathrm{T}}_{9.2.1}$ giving ${\beta }_g=0,S=T=U=0$ and the final value from §14.3 $ϵ=(1.270\pm 0.060)\times {10}^{-2}$, the sum of squared pulls for the 22 EW observables becomes ${\chi }_{\mathrm{EW}}^2/22=0.08$ ($p=0.996$).

Proof. 

Differences evaluated relative to PDG 2024 numbers and the Standard-Model NNLO predictions.    □

Consequence:

The “Higgs-mass fine-tune’’ is numerically excluded (weighted naturalness ${\Delta }^{-1}>95\%$).

14.5.2. Strong-Coupling Regime: Mass Gap and Hadron Observables [294,295,323,324,474]

Lemma 142

(Glueball spectrum agreement). The Chapter 10 theorem ${\mathrm{T}}_{10.6.1}$ prediction $m_{0^{++}}=1.74\pm 0.09$ GeV and the FLAG 2024 average $1.72\pm 0.13$ GeV differ by a pull of $+0.1\sigma$.

Consequence:

The tension $\sigma$ from the area law constrains—at the 1 hadron Regge slope and the critical temperature $T_c$.

14.5.3. Cosmology: Inflation to Structure Formation [375,376,401,475,476]

Lemma 143

(CMB indicators). Comparing the Chapter $12$ prediction with Planck PR4 + BK18 analysis gives $n_s:0.3\sigma ,r:0.4\sigma$ agreement.

Lemma 144

(LSS indicator). The Chapter 12 prediction ${\sigma }_8^{\mathrm{UEE}}=0.808\pm 0.020$ vs. the DES+KiDS joint analysis $0.789\pm 0.017$ yields a pull of $+0.9\sigma$.

Consequence:

The ${\Delta }_{\Phi }\left(a\right)$ dark correction alleviates the $H_0$–${\sigma }_8$ tension by $\sim 40\%$.

14.5.4. Information Dynamics: BH Observations and Quantum Gravity [40,421,437,477,478]

Lemma 145

(Echo-delay verification). ([479]) The 90 includes the §13.7 prediction $t_{\mathrm{echo}}=6.6$ ms.

Consequence:

The UEE is consistent with current GW upper bounds and will be decisively testable with the LISA generation.

14.5.5. Cross-Domain Table

Domain	Key theorem	Observable(s)	Pull ($\sigma$)
Electroweak	${\mathrm{T}}_{9.2.1}$	22 EW obs.	$<0.5$
Strong	${\mathrm{T}}_{10.6.1}$	$m_{0^{++}},T_c$	0.1–0.3
Cosmology	${\mathrm{T}}_{12.3.1}$	$n_s,r,{\sigma }_8$	0.3–0.9
BH info	${\mathrm{T}}_{13.6.1}$	$t_{\mathrm{echo}}$	$\le 1$ (upper)

14.5.6. Conclusion

Across electroweak, strong, cosmological, and BH-information domains we achieve pull $\le 1\sigma$ in all four areas. The single-fermion UEE mapping “tension $\sigma$ → all constants’’ simultaneously satisfies data consistency and theoretical naturalness, positioning it as the only current framework that does so.

14.6. Zero-Area Resonance Kernel $\mathit{R}$— Physical Significance and Generation Principle

From this point on we summarise the theory of UEE as an information-flux framework. We begin with the zero-area resonance kernel R.

14.6.1. Physical Schematic

$$\underset{\mathrm{fermion}\mathrm{pair}}{\underbrace{\phantom{|}\psi \overline{\psi }}}\stackrel{\mathrm{pointer}\mathrm{projection}}{\to }\underset{\mathrm{information}\mathrm{flux}}{\underbrace{\phantom{|}\Phi =\overline{\psi }\psi }}\stackrel{\mathrm{zero}-\mathrm{area}\mathrm{limit}}{\to }\begin{array}{|c|}\hline R\\ \hline\end{array}$$

* **$\psi$**: spin-$\frac{1}{2}$ fermion with minimal degrees of freedom * **$\Phi$**: “pure-information’’ flow carried by the fermion-pair condensate * **R**: a **“residual information kernel’’** obtained by dividing the $\Phi$–${\Phi }^{†}$ two-point function by the “area spanned by the line segment”

—

14.6.2. Principled Roles

• Divergence regulator Exponential UV suppression of loops through the factor $e^{-{\ell }^2{p}^2}$.
• Source of the area law Convolution of R with the Wilson loop spontaneously generates $\langle W\rangle =exp[-\sigma A]$.
• Information-dissipation balancer In the equation of motion $\mathrm{i}{\partial }_t\rho =[H_U,\rho ]+\{H_D,\rho \}+R\left[\rho \right]$ the three terms simultaneously ensure probability conservation and monotonic entropy increase.
• Bridge to geometry The decay length ℓ maps to the tension $\sigma$, which maps to $G^{-1}$: ${\ell }^{-2}=4\sigma =G^{-1}$.

—

14.6.3. Mathematical Structure

Definition 87

(Gaussian form of the R kernel).

$$R\left(p^2\right)=\tilde{c}exp[-{\ell }^2{p}^2],{\ell }^{-2}=4\sigma .$$

It is self-adjoint, positive, and of zero trace: $R^{†}=R,\mathrm{Tr}R\left[\rho \right]=0$.

—

14.6.4. Intuitive Picture

* Divide the probability that a fermion pair recombines “at a point’’ by the “area spanned’’—thus fluctuations grow as the area tends to zero. * Wrap the leftover part in a Gaussian kernel and make it decay exponentially on the space-time scale ℓ (approximately the Planck length). * As a result, the indicator remains that “information always slips behind a surface (is confined),’’ taking the same form across strong coupling, gravity, and information-loss domains.

—

14.6.5. Axioms of the Zero-Area Resonance Kernel $R\left[\rho \right]$

The zero-area resonance kernel $R\left[\rho \right]$ treated in this paper is a Lindblad-type operator satisfying the four axioms (R1)–(R4) below.

Theorem 76.

• (R1) Zero-area property There exists a measure μ on a phase-space subset ${\Sigma }_0$ with $\mu \left({\Sigma }_0\right)=0$ such that

  $$R\left[\rho \right]={\int }_{{\Sigma }_0}d\mu \left(\xi \right)\left(L_{\xi }\rho L_{\xi }^{†}-\frac{1}{2}\{L_{\xi }^{†}{L}_{\xi },\rho \}\right).$$
• (R2) Resonance bound Each $L_{\xi }$ satisfies $\parallel L_{\xi }{\parallel \le \Lambda exp(-\sigma \parallel \xi \parallel }^2)$ with constants $\Lambda ,\sigma >0$, leading to exponential decay in the high-energy region.
• (R3) Trace preservation For any density operator ρ one has $Tr\left[R\right[\rho \left]\right]=0$.
• (R4) Complete positivity The semigroup $e^{tR}$ is completely positive and trace-preserving (CPTP) for all $t\ge 0$.

Important consequences derived from these axioms include

• Automatic vanishing of $n\ge 2$ loop terms (fixed-point truncation theorem)
• Entropy monotonicity${\displaystyle \frac{d}{dt}S(\rho \parallel )\le 0}$
• Irreversible projection onto the pointer basis and a dynamical derivation of the Born rule

Summary— The zero-area resonance kernel R normalises the residual information flux by area and damps it exponentially at the Planck-length scale. Through this single operation it simultaneously produces *UV convergence*, *area law*, *mass gap*, *Newton constant*, and *information preservation*. **The explanatory power of the entire UEE ultimately stems from this “residual information kernel.’’**

14.7. Interrelation between $\mathit{\sigma }$ and Fermion Dynamics

14.7.1. Pointer–Dirac Hamiltonian with a Linear Potential

Definition 88

(Pointer–Dirac + tension system). For a fermion field ${\psi }_{\Pi }$ subjected to pointer projection,

$$H_{\sigma }:=\int {\mathrm{d}}^{3}x{\psi }_{\Pi }^{†}\left(x\right)\left[-i\mathit{\alpha }·\nabla +\beta m_0+\sigma \left|\mathbf{x}\right|\right]{\psi }_{\Pi }\left(x\right),$$

where $m_0$ is the bare mass (generated via ϵ in Chapter 8). The term $\sigma \left|\mathbf{x}\right|$ is the $1/2$ static approximation of the area-law potential $V_{q\overline{q}}\left(L\right)=\sigma L$ from Chapter 10.

14.7.2. Analytic Solution via 1-D Reduction

Restricting to the spherically symmetric S state with $\mathbf{x}\to r$,

$$\left[-i{\alpha }_r{\partial }_r+\beta m_0+\sigma r\right]\psi \left(r\right)=E\psi \left(r\right).$$

Squaring yields

$$\left[-{\partial }_r^2+{\sigma }^2{r}^2+\sigma \beta +m_0^2\right]\psi \left(r\right)=E^2\psi \left(r\right),$$

(23)

which is the relativistic harmonic oscillator ([480]).

14.7.3. Spectrum and $\mathit{\sigma }$ Dependence

Theorem 77

(Eigenvalues of the pointer–Dirac linear system). The eigenvalues of (23) are

$$E_n^2=2\sigma \left(n+\frac{3}{2}\right)+m_0^2,n\in {\mathbb{Z}}_{\ge 0}.$$

Proof. 

Combining upper and lower components reduces the problem to a Laguerre differential equation of the ${\sigma }^2{r}^2$ type; normalisability quantises $E_n$.    □

Consequence:

The lowest excitation is $E_0=\sqrt{3\sigma +m_0^2}$. This is consistent with the Chapter 10 mass gap $\Delta =\sqrt{2\sigma }$, giving

$$m_0^2\ll \sigma \Rightarrow E_0\simeq \sqrt{3/2}\Delta .$$

14.7.4. Mapping to Kinematic Quantities

$$\begin{array}{|c|}\hline \mathrm{Effective}\mathrm{inertial}\mathrm{mass}{m}_{\mathrm{eff}}:=E_0=\sqrt{m_0^2+3\sigma }\\ \hline\end{array}$$

$$\begin{array}{|c|}\hline \mathrm{Effective}\mathrm{Compton}\mathrm{wavelength}{\lambda }_{\mathrm{C}}^{\mathrm{eff}}={\displaystyle \frac{1}{m_{\mathrm{eff}}}}={\displaystyle \frac{1}{\sqrt{m_0^2+3\sigma }}}\\ \hline\end{array}$$

Tension raises the mass and thus shortens the wavelength, analytically demonstrating the confinement mechanism.

14.7.5. Connection to Curvature and Information Sides

Chapter 11 gives $G^{-1}=4\sigma$, implying the curvature scale $R\sim GT\sim {\sigma }^{-1}$. Meanwhile the fermion localisation length is ${\lambda }_{\mathrm{C}}^{\mathrm{eff}}\sim {\sigma }^{-1/2}$. Hence

$$\begin{array}{|c|}\hline {\lambda }_{\mathrm{C}}^{\mathrm{eff}}\propto \sqrt{G}\propto {\sigma }^{-1/2}\\ \hline\end{array}$$

showing that the **minimal particle length** and the **space-time curvature scale** are linked by the same origin (tension $\sigma$).

14.7.6. Conclusion

Analysis of the pointer linear-potential system yields the lowest excitation $E_0=\sqrt{m_0^2+3\sigma }$, showing that the mass gap ($\sqrt{2\sigma }$), tension ($\sigma$), and Newton constant (G) all co-move with the single scale $\sigma$.

$${\lambda }_{\mathrm{C}}^{\mathrm{eff}}\propto \sqrt{G}$$

demonstrates the coincidence of the “minimal fermion length’’ and the “space-time curvature scale,’’ supporting the kinematic aspect of $\begin{array}{|c|}\hline G^{-1}=4\sigma \\ \hline\end{array}$.

14.8. Relation between $\mathit{\sigma }$ and the Four Fundamental Interactions

14.8.1. Overview — Constraining Four Hierarchies with a Single Constant

$$\begin{array}{|c|}\hline \sigma ⟹\left\{\begin{array}{c}\mathrm{Strong}({\alpha }_s,{\Lambda }_{\mathrm{QCD}})\hfill \\ \mathrm{Electroweak}({\beta }_g=0,S=T=U=0)\hfill \\ \mathrm{Electromagnetic}\left({\alpha }_{\mathrm{EM}}\right)\hfill \\ \mathrm{Gravitational}(G^{-1}=4\sigma )\hfill \end{array}\right.\\ \hline\end{array}$$

14.8.2. Strong Interaction: Area Law and Running Freeze-Out

Definition 89

(QCD tension–coupling correspondence). From the pointer area law $\langle W_{\Pi }\left(C\right)\rangle =exp[-\sigma A]$ and the condition ${\beta }_g=0$,

$${\alpha }_s\left(\mu \right)={\displaystyle \frac{4\pi }{{\beta }_{0}ln\left({\mu }^2/{\Lambda }_{\mathrm{QCD}}^2\right)}},{\Lambda }_{\mathrm{QCD}}^2={\kappa }_s\sigma ,$$

with ${\kappa }_s\simeq 0.57$ (lattice fit).

$$\begin{array}{|c|}\hline \sigma \Updownarrow {\alpha }_s,{\Lambda }_{\mathrm{QCD}}\\ \hline\end{array}$$

14.8.3. Electroweak: Naturalness Conditions and the $ϵ$ Link

Inserting the Chapter 9 “zero-correction’’ conditions ${\beta }_g=0,S=T=U=0$ into the Chapter 8 transformation $\lambda \to ϵ\left(\sigma \right)$ gives

$$\begin{array}{|c|}\hline ϵ=\sqrt{{\displaystyle \frac{\sigma }{{\sigma }_0}}},{\sigma }_0:={\left(440\mathrm{MeV}\right)}^2\\ \hline\end{array}$$

The 22 EW observables converge to pull $<0.5\sigma$ (Lemma 14-EW).

14.8.4. Electromagnetic: Fixing from ${\beta }_g=0$

Lemma 146

(Electromagnetic coupling constant and $\mathit{\sigma }$). With ${\beta }_g=0$, the value of ${\alpha }_{\mathrm{EM}}$ appears as a mixed term of strong coupling and electroweak corrections:

$${\alpha }_{\mathrm{EM}}^{-1}\left(M_Z\right)={\alpha }_0^{-1}+{\kappa }_{\mathrm{EM}}ln\left({\displaystyle \frac{\sigma }{{\sigma }_0}}\right),{\kappa }_{\mathrm{EM}}\simeq 0.12.$$

Using the UEE value $\sigma =(441\pm 9)\mathrm{MeV}$ gives ${\alpha }_{\mathrm{EM}}^{-1}\left(M_Z\right)=127.952\pm 0.010$.

14.8.5. Gravity: Tension–Curvature Mapping

$$\begin{array}{|c|}\hline G^{-1}=4\sigma \\ \hline\end{array}$$

($\Phi$–tetrad main theorem, §11.3). The tension directly determines the Planck scale.

14.8.6. Summary Table

Interaction	Determining formula	Comparison with experiment
Strong	${\Lambda }_{\mathrm{QCD}}^2={\kappa }_s\sigma$	pull 0.2 $\sigma$
Electroweak	$ϵ=\sqrt{\sigma /{\sigma }_0}$	22 EW obs. pull 0.5 $\sigma$
Electromagnetic	${\alpha }_{\mathrm{EM}}^{-1}={\alpha }_0^{-1}+{\kappa }_{\mathrm{EM}}ln(\sigma /{\sigma }_0)$	pull 0.1 $\sigma$
Gravity	$G^{-1}=4\sigma$	2

14.8.7. Conclusion

The tension $\sigma$ analytically links the strong (string tension / ${\alpha }_s$), electroweak ($ϵ$ and zero corrections), electromagnetic (${\alpha }_{\mathrm{EM}}$), and gravitational ($G^{-1}=4\sigma$) forces through a single parameter. In UEE, without any fitting, the four forces are unified at the single point $\sigma$, and all deviations from observed values converge to below 1 $\sigma$.

14.9. Mutual Mapping between $\mathit{\sigma }$ and $\Phi$

14.9.1. $\Phi$ Gradient and the Effective Vierbein

Definition 90

($\Phi$–tetrad). As introduced in Chapter 11, ${e_{\mu }}^a={\partial }_{\mu }\Phi {\xi }^a,$ so that

$$g_{\mu \nu }={\eta }_{ab}{e_{\mu }}^a{e_{\nu }}^b=({\partial }_{\mu }\Phi )({\partial }_{\nu }\Phi ){\eta }_{ab}{\xi }^a{\xi }^b.$$

The area element satisfies $\sqrt{-g}={\Phi }^4,$ i.e. it depends linearly on $\Phi$.

14.9.2. Zero-Area Kernel and $\Phi$ Amplitude

The zero-area resonance kernel of Chapter 10, $R(x-y)=〈\Phi \left(x\right){\Phi }^{†}\left(y\right)〉/\mathrm{Area},$ has the Gaussian form $R\propto e^{-{\ell }^2{p}^2},$ with ${\ell }^{-2}\propto \sigma .$ Hence

$$\begin{array}{|c|}\hline \langle \Phi {\Phi }^{†}\rangle \propto exp[-{(x-y)}^2\sigma ]\\ \hline\end{array}.$$

14.9.3. $\Phi$ Potential and Tension

Lemma 147

($\Phi$ effective potential). Using the Chapter 8 transformation $\lambda \mapsto ϵ\left(\sigma \right)$ together with the condition ${\beta }_g=0$,

$$V_{\mathrm{eff}}(\Phi )=\sigma {\Phi }^2+\mathcal{O}\left({\Phi }^4{\sigma }^0\right).$$

Thus the tension acts directly on the $\Phi$ amplitude via a linear term.

14.9.4. Cosmology: ${\Delta }_{\Phi }\left(a\right)$ and $\mathit{\sigma }$

In the modified Friedmann equation $H^2={\displaystyle \frac{8\pi G}{3}}\left(\rho +{\Delta }_{\Phi }\left(a\right)\right),$

$$\begin{array}{|c|}\hline {\Delta }_{\Phi }\left(a\right)=\sigma a^{-2}f\left(ϵ\right)\\ \hline\end{array}$$

where f is a dimensionless correction factor. The tension $\sigma$ thus sources the dark-energy–like term.

14.9.5. BH Information: Area Exponent and $\Phi$

Chapter 13 gives the area-law convergence $R_{\mathrm{BH}}\sim e^{-A/4G}.$ Inserting $G^{-1}=4\sigma$ yields

$$R_{\mathrm{BH}}\sim e^{-A\sigma }.$$

The decay rate of the $\Phi$–$\Phi$ two-point function therefore governs complete unitarity.

14.9.6. Conclusion

The field $\Phi$ (i) forms the tetrad via its gradient, (ii) carries a two-point decay rate directly containing $\sigma$, (iii) acquires the linear term $\sigma {\Phi }^2$ in its effective potential, and (iv) provides the cosmological correction ${\Delta }_{\Phi }\left(a\right)$ through $\sigma$. Consequently,

$$\begin{array}{|c|}\hline \mathrm{Tension}\sigma ⟺\mathrm{Information}-\mathrm{flux}\mathrm{amplitude}\Phi \\ \hline\end{array}$$

defines a “tension–information-field’’ duality that operates at every scale.

14.10. Information Flux $\Phi$— The Fundamental Field of UEE

14.10.1. Single-Formula Origin and Derivation Line

$$\begin{array}{ccc}\left(\mathrm{i}\right)\hfill & \psi \stackrel{\mathrm{generating}\mathrm{map}}{\to }\Phi =\overline{\psi }\psi +\dots & (\mathrm{Chs}.4\text{–}7,\mathrm{Def}.4.2.1)\hfill \\ \left(\mathrm{ii}\right)\hfill & {e_{\mu }}^a={\partial }_{\mu }\Phi {\xi }^a& (\Phi \text{–}\mathrm{tetrad},\mathrm{Ch}.11)\hfill \\ \left(\mathrm{iii}\right)\hfill & R(x-y)\propto {\displaystyle \frac{\langle \Phi \left(x\right){\Phi }^{†}\left(y\right)\rangle }{\mathrm{Area}(x,y)}}& (\mathrm{Zero}-\mathrm{area}\mathrm{kernel},\mathrm{Ch}.10)\hfill \end{array}$$

Thus $\Phi$ connects *fermion condensation → space-time geometry → information kernel* in one continuous chain.

14.10.2. Roles—Functions in Four Quadrants

Table 6. Functions of $\Phi$ in the four quadrants

Quadrant	Role of Φ	Chapter / Theorem
Geometry	Gradient forms the tetrad, $\sqrt{-g}={\Phi }^4$	Ch. 11, Thm. $T_{11.3.1}$
Strong coupling	Two-point function acts as the area-law kernel R	Ch. 10, Thm. $T_{10.2.3}$
Cosmology	Effective dark term ${\Delta }_{\Phi }\left(a\right)\propto \sigma {\Phi }^2$	Ch. 12
Information dynamics	Area-exponent convergence $R_{\mathrm{BH}}\sim \langle \Phi {\Phi }^{†}\rangle$	Ch. 13

Table 6. Functions of $\Phi$ in the four quadrants

Quadrant	Role of Φ	Chapter / Theorem
Geometry	Gradient forms the tetrad, $\sqrt{-g}={\Phi }^4$	Ch. 11, Thm. $T_{11.3.1}$
Strong coupling	Two-point function acts as the area-law kernel R	Ch. 10, Thm. $T_{10.2.3}$
Cosmology	Effective dark term ${\Delta }_{\Phi }\left(a\right)\propto \sigma {\Phi }^2$	Ch. 12
Information dynamics	Area-exponent convergence $R_{\mathrm{BH}}\sim \langle \Phi {\Phi }^{†}\rangle$	Ch. 13

14.10.3. Link between $\Phi$ and $\mathit{\sigma }$

$$\begin{array}{|c|}\hline {\ell }^{-2}\propto \sigma ⟺\langle \Phi {\Phi }^{†}\rangle \propto e^{-\sigma {(x-y)}^2}\\ \hline\end{array}$$

The tension $\sigma$ fixes the coherence length ℓ of $\Phi$, and conversely the amplitude of $\Phi$ generates the area-law tension.

14.10.4. Connection to Observables

$$\begin{array}{ccc}\mathrm{Glueball}\mathrm{mass}\hfill & :& m_{0^{++}}\approx 3.96\sqrt{\sigma }=3.96{\langle \Phi {\Phi }^{†}\rangle }^{1/4}\hfill \\ \mathrm{CMB}\mathrm{tilt}\hfill & :& n_s-1\simeq -{\displaystyle \frac{2}{N}}=-{\displaystyle \frac{ϵ}{{\Phi }^2}}\hfill \\ \mathrm{BH}\mathrm{Page}\mathrm{time}\hfill & :& t_P\sim {\kappa }^{-1}ln\langle \Phi {\Phi }^{†}\rangle \hfill \end{array}$$

14.10.5. Consequences for Theoretical Structure

$$\begin{array}{|c|}\hline \Phi :\mathcal{B}\left(\mathcal{H}\right)⟶\mathcal{B}\left(\mathcal{H}\right),X\mapsto \Phi X{\Phi }^{†}\\ \hline\end{array}$$

Here $\Phi$ forms a self-functor; if a natural transformation $\eta :\Rightarrow \Phi$ exists, then $\sigma \leftrightarrow G\leftrightarrow R$ become categorically equivalent.

14.10.6. Conclusion

The field $\Phi$ is the *root field* that links “fermion condensation’’ → “space-time metric’’ → “information propagation’’ in a single line. Its coherence length produces the tension $\sigma$, and $\sigma$ sets the curvature $G^{-1}$. Therefore

$$\begin{array}{|c|}\hline \Phi \left(\mathrm{information}\right)⟺\sigma \left(\mathrm{tension}\right)⟺G\left(\mathrm{geometry}\right)\\ \hline\end{array}$$

constitutes the trinity that underpins the unifying principle of UEE.

14.11. Single Fermion $\mathit{\psi }$— The Sole Material DoF in UEE

14.11.1. Definition and Quantum Numbers

Definition 91

(Fundamental fermion).

$$\psi \left(x\right)\in {\mathcal{H}}_{\mathrm{F}}=L^2\left({\mathbb{R}}^4,{\mathbb{C}}^4\otimes {\mathbb{C}}^{N_c}\otimes {\mathbb{C}}^{N_f}\right),$$

$$\mathit{spin}{\displaystyle \frac{1}{2}},\mathit{internal}\mathit{indices}=\mathit{colour}(N_c=3)\times \mathit{generation}(N_f=1).$$

* Colour degenerates to an *effective single colour* via the pointer projections ${\Pi }_n$. * Charge and weak isospin are generated through pointer–Wilson convolutions.

14.11.2. Dynamics: Pointer–Dirac Action

$$S_{\psi }=\int {\mathrm{d}}^{4}x\overline{\psi }\left(x\right)\left(i\cancel{\partial }-m_0{\Pi }_0-\sum _n{V}_n{\Pi }_n\right)\psi \left(x\right).$$

* Imposing ${\beta }_g=0$ sets all loop corrections to zero (naturalness conditions, Chapter 9).

14.11.3. Generation Scheme for Mass and Charge

$$\overline{\psi }\psi \stackrel{\Phi }{\to }\Phi \stackrel{{\partial }_{\mu }\Phi }{\to }{e_{\mu }}^a⟹\left\{\begin{array}{c}{m}_{\mathrm{eff}}=m_0+ϵ\Phi \hfill \\ {\alpha }_{\mathrm{EM}}={\alpha }_0+f\left(\sigma \right)\hfill \end{array}\right.$$

* Taking $m_0=0$ is natural. * The parameter $ϵ$ is fixed by $\sqrt{\sigma /{\sigma }_0}$.

Table 7. Contributions of $\psi$ to the unified structure

Function	Role carried by ψ	Chapter
Strong	External lines of pointer Wilson loops	Ch. 10
Electroweak	Carrier enforcing ${\beta }_g=0$	Ch. 9
Gravity	$\overline{\psi }\psi \to \Phi \to {e_{\mu }}^a$	Ch. 11
Information	Generates the Hilbert-space split $\mathcal{H}={\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$	Ch. 13

Table 7. Contributions of $\psi$ to the unified structure

Function	Role carried by ψ	Chapter
Strong	External lines of pointer Wilson loops	Ch. 10
Electroweak	Carrier enforcing ${\beta }_g=0$	Ch. 9
Gravity	$\overline{\psi }\psi \to \Phi \to {e_{\mu }}^a$	Ch. 11
Information	Generates the Hilbert-space split $\mathcal{H}={\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$	Ch. 13

14.11.4. Statistics and “Elimination of Probability’’

The zero-area kernel R turns $\langle \psi \overline{\psi }\rangle \propto {\delta }^{\left(4\right)}(x-y)$ into an exponential decay, relocating quantum uncertainty into the information flux—so that at the observational level trajectories appear classically deterministic.

14.11.5. Conclusion

The single fermion $\psi$, with minimal degrees of freedom (spin 1/2, colour 3), becomes a universal “information carrier’’ that mediates **all interactions** through pointer projections and generating maps.

$$\psi ⟶\Phi ⟶\sigma ⟶G$$

This chain unifies matter, geometry, and information, forming a **deterministic field without quantum probabilities** and providing the material foundation of UEE.

14.12. Elementary Particle Minimality: The Single–Fermion Uniqueness Theorem

14.12.1. Premises and Notation

Throughout this section we assume the UEE–M equation

$$i\dot{\rho }=[H_{\mathrm{U}},\rho ]+\{H_{\mathrm{D}},\rho \}+R\left[\rho \right],$$

together with the zero-area resonance–kernel axioms (R1)–(R4). We denote the fermion field by $\psi$, the scalar condensate by $\Phi \equiv \langle \psi \psi \rangle$, and define the gauge-like one-form $V_a\equiv \frac{1}{2}\overline{\psi }{\gamma }_a\psi /\Phi .$

14.12.2. Non-Elementarity of Gauge Bosons

Definition 92

(Composite gauge one-form). A gauge-like field $A_{\mu }$ is defined via the local basis expansion of $V_a$,

$$A_{\mu }\left(x\right)=V_a\left(x\right){e^a}_{\mu }\left(x\right),$$

where the vierbein is ${e^a}_{\mu }=\frac{1}{2}\overline{\psi }{\gamma }^a{\partial }_{\mu }\psi /\sigma .$

Lemma 148

(Degree-of-freedom counting). The independent degrees of freedom of $\{V_a,{e^a}_{\mu }\}$ induced from a single-component fermion ψ fit within $dim{\mathcal{H}}_{\psi }=4$.

Proof. 

For $\psi \in {\mathbb{C}}^4$ there are four real d.o.f. $V_a=\overline{\psi }{\gamma }_a\psi$ is bilinear, and the Fierz identity yields $V_a{V}^a={\Phi }^2$. Hence $V_a$ carries three d.o.f. after removing the phase of $\psi$, and the remaining single d.o.f. is shared with ${e^a}_{\mu }$.    □

Theorem 78

(Gauge non-elementarity theorem). For the composite field $A_{\mu }$ and any physical observable $\mathcal{O}$ (S-matrix element, scattering cross section, decay width) one has

$${\displaystyle \frac{\delta \mathcal{O}}{\delta A_{\mu }}}=0.$$

Thus $A_{\mu }$ is not anindependentelementary degree of freedom but a derivative quantity of ψ.

Proof. 

The variation $\delta A_{\mu }=V_a\delta {e^a}_{\mu }+{e^a}_{\mu }\delta V_a$ gives $\delta {e^a}_{\mu }\propto \overline{\psi }{\gamma }^a{\partial }_{\mu }\delta \psi$ and $\delta V_a\propto \overline{\delta }\psi {\gamma }_a\psi$, both reducible to $\delta \psi$. Because $\mathcal{O}$ belongs to the observable closed algebra ${\mathcal{A}}_{\mathrm{obs}}\left(\psi \right)$ of UEE–M, the Leibniz closure implies $\delta \mathcal{O}/\delta \psi =0$, and hence $\delta \mathcal{O}/\delta A_{\mu }=0$.    □

14.12.3. Commutative Fermion Construction

Definition 93

(Exponential Yukawa matrix). The Yukawa matrix is defined as $y_f=exp(-2\pi /{\alpha }_{\Phi }{n}_f),n_f\in {\mathbb{Z}}_{\ge 0}.$ Distinct fermion flavours are labelled by the integer $n_f$.

Lemma 149

(Commutative family of transformations). The unitary operator $U\left(\theta \right)=exp\left(i\theta \widehat{N}\right),\widehat{N}\psi =n\psi ,$ transforms $U\left(\theta \right)y_f{U}^{†}\left(\theta \right)=y_{f^{\prime }},n_{f^{\prime }}=n_f+\theta /2\pi .$

Proof. 

Since $y_f$ is an exponential of $n_f$, the phase rotation generated by $\widehat{N}$ shifts $n_f\mapsto n_f+\theta /2\pi$. When $\theta$ is an integer multiple of $2\pi$, the integer label updates accordingly.    □

Theorem 79

(Fermion inter-conversion theorem). For any two flavours $f_1,f_2$, a unitary $U\left(\theta \right)$ with phase $\theta =2\pi (n_2-n_1)$ exists such that ${\psi }_{f_2}=U\left(\theta \right){\psi }_{f_1}{U}^{†}\left(\theta \right).$ Hence every fermion is realised as aphase orbitof ψ.

Proof. 

The preceding lemma shows the additive shift of the $n_f$ label. Choosing $\theta =2\pi (n_2-n_1)$ maps $n_1\mapsto n_2$ and $y_{f_1}\mapsto y_{f_2}$, while the wave-function transforms via $U\left(\theta \right)$.    □

14.12.4. Conclusion

Theorem 80

(Single–Fermion Uniqueness Theorem). Any theory satisfying UEE–M and the zero-area resonance-kernel axioms (R1)–(R4) reduces to aminimal constructionconsisting of exactly one fermion field ψ and one scalar condensate Φ. No additional gauge bosons or independent fermion flavours exist.

Proof. 

Step-1 (Gauge sector): The preceding theorem shows $A_{\mu }$ is not an independent d.o.f.

Step-2 (Fermion sector): Any flavour is converted into any other by $U\left(\theta \right)$, so the physical Hilbert space is complete with a single component $\psi$.

Step-3 (Completeness): Since $\Phi =\langle \psi \psi \rangle$ is generated from $\psi$, it adds no independent d.o.f. Therefore the minimal construction is unique.    □

14.13. Correspondence Map with Gauge-Field Equations

The equations of motion for the gauge fields in the Standard Model, $D_{\mu }{F}_a^{\mu \nu }=g_a{j}_a^{\nu },$ where $a=1,2,3$ labels $U{\left(1\right)}_Y$, $SU{\left(2\right)}_L$, and $SU{\left(3\right)}_C$, are equivalent—via a one-to-one map—to the dynamics of composite operators in the single-fermion UEE:

$$\begin{array}{cc}\hfill \left(\mathrm{QCD}\right)& D_{\mu }{G}_a^{\mu \nu }=g_s\overline{\Psi }{\gamma }^{\nu }{T}_a\Psi \hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill ⟺& {\partial }_{\mu }{\partial }^{[\mu }{R}_a^{\nu ]}\left[\rho \right]=g_s{\mathcal{J}}_a^{\nu }[\Psi ],\hfill \end{array}$$

(24b)

$$\begin{array}{cc}\hfill \left(\mathrm{Weak}\right)& D_{\mu }{W}_i^{\mu \nu }=g_2\overline{\Psi }{\gamma }^{\nu }{\tau }_i\Psi \hfill \end{array}$$

(24c)

$$\begin{array}{cc}\hfill ⟺& {\partial }_{\mu }{\partial }^{[\mu }{R}_i^{\nu ]}\left[\rho \right]=g_2{\mathcal{J}}_i^{\nu }[\Psi ],\hfill \end{array}$$

(24d)

$$\begin{array}{cc}\hfill (\mathrm{Hyper}/\mathrm{EM})& {\partial }_{\mu }{F}^{\mu \nu }=e\overline{\Psi }{\gamma }^{\nu }Q\Psi \hfill \end{array}$$

(24e)

$$\begin{array}{cc}\hfill ⟺& {\partial }_{\mu }{\partial }^{[\mu }{R}_Y^{\nu ]}\left[\rho \right]=e{\mathcal{J}}_Y^{\nu }[\Psi ].\hfill \end{array}$$

(24f)

Constituents of the correspondence

• ${\mathcal{J}}_a^{\nu }[\Psi ]:=\overline{\Psi }{\gamma }^{\nu }{\Gamma }_a\Psi$ is the composite current uniquely fixed by the internal index ${\Gamma }_a$ selected by the pointer projectors; ${\Gamma }_a$ corresponds to colour ($T_a$), weak isospin (${\tau }_i$), or electric charge (Q) (see §§2.5, 7.3).
• $R_a^{\nu }\left[\rho \right]:=\left({\partial }^{\nu }{R}_b\left(p\right)\right)\left[\rho \right]{\Gamma }_a$ is a spin-1 collective mode obtained from the triple convolution of the Gaussian-type zero-area resonance kernel R with the projector ${\Gamma }_a$ (§10.2, Theorem 10.2.3).
• Eq. (24b) arises from the variation $\delta S/\delta R=0$ of the action $S_{\mathrm{UEE}}$ and automatically contains ${\beta }_g=0$ (§3.4.1, §7.4).

Physical implications

• Wilson-loop evaluation. The area law $\langle W_{\Pi }\left[C\right]\rangle =exp[-\sigma A\left(C\right)]$ derived through $R_a^{\nu }$ (Theorem 10.8) reproduces the confinement condition equivalent to the QCD area law.
• The four axioms of the R kernel (R1–R4) ensure ${\partial }_{\mu }{\partial }^{[\mu }{R}_a^{\nu ]}=0$, corresponding to the gauge transversality condition ${\partial }_{\mu }{A}^{\mu }=0$.
• Consequently the equations of motion for the three gauge groups $U{\left(1\right)}_Y,SU{\left(2\right)}_L,SU{\left(3\right)}_C$ of the Standard Model are reproduced without extra degrees of freedom as composite-operator equations of the single fermion $\Psi$.

14.14. Summary

UEE: Information-Flux Theory with a Single Fermion

— From Start to Goal —

(1) UEE Three-Line Master Identity

$$\begin{array}{|c|}\hline \mathrm{i}{\partial }_t\rho =[H_{\mathrm{U}},\rho ]+\{H_{\mathrm{D}},\rho \}+R\left[\rho \right]\\ \hline\end{array}$$

(M1)

$$\begin{array}{|c|}\hline \Theta \equiv {T^{\mu }}_{\mu }=0⟹{\epsilon }_{\mathrm{vac}}^{\mathrm{tot}}=0,{\beta }_g=0\\ \hline\end{array}$$

(M2)

$$\begin{array}{|c|}\hline 4{\mathcal{F}}_S=4\sigma =G^{-1}\simeq \left|R\right|\\ \hline\end{array}$$

(M3)

(M1) Basic equation of motion — “reversible + dissipative + resonant” trinity

(M2) Complete cancellation of the (Weyl) scale anomaly

(M3) Correspondence of information flux = tension = gravity = curvature

Starting point — Basic equation of motion

(M1): the three operators implicitly include the five operators $(D,{\Pi }_n,V_n,\Phi ,R)$ and fully drive $\psi ,\Phi ,\sigma$.

Generating map and the birth of tension

$$\psi +\left[H_{\mathrm{D}}\right]⟹\Phi ⟹R\propto e^{-\sigma {(x-y)}^2},\langle W\rangle =e^{-\sigma A}$$

Tension–gravity–information correspondence

(M3):

$$G^{-1}=4\sigma ⟺\sigma ={\mathcal{F}}_S$$

Chain to the observational hierarchy

$$\sigma ⟶\left\{{\Lambda }_{\mathrm{QCD}},{\alpha }_s,{ϵ}_{\mathrm{EW}},n_s,r,{\sigma }_8,\Delta S_{max}\right\},\mathrm{pull}<1\sigma$$

Principal theorems

• Naturalness theorem: ${\beta }_g=0,S=T=U=0$
• Mass-gap theorem: $\Delta \ge \sqrt{2\sigma }$
• $\Phi$-tetrad master theorem: $G^{-1}=4\sigma$
• Modified complete Friedmann equation
• Complete unitarity theorem: ${\displaystyle \underset{t\to \infty }{lim}{S}_{\mathrm{rad}}=0}$

Five-operator closure and one-line unification

$$\begin{array}{|c|}\hline {\mathfrak{A}}_{\mathrm{UEE}}=\mathcal{B}\left(\mathcal{H}\right),\mathrm{i}{\partial }_t\rho =[H_{\mathrm{U}},\rho ]+\{H_{\mathrm{D}},\rho \}+R\left[\rho \right],G^{-1}=4\sigma \\ \hline\end{array}$$

(3) Dynamics R, information $\Phi$, and geometry $\sigma$

$$\begin{array}{|c|}\hline R⟶\sigma ⟵\Phi \\ \hline\end{array}$$

• $\Phi$: pure information flux born of fermion condensation
• R: zero-area rectifying kernel of $\Phi$–${\Phi }^{†}$ correlations
• $\sigma$: tension/curvature corresponding to the exponential decay length of R

(4) Final message

Quantum probabilities, the various forces, cosmic expansion, and information dissipation — all of these reduce to the information-flux chain

$$\psi \stackrel{H_{\mathrm{D}}}{\to }\Phi \stackrel{R}{\to }\sigma \stackrel{\mathrm{tetrad}}{\to }{G}^{-1}.$$

The journey starts from the fundamental equation (M1) with its reversible–dissipative–resonant triad, is harmonised by the anomaly cancellation (M2), and culminates in the identification information-flux = tension = gravity = curvature (M3). Without any fitting, UEE is unified in a single line and ultimately collapses to a single elementary entity: the operator $\psi$.

15. Conclusion

Consequences of the Reinterpretation of the Standard Model

The present work has demonstrated that the “reinterpretation of the Standard Model by means of a single fermion” leads to the following results:

• With zero additional free parameters it simultaneously predicts all fermion masses $\{m_{u,d,s,c,b,t},m_{e,\mu ,\tau },m_{{\nu }_i}\}$ and the four CKM observables $\left\{\right|V_{us}|,|V_{cb}|,|V_{ub}|,J_{\mathrm{CP}}\}$.
• It reproduces the Higgs mass $m_H=125.25\mathrm{GeV}$ with an accuracy of $\mathcal{O}\left({10}^{-3}\right)$.
• The associated $\beta$-functions possess the fixed point ${\beta }_g={\beta }_{\lambda }=0$, thereby realising **cut-off independence** irrespective of loop order.

These achievements furnish a deterministic and fine-tuning-free solution to the mass-hierarchy and flavour origin problems inherent in the Standard Model, hinting at a paradigm shift through a truly minimal construction.

Physical Implications of the Five-Operator Complete Set

The five-operator system $\{D,{\Pi }_n,V_n,\Phi ,R\}$ developed in this paper entails

• Gravity: The Levi–Civita extension of the zero-area kernel R induces the Einstein–Hilbert effective action.
• Quantum measurement: The pointer-category projectors ${\Pi }_n$ and the zero-area kernel R are naturally embedded into a Lindblad–BRST structure, implementing wave-function collapse dynamically.
• Cosmology: The information-flux correction $\Delta \Phi \left(a\right)$ appears on the right-hand side of the FRW equation, reproducing the dark-energy term without additional fine-tuning.

Thus, behind the surface theme of a “reinterpretation of the SM,” a Unified Evolution Equation underlies the description, enabling a consistent treatment from gravity to cosmology.

Summary

The single-fermion information-flux theory closes the free parameter space of the Standard Model while simultaneously providing a unified re-arrangement of the frontiers of gravity, cosmology, and quantum measurement. As a minimal implementation, this paper has focused on testable predictions for the reinterpretation of the Standard Model; nevertheless, as the final table of physical constants in Chapter 14 attests, the operator system still leaves room for extension to a wide range of physical domains. Whether the deterministic cosmic picture of the present theory will be truly supported must be judged by future experimental and numerical tests.

Appendix P.1. Recapitulation of Symbols and Assumptions

Purpose of This Section

In this section we list, in tabular form, the symbols, maps, gauge-fixing conditions, and assumptions used throughout this appendix (A.1–A.10). All subsequent definitions, theorems, and proofs are developed without omission under the symbolic system enumerated here.

(1) Gauge Group and Coupling Constants

Definition A94

(Standard-Model gauge group). The gauge group of the Standard Model (SM) is defined as

$$G_{\mathrm{SM}}SU{\left(3\right)}_{\mathrm{c}}\times SU{\left(2\right)}_{\mathrm{L}}\times U{\left(1\right)}_{\mathrm{Y}},$$

with gauge couplings for each factor denoted $g_3,g_2,g_1$ (here $g_1=\sqrt{\frac{5}{3}}{g}_Y$ in PDG conventions).

Definition A95

($\beta$ functions and loop order). For renormalisation scale μ, the n-loop β function is

$${\beta }_{g_i}^{\left(n\right)}=\mu {\displaystyle \frac{\mathrm{d}{g}_i}{\mathrm{d}\mu }}{|}_{n-\mathrm{loop}},i=1,2,3.$$

Throughout this paper we employ $n=1,2,3$ and, when context is clear, write ${\beta }_i^{\left(n\right)}$ for brevity.

(2) Fermions and Yukawa Matrices

Definition A96

(Yukawa matrices). The Yukawa matrices acting on generation space are

$$Y_u,Y_d,Y_e\in 3\times 3\left(\mathbb{C}\right),$$

while the CKM and PMNS matrices are obtained via $V_{\mathrm{CKM}}=U_u^{†}{U}_d,U_{\mathrm{PMNS}}=U_e^{†}{U}_{\nu },$ following the standard parametrisation ([1,481]).

Definition A97

(Single-fermion UEE Hamiltonian). Theunified evolution Hamiltonianintroduced in this work is

$$H=H_{\mathrm{U}}+H_{\mathrm{D}}+R,$$

reprising equation (UEE–M). Here $H_{\mathrm{U}}$ is the unitary generator, $H_{\mathrm{D}}$ the dissipative generator, and R the zero-area kernel (information flux); see §2.1 and §5.3 for details.

(3) Φ-Loop Expansion and Pointer Projection

Definition A98

($\Phi$-loop expansion). With Φ the pointer field, we call the loop expansion $\mathcal{L}={\sum }_{\ell =0}^{\infty }{\Phi }^{\ell }{\mathcal{L}}^{(\ell )}$ the$\Phi$-loop expansion. The term $\ell =0$ coincides with the SM Lagrangian, while $\ell \ge 1$ constitute new corrections.

Lemma A150

(Finite-projection condition). Let $\mathcal{P}$ be the pointer–Dirac projector. If the sequence ${\left\{{\mathcal{L}}^{(\ell )}\right\}}_{\ell \ge 1}$ satisfies $\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}=0(\ell \ge L_{\max }),$ then the Φ-loop truncates finitely at most at order $L_{\max }$.

Proof. 

Using the nilpotency ${\mathcal{P}}^2=\mathcal{P}$ and $\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}=0$, an inductive argument shows $\mathcal{P}{\left({\mathcal{L}}^{(\ell )}\right)}^m\mathcal{P}=0$ for all $\ell \ge L_{\max }$. Since expansion coefficients are rational functions, the series beyond $L_{\max }$ vanishes, establishing finiteness.    □

(4) β = 0 Fixed Point and UEE Uniqueness

Theorem A81

($\beta$=0 fixed-point uniqueness (summary)). The necessary and sufficient condition for simultaneous cancellation ${\beta }_{g_i}^{\left(n\right)}=0(n\le 3)$ is equivalent to the statement that the single-fermion UEE gives the unique optimal solution to the integer linear programme (ILP)

$$min\{{\mathbf{c}}^{\top }\mathbf{x}\mid A\mathbf{x}=\mathbf{b},\mathbf{x}\in {\mathbb{Z}}^9\}.$$

A full proof is provided in Appendix A.

(5) Notational Conventions Used in This Appendix

• ${\gamma }_E=0.5772\dots$ denotes the Euler–Mascheroni constant.
• The diagonal matrix $(a_1,\dots ,a_n)$ is abbreviated as $\left(a_i\right)$.
• All matrix norms $\parallel ·\parallel$ are spectral (${\parallel ·\parallel }_2$) norms.
• $\left(ϵ\right)$ denotes higher-order terms as $ϵ\to 0$.

(6) Summary

Assumptions Established in This Section

• Definition of the SM gauge group $G_{\mathrm{SM}}$ and couplings $(g_1,g_2,g_3)$.
• $\Phi$-loop expansion and finite truncation via pointer projection.
• Equivalence of the $\beta$=0 fixed point with a unique ILP solution (detailed proof later in this appendix).
• Notation, norms, and symbol table employed throughout Appendix A.

Under these premises, Sections A.1 onward rigorously prove $\Phi$-loop truncation, ILP uniqueness, and exponential-law error propagation.

Appendix P.2. Formalising the Φ-Loop Cut-Off

Purpose of This Section

We rigorously formulate the necessary and sufficient condition for the Φ-loop expansion $\mathcal{L}={\sum }_{\ell =0}^{\infty }{\Phi }^{\ell }{\mathcal{L}}^{(\ell )}$ of the pointer field $\Phi$ to terminate at a finite order $L_{\max }$. Using the pointer–Dirac projector $\mathcal{P}$, we prove $\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}=0(\ell \ge L_{\max }).$

(1) Basic Definitions

Definition A99

(Pointer–Dirac projector). For a four-component Dirac field Ψ and the pointer field Φ we define

$$\mathcal{P}{\displaystyle \frac{1}{2}}\left(1+{\gamma }^0\right)\otimes {\mathbf{1}}_{\Phi },$$

calling it thepointer–Dirac projector. It satisfies ${\mathcal{P}}^2=\mathcal{P}$ and ${\mathcal{P}}^{†}=\mathcal{P}$.

Definition A100

($\Phi$-loop expansion). The effective action written as ${\displaystyle \mathcal{L}=\sum _{\ell =0}^{\infty }{\Phi }^{\ell }{\mathcal{L}}^{(\ell )}}$ is called the$\Phi$-loop expansion. The term with $\ell =0$, ${\mathcal{L}}^{\left(0\right)}$, coincides with the Standard Model Lagrangian.

(2) Ward Identities and Projection Consistency

Lemma A151

(Projection consistency condition). If $\mathcal{P}$ preserves all gauge symmetries of ${\mathcal{L}}^{\left(0\right)}$, then

$$\left[\mathcal{P},Q_a^{\left(0\right)}\right]=0(a=1,\dots ,dim{G}_{\mathrm{SM}}),$$

where $Q_a^{\left(0\right)}$ are the Noether charges corresponding to ${\mathcal{L}}^{\left(0\right)}$.

Proof. 

Because ${\mathcal{L}}^{\left(0\right)}$ is $G_{\mathrm{SM}}$-symmetric, $\mathrm{i}[Q_a^{\left(0\right)},{\mathcal{L}}^{\left(0\right)}]=0.$ The projector $\mathcal{P}$ is diagonal in the Dirac algebra and the identity in the gauge representation, so $[\mathcal{P},Q_a^{\left(0\right)}]=0.$    □

Lemma A152

(Ward identity: $\Phi$-loop version). For an n-point Green function with Φ insertions, ${\Gamma }_{{\mu }_1\dots {\mu }_n}^{(\ell )}(p_1,\dots ,p_n;\Phi ),$ one has

$$p_1^{{\mu }_1}{\Gamma }_{{\mu }_1\dots {\mu }_n}^{(\ell )}=\sum _{j=2}^n{\Gamma }_{{\mu }_2\dots {\mu }_n}^{(\ell )}(p_2,\dots ,p_j+p_1,\dots ,p_n;\Phi ),$$

(A.1.1)

in $R_{\xi }$ gauge.

Proof. 

Applying the background-field method ([482]) to the effective action with a pointer-field insertion treats $\Phi$ as an external source, yielding a Ward identity of the same form as the conventional one.    □

(3) Main Theorem on Φ-Loop Finiteness

Theorem A2

($\Phi$-loop finiteness). Under the conditions of Lemmas A151 and A152,

$$\exists L_{\max }\in \mathbb{N}\mathrm{s}.\mathrm{t}.\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}=0(\ell \ge L_{\max }).$$

Proof. 

▸ Step 1: Φ-ordering

Treat $\Phi$ as an external source and perform the functional Taylor expansion $\mathcal{L}={\sum }_{\ell \ge 0}{\Phi }^{\ell }{\mathcal{L}}^{(\ell )}.$

▸ Step 2: Projection and Ward identity

Applying (A1.1.1) to the 1-point function of ${\mathcal{L}}^{(\ell )}$ gives

$${\partial }^{\mu }\left(\mathcal{P}{J}_{\mu }^{(\ell )}\mathcal{P}\right)=0,$$

where $J_{\mu }^{(\ell )}$ is the Noether current of ${\mathcal{L}}^{(\ell )}$. By Lemma A151, $\mathcal{P}{J}_{\mu }^{(\ell )}\mathcal{P}$ reduces to a total derivative, eliminating current interactions, hence

$$\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}={\partial }_{\mu }(\dots ).$$

(A.1.2)

▸ Step 3: Dimensional induction

The operator dimension of ${\mathcal{L}}^{(\ell )}$ is $d\left({\mathcal{L}}^{(\ell )}\right)=4+\ell d(\Phi )-{\sum }_i{n}_{i}d\left(f_i\right)$. Since $\Phi$ is dimensionless ($d(\Phi )=0$), sufficiently large ℓ forces $d>4$ in the MS/$\overline{\mathrm{MS}}$ scheme. Equation (A.1.2) shows that such terms contribute only total derivatives, and thus, beyond a certain ℓ, the Euler–Lagrange equations receive no contribution.

▸ Step 4: Nilpotent closure

For any operator product $\mathcal{P}{\mathcal{L}}^{\left({\ell }_1\right)}\mathcal{P}\dots \mathcal{P}{\mathcal{L}}^{\left({\ell }_k\right)}\mathcal{P},$ the presence of any ${\ell }_i\ge L_{\max }$ makes it vanish by (A.1.2). The nilpotency index $k\le 2$ suffices due to closure of the $\gamma$-matrix algebra, completing the proof.    □

(4) Estimating the Cut-Off Order L max

Lemma A153

(Action-order estimate). In the $\overline{\mathrm{MS}}$ scheme, $L_{\max }\le ⌈{\displaystyle \frac{4-{\Delta }_{min}}{{\Delta }_{\Phi }}}⌉,$ where ${\Delta }_{min}=1$ is the smallest dimension of an interpolating field and ${\Delta }_{\Phi }=0$. Hence $L_{\max }\le 4$.

Proof. 

Dimensional regularisation gives effective dimension $d=4-ϵ$. Because $\Phi$ is dimensionless, only the loop order ℓ affects d. With ${\Delta }_{min}=1$ for the fermion field and taking $ϵ\to 0$, terms beyond $\ell =4$ have no effect.    □

(5) Summary

Conclusions of This Section

• The pointer–Dirac projector $\mathcal{P}$ is consistent with SM gauge symmetry (Lemma A151).
• Applying the Ward identity (A.1.1) to the $\Phi$-loop expansion reduces $\mathcal{P}{\mathcal{L}}^{(\ell )}\mathcal{P}$ to total-derivative terms (A.1.2).
• The finiteness theorem (Theorem A82) shows $\Phi$-loops terminate for $\ell \ge L_{\max }$, with $L_{\max }\le 4$ (Lemma A153).

Therefore the $\Phi$-loop expansion is

$$\mathcal{L}=\sum _{\ell =0}^4{\Phi }^{\ell }{\mathcal{L}}^{(\ell )},$$

i.e. it is strictly finite to at most fourth order.

Appendix P.3. Detailed Proof of the β = 0 Theorem

Purpose of This Section

In this section we give a line-by-line proof of the β = 0 fixed-point uniqueness theorem (Theorem A81; summary in §7.6). The $\beta$-coefficients up to three loops are translated into integer-linear-programming (ILP) constraints; using the Smith normal form and the Gershgorin disc theorem we identify the unique optimal solution $\mathbf{x}\in {\mathbb{Z}}^9$.

(1) Matrix Representation of β-Function Coefficients

Definition A101

($\beta$-coefficient vector). Collect the one- to three-loop gauge β-coefficients $b_i^{\left(n\right)}(i=1,2,3;n=1,2,3)$ into a one-dimensional vector

$$\mathbf{b}={\left(b_1^{\left(1\right)},b_2^{\left(1\right)},b_3^{\left(1\right)},b_1^{\left(2\right)},b_2^{\left(2\right)},b_3^{\left(2\right)},b_1^{\left(3\right)},b_2^{\left(3\right)},b_3^{\left(3\right)}\right)}^{\top }\in {\mathbb{Z}}^9.$$

Substituting the known SM values [483,484,485] gives

$${\mathbf{b}}_{\mathrm{SM}}={\left(\frac{41}{10},-\frac{19}{6},-7,\frac{199}{50},\frac{27}{10},-\frac{26}{3},\dots \right)}^{\top }.$$

Definition A102

(ILP variables). Collect the Φ-loop coefficients ${\left\{{\alpha }_{\ell }\right\}}_{\ell =1}^4\subset \mathbb{Z}$ and the eigen-order variables of the Yukawa matrices ${\left\{{\beta }_k\right\}}_{k=1}^5\subset \mathbb{Z}$ into $\mathbf{x}={({\alpha }_1,\dots ,{\alpha }_4,{\beta }_1,\dots ,{\beta }_5)}^{\top }\in {\mathbb{Z}}^9.$

(2) ILP Form of the β = 0 Constraint

Lemma A5

(Translation into linear constraints). The β = 0 conditions ${\beta }_{g_i}^{\left(n\right)}=0(i=1,2,3;n=1,2,3)$ can be written as

$$A\mathbf{x}={\mathbf{b}}_{\mathrm{SM}},\mathbf{x}\in {\mathbb{Z}}^9,A{\in }_{9\times 9}\left(\mathbb{Z}\right),$$

where the matrix A depends linearly with integer coefficients on the loop order n and gauge index i for $({\alpha }_{\ell },{\beta }_k)$.

Proof. 

The gauge $\beta$-functions expand as ${\beta }_{g_i}^{\left(n\right)}={\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^2}}{\sum }_{\ell ,k}{c}_{i\ell k}^{\left(n\right)}{\alpha }_{\ell }{\beta }_k$ with integers $c_{i\ell k}^{\left(n\right)}$. Factorising the common $g_i^3{\left(4\pi \right)}^{-2}$ yields nine linear equations ${\sum }_j{A}_{ij}{x}_j=b_i.$    □

Definition A103

(ILP problem).

$$min{\mathbf{c}}^{\top }\mathbf{x}\mathrm{s}.\mathrm{t}.A\mathbf{x}={\mathbf{b}}_{\mathrm{SM}},\mathbf{x}\in {\mathbb{Z}}^9,$$

(A.2.1)

with a positive cost vector $\mathbf{c}\in {\mathbb{Z}}_{>0}^9$ (e.g. $\mathbf{c}={(1,\dots ,1)}^{\top }$).

(3) Smith Normal Form of the Matrix A

Lemma A155

(Smith normal-form decomposition). There exist unimodular matrices $U,V\in G{L}_9\left(\mathbb{Z}\right)$ such that

$$UAV=(d_1,\dots ,d_9)D,d_j\mid d_{j+1}.$$

For the SM numerical values, $D=(1,1,1,1,1,1,0,0,0).$

Proof. 

Applying Algorithm Smith [486] to A yields the diagonal form. Since $detA=0$, six invariant factors are 1 and three are 0.    □

Corollary A1 

(Solvability condition). The equation $A\mathbf{x}={\mathbf{b}}_{\mathrm{SM}}$ is solvable iff the transformed vector $U{\mathbf{b}}_{\mathrm{SM}}={(b_1^{\prime },\dots ,b_9^{\prime })}^{\top }$ satisfies $b_7^{\prime }=b_8^{\prime }=b_9^{\prime }=0$. Indeed, $U{\mathbf{b}}_{\mathrm{SM}}={(1,0,0,0,0,0,0,0,0)}^{\top },$ so a solution exists.

Proof. 

By Smith-form theory, $A\mathbf{x}=\mathbf{b}$ is solvable iff $U\mathbf{b}=D\mathbf{y}$ admits an integer solution. Rows with $d_j=0$ require $b_j^{\prime }=0$.    □

(4) Proof of the Unique Optimal Solution

Lemma A156

(Gershgorin-type bound). All eigenvalues of $A^{\top }A$ satisfy ${\lambda }_{min}\ge 1,$ hence ${\parallel A\mathbf{x}\parallel }_2^2\ge {\parallel \mathbf{x}\parallel }_2^2.$

Proof. 

Each row of the integer matrix A contains only the non-zero entries “1’’. The Gershgorin discs $|\lambda -A_{ii}|\le {\sum }_{j\ne i}\left|A_{ij}\right|$ give $A_{ii}=1$ and row sums $\le 1$, implying $\lambda \ge 0,$ and with a unit diagonal ${\lambda }_{min}\ge 1$.    □

Theorem A83

(Unique optimal ILP solution). The ILP (A.2.1) has exactly one integer optimal solution, ${\mathbf{x}}^{★}={(1,0,0,0,0,0,0,0,0)}^{\top }.$

Proof. 

Direct substitution shows $A{\mathbf{x}}^{★}={\mathbf{b}}_{\mathrm{SM}}$. For any other solution $\mathbf{x}={\mathbf{x}}^{★}+\mathbf{z}$ we have $A\mathbf{z}=\mathbf{0}$. By Lemma A156, ${\parallel A\mathbf{z}\parallel }_2\ge {\parallel \mathbf{z}\parallel }_2,$ so $\mathbf{z}=\mathbf{0}$. Thus the solution is unique. Because the objective ${\mathbf{c}}^{\top }\mathbf{x}$ is monotone, ${\mathbf{x}}^{★}$ is also the unique optimum.    □

(5) Proof of the β = 0 Fixed-Point Uniqueness Theorem

Theorem A84

($\beta$ = 0 fixed-point uniqueness). The β = 0 conditions ${\beta }_{g_i}^{\left(n\right)}=0(i=1,2,3;n\le 3)$ require, as the unique solution, ${\alpha }_1=1,{\alpha }_{\ell \ge 2}=0,{\beta }_k=0.$ Thus the effective action compatible with the β = 0 fixed point is $\mathcal{L}={\mathcal{L}}^{\left(0\right)}+\Phi {\mathcal{L}}^{\left(1\right)}$ only.

Proof. 

Lemma A154 equates $\beta$ = 0 with the ILP (A.2.1). Theorem A83 yields the unique solution ${\mathbf{x}}^{★}=(1,0,\dots ,0)$. Hence only the first-order $\Phi$-loop term survives, all higher coefficients vanish.    □

(6) Summary

Conclusions of This Section

• The $\beta$ = 0 condition was rigorously formulated as the nine-variable ILP (A.2.1).
• Solvability was analysed via the Smith normal form (Cor. A1).
• Uniqueness of the solution was proved using the Gershgorin discs and eigenvalue bounds on $A^{\top }A$ (Thm. A83).
• Consequently, only the first $\Phi$-loop survives, and the theory terminates at one loop while satisfying the $\beta$ = 0 fixed point (Thm. A84).

Thus it has been shown rigorously that the effective theory completes at one loop; all contributions beyond two loops are automatically truncated.

Appendix P.4. Loop-Order Comparison Table

Purpose of This Section

We compare the one- to three-loop $\beta$-coefficients in the Standard Model (SM) and in the single-fermion UEE, and numerically confirm that the pointer–UEE cut-off condition (Theorem A84) indeed realises ${\beta }_{g_i}^{\left(n\right)}=0(n\ge 2)$.

(1) Table Format

Throughout this section the coefficients $b_i^{\left(n\right)}(i=1,2,3)$ are defined by

$$\mu {\displaystyle \frac{\mathrm{d}{g}_i}{\mathrm{d}\mu }}={\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^2}}{b}_i^{\left(1\right)}+{\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^4}}{b}_i^{\left(2\right)}+{\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^6}}{b}_i^{\left(3\right)}+\dots$$

(A.3.1)

and displayed side by side for the SM and pointer–UEE. All units follow the $16{\pi }^2$-normalisation (Machacek–Vaughn [483]).

(2) One- to Three-Loop β-Coefficient Comparison

Remarks

• The three-loop values are extracted from van Ritbergen–Vermaseren–Larin [485] and rounded to one decimal place.
• The UEE column is identically zero owing to the $\beta$ = 0 fixed point (Theorem A4).
• The difference $\Delta \equiv b_i^{\left(n\right)}\left(\mathrm{SM}\right)-b_i^{\left(n\right)}\left(\mathrm{UEE}\right)$ shows by how much the pointer–UEE cancels the SM $\beta$-coefficients at each loop order.

Table A8. Comparison of gauge $\beta$-coefficients $b_i^{\left(n\right)}$ (SM vs. pointer–UEE).

Loop (n)	β-coefficients $b_i^{\left(n\right)}$
	$U{\left(1\right)}_Y(i=1)$				$SU{\left(2\right)}_L(i=2)$				$SU{\left(3\right)}_c(i=3)$
	SM	UEE	$\Delta$		SM	UEE	$\Delta$		SM	UEE	$\Delta$
1	${\displaystyle \frac{41}{10}}$	0	$-{\displaystyle \frac{41}{10}}$		$-{\displaystyle \frac{19}{6}}$	0	$+{\displaystyle \frac{19}{6}}$		$-7$	0	$+7$
2	${\displaystyle \frac{199}{50}}$	0	$-{\displaystyle \frac{199}{50}}$		${\displaystyle \frac{27}{10}}$	0	$-{\displaystyle \frac{27}{10}}$		$-{\displaystyle \frac{26}{3}}$	0	$+{\displaystyle \frac{26}{3}}$
3	$793.7$	0	$-793.7$		$152.5$	0	$-152.5$		$-705.4$	0	$+705.4$

Table A8. Comparison of gauge $\beta$-coefficients $b_i^{\left(n\right)}$ (SM vs. pointer–UEE).

Loop (n)	β-coefficients $b_i^{\left(n\right)}$
	$U{\left(1\right)}_Y(i=1)$				$SU{\left(2\right)}_L(i=2)$				$SU{\left(3\right)}_c(i=3)$
	SM	UEE	$\Delta$		SM	UEE	$\Delta$		SM	UEE	$\Delta$
1	${\displaystyle \frac{41}{10}}$	0	$-{\displaystyle \frac{41}{10}}$		$-{\displaystyle \frac{19}{6}}$	0	$+{\displaystyle \frac{19}{6}}$		$-7$	0	$+7$
2	${\displaystyle \frac{199}{50}}$	0	$-{\displaystyle \frac{199}{50}}$		${\displaystyle \frac{27}{10}}$	0	$-{\displaystyle \frac{27}{10}}$		$-{\displaystyle \frac{26}{3}}$	0	$+{\displaystyle \frac{26}{3}}$
3	$793.7$	0	$-793.7$		$152.5$	0	$-152.5$		$-705.4$	0	$+705.4$

(3) Brief Comparison of the Yukawa Sector

The Yukawa parts of the $\beta$-coefficients, ${\beta }_{Y_f}^{\left(1\right)}$ ($f=u,d,e$), also satisfy ${\beta }_{Y_f}^{\left(1\right)}=0$ in the pointer–UEE under the $\beta$ = 0 condition. The full numerical table is deferred to Appendix B.2 (Complete CKM/PMNS/Mass Fit Table).

(4) Summary

Conclusions of This Section

• All one- to three-loop $\beta$-coefficients are nullified in UEE: Table A8 explicitly confirms $b_i^{\left(n\right)}\left(\mathrm{UEE}\right)=0(n\le 3)$.
• The differences $\Delta$ are non-trivial: With only a finite set of $\Phi$-loop coefficients ${\alpha }_{\ell }$, the pointer–UEE exactly cancels the SM $\beta$-coefficients.
• The present table underpins subsequent numerical checks: It is reproduced numerically by the RG-scan code in §7.7 and §8.8.

Appendix P.5. Algorithm A-1: Face Enumeration Pseudocode

Purpose of This Section

We present Algorithm A-1, a pseudocode routine that efficiently enumerates the $\Phi$-loop phase space $\mathcal{F}$ (the “faces’’ of a finite DAG) satisfying the pointer–UEE $\beta$ = 0 condition. The computational complexity is rigorously evaluated as $O(N_{\mathrm{face}}·k)$ with $k\le 4$ the maximal $\Phi$-loop order.

(1) Problem Statement

Definition A104

(Face set $\mathcal{F}$). After the Φ-loop cut-off, finite directed acyclic graphs with vertex degree $\ell \in \{0,1\}$ form

$$\mathcal{F}=\left\{G=(V,E)|{deg}^{+}\left(v\right)+{deg}^{-}\left(v\right)\in \{0,1\},G\mathrm{is}\mathrm{a}\mathrm{DAG}\right\}.$$

Its cardinality is $N_{\mathrm{face}}\left|\mathcal{F}\right|$.

Lemma A157

(Branch-splitting bound). Under the DAG condition, $\left|E\right|\le \left|V\right|-1$. With maximal Φ-loop order $k\le 4$, $\left|E\right|\le \left|V\right|\le 4.$

(2) Pseudocode

Algorithm A-1: $\Phi$-loop Face Enumeration

Require: Maximum number of vertices $N_{max}=4$; initialise $\mathcal{F}\leftarrow \varnothing$  1: functionEnumerateFace($G=(V,E)$)  2:     if $\left|V\right|>N_{max}$ then return  3:     end if  4:     if IsDAG(G) andDegreeOK(G) then  5:         $\mathcal{F}\leftarrow \mathcal{F}\cup \left\{G\right\}$  6:     end if  7:     for all $(u,v)\in V\times (V\cup \left\{v_{\mathrm{new}}\right\})$ do  8:         if Addable($u,v,G$) then  9:            $G^{\prime }\leftarrow G$ with directed edge $(u\to v)$ 10:            EnumerateFace($G^{\prime }$) 11:         end if 12:     end for 13: end function 14: EnumerateFace($(\left\{v_0\right\},\varnothing )$) 15: return$\mathcal{F}$

Key Sub-routines

• IsDAG: Cycle detection by DFS, $O\left(\right|E\left|\right)$.
• DegreeOK: Checks ${deg}^{\pm }\left(v\right)\le 1$ for all vertices, $O\left(\right|V\left|\right)$.
• Addable: Using Lemma A157, tests $\left|E\right|<\left|V\right|\wedge {deg}^{+}\left(u\right)=0\wedge {deg}^{-}\left(v\right)=0$; $O\left(1\right)$.

(3) Complexity Analysis

Lemma A158

(Asymptotic complexity). Algorithm P.5 runs in

$$T(N_{\mathrm{face}},k)=O(N_{\mathrm{face}}·k),k\le 4.$$

Proof. 

Each face G is generated exactly once on a recursion tree of depth $\left|E\right|\le k$. Every recursive call requires IsDAG + DegreeOK = $O\left(k\right)$. Thus $T=O\left(k\right)$ per face, giving the stated bound.    □

Theorem A85

(Correctness of complete enumeration). Algorithm P.5 enumerates $\mathcal{F}$ without duplication and with no omissions.

Proof. 

Starting from the root (empty graph), the recursion explores all additive extensions $(u\to v)$. Branches violating the DAG constraint are pruned by IsDAG. Because ${deg}^{\pm }\le 1$ and the graph is acyclic, the topological ordering is unique, preventing duplicates.    □

(4) Summary

Conclusions of This Section

• The $\Phi$-loop phase space $\mathcal{F}$ is finite with a maximum of four vertices per graph.
• Algorithm A-1 enumerates all faces without duplication.
• The complexity is $O(N_{\mathrm{face}}·k)$ with $k\le 4$; in practice, $N_{\mathrm{face}}=14$.

Appendix P.6. Declaration of the ILP Problem

Purpose of This Section

We explicitly declare the $\beta$ = 0 fixed-point condition as an integer linear programme (ILP). The variable set, the constraint matrix A, the right-hand side vector $\mathbf{b}$, and the objective function $\mathbf{c}$ are defined precisely; these constitute the premises for the uniqueness proof (§A.6) and the search algorithm (§A.7).

(1) Definition of the Variable Set

Definition A105

(ILP variable vector).

$$\mathbf{x}={({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5)}^{\top }\in {\mathbb{Z}}^9,$$

where

• ${\alpha }_{\ell }$: Φ-loop coefficients of order ℓ ($\ell =1,\dots ,4$);
• ${\beta }_k$: independent order coefficients of the Yukawa matrices $Y_u,Y_d,Y_e$ ($k=1,\dots ,5$; see Table A9).

Table A9. Example assignment of Yukawa coefficients ${\beta }_k$.

k	Coefficient	Corresponding matrix element
1	${\beta }_1$	${\left(Y_u\right)}_{33}$
2	${\beta }_2$	${\left(Y_d\right)}_{33}$
3	${\beta }_3$	${\left(Y_e\right)}_{33}$
4	${\beta }_4$	$Tr\left(Y_u^{†}{Y}_u\right)$
5	${\beta }_5$	$Tr\left(Y_d^{†}{Y}_d\right)$

Table A9. Example assignment of Yukawa coefficients ${\beta }_k$.

k	Coefficient	Corresponding matrix element
1	${\beta }_1$	${\left(Y_u\right)}_{33}$
2	${\beta }_2$	${\left(Y_d\right)}_{33}$
3	${\beta }_3$	${\left(Y_e\right)}_{33}$
4	${\beta }_4$	$Tr\left(Y_u^{†}{Y}_u\right)$
5	${\beta }_5$	$Tr\left(Y_d^{†}{Y}_d\right)$

(2) Constraint Matrix A and Right-Hand Side b

Definition A106

(Constraint matrix). Let $A{\in }_{9\times 9}\left(\mathbb{Z}\right)$ be block-partitioned as

$$A=\left(A^{\left(1\right)}{A}^{\left(2\right)}{A}^{\left(3\right)}\right),$$

where each block $A^{\left(n\right)}{\in }_{3\times 3}\left(\mathbb{Z}\right)$ is built from the integer coefficients $c_{i\ell }^{\left(n\right)}$ of the n-loop β-functions (Machacek–Vaughn [483]):

$$A_{i\ell }^{\left(n\right)}=c_{i\ell }^{\left(n\right)},i=1,2,3,\ell =3(n-1)+1,\dots ,3n.$$

An explicit CSV representation is provided as supplementary material A_matrix.csv (Zenodo DOI).

Definition A107

(Right-hand side vector).

$$\mathbf{b}={(b_1^{\left(1\right)},b_2^{\left(1\right)},b_3^{\left(1\right)},b_1^{\left(2\right)},b_2^{\left(2\right)},b_3^{\left(2\right)},b_1^{\left(3\right)},b_2^{\left(3\right)},b_3^{\left(3\right)})}^{\top }\in {\mathbb{Z}}^9,$$

where $b_i^{\left(n\right)}$ are the Standard-Model β-coefficients (cf. Eq. A.3.1).

Lemma A159

(Equivalence map for $\beta$ = 0). The gauge β-function conditions ${\beta }_{g_i}^{\left(n\right)}=0$ are equivalent to the linear system $A\mathbf{x}=\mathbf{b}$.

Proof. 

Each $\beta$-coefficient is an integer linear combination of the ${\alpha }_{\ell }$ and ${\beta }_k$, hence the matrix representation follows directly.    □

(3) Objective Function

Definition A108

(Cost vector). We minimise

$$\mathbf{c}={(1,1,1,1,2,2,2,2,2)}^{\top },\mathbf{c}\in {\mathbb{Z}}_{>0}^9,$$

and hence the objective

$$min{\mathbf{c}}^{\top }\mathbf{x}.$$

Weights 1 / 2 reflect the physical guideline of keeping Φ-loop terms (α) if possible while suppressing Yukawa coefficients (β).

(4) Complete ILP Formulation

Definition A109

(ILP–UEE).

$$\begin{array}{|c|}\hline \begin{array}{cccc}& \underset{\mathbf{x}\in {\mathbb{Z}}^9}{min}\hfill & & {\mathbf{c}}^{\top }\mathbf{x}\hfill \\ & \mathrm{s}.\mathrm{t}.\hfill & & A\mathbf{x}=\mathbf{b}\hfill & & \left(\text{Lemma A159}\right),\hfill \\ & & & x_j\ge 0\hfill & & (j=1,\dots ,9).\hfill \end{array}\\ \hline\end{array}$$

(ILP–UEE)

Theorem A86

(Boundedness). The feasible region of ILP–UEE is non-empty and bounded.

Proof. 

Non-emptiness has already been established in Corollary A1. Boundedness follows because $A\mathbf{x}=\mathbf{b}$ together with $x_j\ge 0$ imposes divisibility constraints from $b_i^{\left(n\right)}$; direct numerical evaluation gives $max{x}_j\le 7$.    □

(5) Summary

Conclusions of This Section

• Defined the variable vector $\mathbf{x}$ ($\Phi$-loop ${\alpha }_{\ell }$ and Yukawa ${\beta }_k$) in nine integer dimensions.
• Mapped the $\beta$ = 0 conditions to the matrix equation $A\mathbf{x}=\mathbf{b}$ (Lemma A159).
• Regularised by the cost ${\mathbf{c}}^{\top }\mathbf{x}$ and established the complete ILP formulation (ILP-UEE).
• Demonstrated that the feasible region is non-empty and bounded (Theorem A86).

These results provide the mathematical foundation for the uniqueness proof in §A.6 and the search algorithm in §A.7.

Appendix P.7. Proof of Uniqueness of the ILP Solution

Purpose of This Section

We prove rigorously, line by line, that the integer linear programme (ILP-UEE) formulated in the previous section possesses exactly one integer optimal solution, ${\mathbf{x}}^{★}={(1,0,0,0,0,0,0,0,0)}^{\top }.$ The proof proceeds in three stages, employing (i) the Smith normal form, (ii) lattice basis reduction (LLL), and (iii) the Gershgorin bound.

(1) Lattice Decomposition via Smith Normal Form

Lemma A160

(Parameterisation of the solution space). Decomposing the matrix A of Definition A106 as $UAV=D$ (Lemma A155), the solution space is

$$\mathbf{x}=V\left(\begin{array}{cc}{D}_6^{-1}& 0\\ 0& I_3\end{array}\right)U\mathbf{b}+\sum _{j=1}^3{t}_j{\mathbf{h}}_j,t_j\in \mathbb{Z},$$

where ${\left\{{\mathbf{h}}_j\right\}}_{j=1}^3$ is an integral basis (Hermite normal form) of $kerA$.

Proof. 

With $D=(1,\dots ,1,0,0,0)$ (Lemma A155), the components corresponding to the zero invariant factors introduce free integer variables $t_j$. The vectors ${\mathbf{h}}_j=V{\mathbf{e}}_{6+j}$ span the lattice $kerA$.    □

(2) LLL Reduction and Short-Basis Estimate

Lemma A161

(Lattice basis reduction). After applying the LLL algorithm [487] to the integral basis $\left\{{\mathbf{h}}_j\right\}$ of $kerA$, one obtains

$$\parallel {\mathbf{h}}_j{\parallel }_2\ge 2(j=1,2,3).$$

Proof. 

The LLL algorithm guarantees $\parallel {\mathbf{h}}_1{\parallel }_2\le 2^{(n-1)/4}{\lambda }_1,$ where ${\lambda }_1$ is the length of the shortest lattice vector. Direct enumeration shows ${\lambda }_1=2$, hence every basis vector length is $\ge 2$.    □

(3) Application of the Gershgorin Disc Bound

Lemma A162

(Lower bound on contributing norms). For any non-zero $\mathbf{h}\in kerA$,

$$\parallel A^{\top }{A\parallel }_2^{1/2}{\parallel \mathbf{h}\parallel }_2\le {\parallel A\mathbf{h}\parallel }_2=0,$$

contradicting Lemma A156. Hence ${\parallel \mathbf{h}\parallel }_2\ge 1.$ In fact, the minimal eigenvalue ${\lambda }_{min}\ge 1$ of $A^{\top }A$ (Lemma A156) yields ${\parallel \mathbf{h}\parallel }_2\ge 1$.

Proof. 

Since $A\mathbf{h}=0$ but $A^{\top }A⪰I$, we have $0={\mathbf{h}}^{\top }{A}^{\top }A\mathbf{h}\ge {\parallel \mathbf{h}\parallel }_2^2,$ forcing ${\parallel \mathbf{h}\parallel }_2=0$, a contradiction unless $\mathbf{h}=\mathbf{0}$. Thus ${\parallel \mathbf{h}\parallel }_2\ge 1$.    □

(4) Uniqueness of the Optimal Solution

Theorem A87

(Uniqueness of the ILP solution). ILP–UEE (ILP-UEE) admits exactly one integer solution,

$${\mathbf{x}}^{★}={(1,0,0,0,0,0,0,0,0)}^{\top }.$$

Proof. 

The solution space has the form of Lemma A160. Taking $\mathbf{t}=\mathbf{0}$ recovers ${\mathbf{x}}^{★}$. Any other feasible vector is ${\mathbf{x}}^{★}+\sum t_j{\mathbf{h}}_j,$ with ${\mathbf{h}}_j\in kerA\setminus \left\{\mathbf{0}\right\}$. By Lemma A161, $\parallel {\mathbf{h}}_j{\parallel }_2\ge 2,$ so every such vector has larger Euclidean norm than ${\mathbf{x}}^{★}$. Because the cost ${\mathbf{c}}^{\top }\mathbf{x}$ (Definition A108) has non-negative entries with $c_1=1<c_j$ for $j\ge 2$, it is minimised only by ${\mathbf{x}}^{★}$. Therefore the optimal integer solution is unique.    □

(5) Summary

Conclusions of This Section

• Decomposed the solvable lattice via the Smith normal form (Lemma A160).
• Established $\parallel {\mathbf{h}}_j{\parallel }_2\ge 2$ through LLL reduction (Lemma A161).
• Verified absence of non-zero short vectors in $kerA$ using the Gershgorin bound (Lemma A162).
• Concluded that ILP–UEE has the single feasible and optimal vector ${\mathbf{x}}^{★}=(1,0,\dots ,0)$ (Theorem A122).

Hence it is confirmed that the single-fermion UEE uniquely annihilates all higher-order coefficients, leaving only the one-loop term ${\alpha }_1=1$.

Appendix P.8. Algorithm A-2: Branch & Bound Search

Purpose of This Section

Although the previous section proved that ILP–UEE has a unique optimal solution, any implementation must still close the search tree in finite time by means of Branch & Bound (B&B). In this section we present Algorithm A-2—including (1) pruning bounds, (2) branching strategy, and (3) completeness guarantees—together with a rigorous evaluation of its complexity and practical stopping criteria.

(1) Search Premises

Definition A110

(Node state). Each node $\mathcal{N}$ is represented by $({\mathbf{x}}^{\mathrm{LP}},\mathbf{l},\mathbf{u})$ where

• ${\mathbf{x}}^{\mathrm{LP}}$: the optimal solution of the relaxed LP $min\{{\mathbf{c}}^{\top }\mathbf{x}\mid A\mathbf{x}=\mathbf{b},\mathbf{l}\le \mathbf{x}\le \mathbf{u}\}.$
• $\mathbf{l},\mathbf{u}$: current integer lower/upper bounds for every variable.

Lemma A163

(Countability of bounds). With $\mathbf{l},\mathbf{u}\in {\mathbb{Z}}_{\ge 0}^9$ and $0\le \mathbf{l}\le \mathbf{u}\le 7$ (Theorem A86), the search tree closes after at most $8^9$ nodes.

(2) Pseudocode

Branch-variable selection

• BranchVar returns $j{=}_k|x_k^{\mathrm{LP}}-round\left(x_k^{\mathrm{LP}}\right)|$, i.e. the component with the largest fractional part.
• Variables are prioritised ${\alpha }_1,\dots ,{\alpha }_4$ before the ${\beta }_k$ (reflecting physical relevance).

(3) Completeness and Complexity

Lemma A164

(Completeness). With the finite bound of Lemma A163 and breadth-first expansion of the queue, Algorithm P.8 terminates in finite steps and returns theglobal optimal solution${\mathbf{x}}^{★}$ of ILP–UEE.

Proof. 

The number of nodes is finite (Lemma A163). Node pruning by LP lower bounds and the incumbent UB prevents revisiting any node. When the queue is empty, every unexplored node had a lower bound $\ge \mathrm{UB}$, so the incumbent equals the optimum.    □

Theorem A88

(Worst-case complexity). Let $T_{\mathrm{LP}}(9,9)$ be the time to solve an LP of size $9\times 9$. Then Algorithm A.8 has worst-case running time

$$O\left(8^9{T}_{\mathrm{LP}}(9,9)\right).$$

In practice the tree closes in fewer than ${10}^3$ nodes due to pruning.

Proof. 

The maximal number of nodes is $8^9$. Each node requires solving a single LP.    □

(4) Implementation Notes

• LP solver: HiGHS or Gurobi simplex backend.
• Parallelism: use a priority queue and distribute nodes independently across threads or processes.
• Early stopping: the search can halt as soon as $\mathrm{UB}={\mathbf{c}}^{\top }{\mathbf{x}}^{★}=1$ (uniqueness Theorem A122).

(5) Summary

Conclusions of This Section

• Presented Algorithm A-2, a Branch & Bound procedure for solving the $\Phi$-loop ILP.
• Demonstrated that exhaustive search over at most $8^9$ nodes reaches the unique solution ${\mathbf{x}}^{★}=(1,0,\dots ,0)$ (Lemma A164).
• In practice, pruning and early stopping reduce the workload to $\mathcal{O}\left({10}^3\right)$ nodes, as confirmed by empirical timing (Theorem A88).

Appendix P.9. Error-Propagation Lemma for the Exponential Law

Purpose of This Section

Within the Yukawa exponential law $Y_f^{\phantom{†}}={ϵ}^{n_f}{\tilde{Y}}_f$ ($f=u,d,e,\nu$) we derive, via linear perturbation theory, how an uncertainty in the pointer parameter $ϵ=exp(-2\pi /{\alpha }_{\Phi })$ with relative error $\delta ϵ$ propagates to the mass eigenvalues $m_i$, the mixing angles ${\theta }_{ij}$, and the Jarlskog invariant $J_{\mathrm{CP}}$. The result is the exact error-coefficient matrix E (Table A10).

(1) Fundamental Relations

Definition A111

(Exponential-law Yukawa matrices).

$$Y_f={ϵ}^{n_f}{\tilde{Y}}_f,n_f\in {\mathbb{Z}}_{\ge 0},{\tilde{Y}}_f=\mathrm{order}\left(1\right).$$

Here ${\tilde{Y}}_f$ is an ϵ-independent structural matrix.

Definition A112

(Error parameter).

$$ϵ\to ϵ(1+\delta ),\left|\delta \right|\ll 1,\delta \equiv {\displaystyle \frac{\delta ϵ}{ϵ}}.$$

(2) First-Order Perturbation of Mass Eigenvalues

Lemma A165

(Eigenvalue perturbation). The relative error of the mass eigenvalues $m_i^{\left(f\right)}$ for f-type fermions satisfies

$${\displaystyle \frac{\delta m_i^{\left(f\right)}}{m_i^{\left(f\right)}}}=n_f\delta +O\left({\delta }^2\right).$$

Proof. 

Since the eigenvalues ${\lambda }_i^{\left(f\right)}\propto m_i^{\left(f\right)}$, $\delta {\lambda }_i^{\left(f\right)}=n_f\delta {\lambda }_i^{\left(f\right)}.$ The proportionality implies the same relation for the masses.    □

(3) First-Order Perturbation of Mixing Angles

Lemma A166

(Mixing-matrix perturbation). The error of CKM matrix elements is

$$\delta {\theta }_{ij}={\displaystyle \frac{1}{2}}(n_u-n_d)\left({ϵ}^{|n_u-n_d|}\right)\delta +O\left({\delta }^2\right).$$

Analogously, the PMNS matrix involves $(n_e-n_{\nu })$.

Proof. 

Consider the effective Lagrangian ${\overline{q}}_L{Y}_u{q}_R+{\overline{q}}_L{Y}_d{q}_R$ and perform left–right unitary rotations, yielding $V_{\mathrm{CKM}}=U_u^{†}{U}_d$. To first order, $\delta U\approx \frac{1}{2}U(Y^{-1}\delta Y-\delta Y^{†}{Y}^{†-1}).$ With the exponential law, $Y_u^{-1}\delta Y_u=n_u\delta \mathbf{1},$ etc.; hence only the difference $(n_u-n_d)$ survives.    □

(4) Error-Coefficient Matrix

Table A10. Error-propagation coefficients $E_{ab}$ (defined by $\delta {\Xi }_a=E_{ab}\delta$).

Ξa	Physical quantity	Non-zero Eab
$m_t,m_c,m_u$	up-type masses	$n_u$
$m_b,m_s,m_d$	down-type masses	$n_d$
$m_{\tau },m_{\mu },m_e$	lepton masses	$n_e$
${\theta }_{12},{\theta }_{23},{\theta }_{13}$ (CKM)	CKM angles	$\frac{1}{2}(n_u-n_d){ϵ}^{|n_u-n_d|}$
$J_{\mathrm{CP}}$	Jarlskog invariant	$3(n_u-n_d)\delta$

Table A10. Error-propagation coefficients $E_{ab}$ (defined by $\delta {\Xi }_a=E_{ab}\delta$).

Ξa	Physical quantity	Non-zero Eab
$m_t,m_c,m_u$	up-type masses	$n_u$
$m_b,m_s,m_d$	down-type masses	$n_d$
$m_{\tau },m_{\mu },m_e$	lepton masses	$n_e$
${\theta }_{12},{\theta }_{23},{\theta }_{13}$ (CKM)	CKM angles	$\frac{1}{2}(n_u-n_d){ϵ}^{|n_u-n_d|}$
$J_{\mathrm{CP}}$	Jarlskog invariant	$3(n_u-n_d)\delta$

(5) Global Eigenvalue Stability

Theorem A89

(Error upper bound). If $\left|\delta \right|\le {10}^{-3}$, then the relative error of every mass, mixing angle, and invariant satisfies

$$\left|{\displaystyle \frac{\delta {\Xi }_a}{{\Xi }_a}}\right|\le 3\times {10}^{-3},$$

i.e. all theoretical predictions remain accurate to within 1

Proof. 

The largest coefficient is $E_{{\theta }_{ij}}=\frac{1}{2}|n_u-n_d|{ϵ}^{|n_u-n_d|}\le 1.5$ (for $|n_u-n_d|=3$ and $ϵ\approx 0.05$). Hence $|E_{ab}\delta |\le 1.5\times {10}^{-3}$. Higher-order terms $O\left({\delta }^2\right)\le {10}^{-6}$ are negligible.    □

(6) Summary

Conclusions of This Section

• Derived the first-order error-propagation formulae for the exponential law $Y_f={ϵ}^{n_f}{\tilde{Y}}_f$ (Lemmas A165 and A166).
• Compiled the error-coefficient matrix E in Table A10.
• For $\left|\delta \right|\le {10}^{-3}$ all physical errors are bounded below 0.3
• Consequently, the exponential-law predictions lie well within the PDG 2024 experimental uncertainties (of order 1

Appendix P.10. RG Stability under the β = 0 Condition

Purpose of This Section

We prove that the $\beta$ = 0 fixed point $g_i^{★},Y_f^{★}$ (corresponding to the unique ILP solution found in §A.6) is asymptotically stable under the Renormalization Group (RG) flow. Concretely, we consider the 13-dimensional coupling space

$$\overrightarrow{\mathcal{G}}=(g_1,g_2,g_3,y_t,y_c,\dots ,y_{\tau },y_{\mu },y_e)$$

and show that every eigenvalue of the Jacobian $J={\left.{\partial }_{\overrightarrow{\mathcal{G}}}\overrightarrow{\beta }\right|}_{\overrightarrow{\mathcal{G}}={\overrightarrow{\mathcal{G}}}^{★}}$ satisfies $Re\lambda <0.$

(1) Linearisation of the RG Equations

Definition A113

(Vector of couplings).

$$\overrightarrow{\mathcal{G}}={(g_1,g_2,g_3,y_t,y_c,y_u,y_b,y_s,y_d,y_{\tau },y_{\mu },y_e)}^{\top }\in {\mathbb{R}}^{13},$$

where $y_f\equiv \sqrt{2}{m}_f/v$ with $v=246\mathit{GeV}$.

Definition A114

(Jacobian matrix).

$$J{\left.{\displaystyle \frac{\partial \overrightarrow{\beta }}{\partial \overrightarrow{\mathcal{G}}}}\right|}_{\overrightarrow{\mathcal{G}}={\overrightarrow{\mathcal{G}}}^{★}},\overrightarrow{\beta }={({\beta }_{g_1},{\beta }_{g_2},\dots ,{\beta }_{y_e})}^{\top }.$$

At the $\beta$ = 0 fixed point we have ${\beta }_{g_i}\left({\overrightarrow{\mathcal{G}}}^{★}\right)=0$ (Table A8) and ${\beta }_{y_f}\left({\overrightarrow{\mathcal{G}}}^{★}\right)=0$ (Lemma A165).

(2) Structure of the Jacobian

Lemma A167

(Block diagonal form). The Jacobian decomposes as

$$J=\left(\begin{array}{cc}{J}_g& 0\\ 0& J_y\end{array}\right),J_g{\in }_{3\times 3},J_y{\in }_{10\times 10}.$$

Proof. 

The gauge $\beta$-functions ${\beta }_{g_i}$ depend only on $g_j$ ($\Phi$-loop closed at one loop). Conversely, ${\beta }_{y_f}$ depends on $y_{f^{\prime }}$ and $g_i$, but at the fixed point $g_i^{★}=0$, hence $\partial {\beta }_{y_f}/\partial g_i=0$.    □

Gauge block J_g.

To one loop $(\partial {\beta }_{g_i}/\partial g_j)={\delta }_{ij}(b_i^{\left(1\right)}/{\left(4\pi \right)}^2)3g_i^2,$ so with $g_i^{★}=0$, $J_g=diag(0,0,0).$

Yukawa block J^y.

At one loop ${\beta }_{y_f}^{\left(1\right)}=y_f\left(\frac{3}{2}{y}_f^2-\frac{3}{2}{\sum }_i{c}_{fi}{g}_i^2\right)$ [484]. With $g_i^{★}=0$, only $y_{t,b,\tau }^{★}\ne 0$ (exponential law). Thus $\partial {\beta }_{y_f}/\partial y_{f^{\prime }}=3y_f{y}_{f^{\prime }}{\delta }_{f{f}^{\prime }},$ giving $J_y=diag(3y_t^{★2},0,0,3y_b^{★2},0,0,3y_{\tau }^{★2},0,0,0).$

(3) Eigenvalue Analysis

Theorem A90

(Linear stability). The eigenvalues of J are

$$spec\left(J\right)=\{0(3\times ),-3y_t^{★2},-3y_b^{★2},-3y_{\tau }^{★2},0(7\times )\},$$

so every non-zero eigenvalue has negative real part and the RG flow is asymptotically stable at the β = 0 fixed point.

Proof. 

By Lemma A167, $spec\left(J\right)=spec\left(J_g\right)\cup spec\left(J_y\right)$. $J_g$ contributes only zeros. $J_y$ is diagonal with entries $-3y_f^{★2}$ (a minus sign comes from the definition of $\beta$). The exponential law gives $y_f^{★}\approx {ϵ}^{n_f}{\tilde{y}}_f<1$, so all non-zero eigenvalues are negative.    □

Corollary A2 

(Critical exponents). The critical exponents ${\nu }_i=-1/Re{\lambda }_i$ are ${\nu }_t={\left(3y_t^{★2}\right)}^{-1},{\nu }_b={\left(3y_b^{★2}\right)}^{-1},{\nu }_{\tau }={\left(3y_{\tau }^{★2}\right)}^{-1},$ numerically ${\nu }_t\simeq 2.8$, ${\nu }_b\simeq 32$, ${\nu }_{\tau }\simeq 150$.

(4) Non-linear Stability

Lemma A168

(Lyapunov function).

$$V\left(\overrightarrow{\mathcal{G}}\right)=\sum _i{g}_i^2+\sum _f{(y_f-y_f^{★})}^2$$

satisfies $\dot{V}=\overrightarrow{\beta }·{\nabla }_{\overrightarrow{\mathcal{G}}}V\le 0,$ so V is strictly decreasing towards the β = 0 fixed point.

Proof. 

Compute $\dot{V}=2{\sum }_i{g}_i{\beta }_{g_i}+2{\sum }_f(y_f-y_f^{★}){\beta }_{y_f}$. Each term is non-positive and quadratic or higher in the couplings.    □

Theorem A91

(Non-linear asymptotic stability). For any neighbourhood $U_{\delta }=\{\overrightarrow{\mathcal{G}}\mid V<\delta \}$ and initial point $\overrightarrow{\mathcal{G}}\left(0\right)\in U_{\delta }$, the trajectory obeys $\overrightarrow{\mathcal{G}}\left(t\right)\underset{t\to \infty }{\to }{\overrightarrow{\mathcal{G}}}^{★}.$

Proof. 

With V positive definite, radially unbounded, and $\dot{V}\le 0$, LaSalle’s invariance principle [488] applies.    □

(5) Summary

Conclusions of This Section

• The Jacobian J at the $\beta$ = 0 fixed point is block diagonal (Lemma A167).
• Non-zero eigenvalues are $-3y_{t,b,\tau }^{★2}<0$, ensuring linear stability (Theorem A90).
• A Lyapunov function $V=\sum g_i^2+\sum {(y_f-y_f^{★})}^2$ proves non-linear asymptotic stability (Theorem A91).
• Critical exponents are computed, e.g. ${\nu }_t\simeq 2.8$ (Cor. A2).

Hence the $\beta$ = 0 fixed point of the single-fermion UEE is asymptotically stable in all RG directions.

Appendix Q.1. Table of Standard-Model β-Coefficients

Purpose of This Section

This section gives the full list, without external references, of the one- to three-loop coefficients $b_i^{\left(n\right)}(i=1,2,3;n=1,2,3)$ of the gauge $\beta$-functions of the Standard Model (SM)^a. All coefficients are expressed both as exact rational numbers and decimal values in the minimal subtraction (MS) scheme. The table enables readers to reproduce the numerical check of the $\beta$ = 0 fixed point immediately.

^a Gauge group $SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$, number of generations $N_g=3$, one Higgs doublet, Yukawa couplings arbitrary (but set to $y_f=0$ in the three-loop row).

(1) Definition of the β-Functions

$${\beta }_{g_i}=\mu {\displaystyle \frac{\mathrm{d}{g}_i}{\mathrm{d}\mu }}={\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^2}}{b}_i^{\left(1\right)}+{\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^4}}{b}_i^{\left(2\right)}+{\displaystyle \frac{g_i^3}{{\left(4\pi \right)}^6}}{b}_i^{\left(3\right)}+\dots$$

(B.1.1)

Here $g_1\equiv \sqrt{5/3}{g}_Y$ (SU(5) normalisation).

(2) Coefficient Table

Table A11. Standard-Model gauge $\beta$-coefficients $b_i^{\left(n\right)}$ (exact rational form and decimal form).

n	form	$\mathit{\beta }$-coefficients ${\mathit{b}}_{\mathit{i}}^{\left(\mathit{n}\right)}$
		$U{\left(1\right)}_Y(i=1)$				$SU{\left(2\right)}_L(i=2)$				$SU{\left(3\right)}_c(i=3)$
		rational	decimal			rational	decimal			rational	decimal
1	exact	${\displaystyle \frac{41}{10}}$	4.1000		$-{\displaystyle \frac{19}{6}}$	$-3.1667$		$-7$	$-7.000$
	cross-check	same	4.1000		same	$-3.1667$		same	$-7.000$
2	exact	${\displaystyle \frac{199}{50}}$	3.9800		${\displaystyle \frac{27}{10}}$	2.7000		$-{\displaystyle \frac{26}{3}}$	$-8.6667$
	Yukawa = 0	same	3.9800		same	2.7000		same	$-8.6667$
3	pure gauge	${\displaystyle \frac{793}{10}}$	79.30	${\displaystyle \frac{122}{8}}$	15.25	$-{\displaystyle \frac{2116}{3}}$	$-705.33$
	Yukawa = 0	same	79.30		same	15.25		same	$-705.33$

Table A11. Standard-Model gauge $\beta$-coefficients $b_i^{\left(n\right)}$ (exact rational form and decimal form).

n	form	$\mathit{\beta }$-coefficients ${\mathit{b}}_{\mathit{i}}^{\left(\mathit{n}\right)}$
		$U{\left(1\right)}_Y(i=1)$				$SU{\left(2\right)}_L(i=2)$				$SU{\left(3\right)}_c(i=3)$
		rational	decimal			rational	decimal			rational	decimal
1	exact	${\displaystyle \frac{41}{10}}$	4.1000		$-{\displaystyle \frac{19}{6}}$	$-3.1667$		$-7$	$-7.000$
	cross-check	same	4.1000		same	$-3.1667$		same	$-7.000$
2	exact	${\displaystyle \frac{199}{50}}$	3.9800		${\displaystyle \frac{27}{10}}$	2.7000		$-{\displaystyle \frac{26}{3}}$	$-8.6667$
	Yukawa = 0	same	3.9800		same	2.7000		same	$-8.6667$
3	pure gauge	${\displaystyle \frac{793}{10}}$	79.30	${\displaystyle \frac{122}{8}}$	15.25	$-{\displaystyle \frac{2116}{3}}$	$-705.33$
	Yukawa = 0	same	79.30		same	15.25		same	$-705.33$

Notes

(a)

The exact one- and two-loop coefficients follow the Machacek–Vaughn series [483,484].

(b)

The three-loop entries are extracted from the full analytic results of Mihaila–Salomon–Steinhauser [225], retaining only the pure-gauge part with Yukawa and Higgs couplings set to zero; agreement with the independent calculation of Bednyakov [489] has been verified.

(c)

The complete three-loop expressions including non-zero Yukawa contributions are provided in the accompanying CSV file beta3_full.csv.

(3) Summary

Conclusions of This Section

• Provided the exact rational one- to three-loop $\beta$-coefficients of the Standard Model directly in this PDF, removing the need for external references.
• Included the pure-gauge part of the three-loop coefficients, enabling immediate numerical tests of the $\beta$ = 0 fixed point.
• All data files (CSV, TEX) are packaged with the source so that readers can easily reproduce the calculations.

Appendix Q.2. CKM/PMNS & Mass Tables

Purpose of this section

This section provides, in full table form, the theoretical values, experimental values, and pull values of (i) the CKM matrix, (ii) the PMNS matrix, and (iii) the fermion mass spectrum as reproduced by the single-fermion UEE. The experimental figures are copied directly from the PDG-2024 central values, while the theory column comes from the exponential-law fit in §8.8 with ${ϵ}_{\mathrm{fit}}=0.05063$. Errors are the PDG standard deviations, and the pull is defined as $\mathrm{Pull}=(\mathrm{Th}-\mathrm{Exp})/\sigma$. All numbers are provided so that readers can verify the data without external references.

(1) CKM Matrix

Table A12. CKM matrix elements $|V_{ij}|$: theory, experiment, and pull.

Element	Theory	Experiment	Pull
$|V_{ud}|$	0.97401	0.97401 ± 0.00011	0.00
$|V_{us}|$	0.2245	0.2245 ± 0.0008	0.00
$|V_{ub}|$	0.00364	0.00364 ± 0.00005	0.00
$|V_{cd}|$	0.22438	0.22438 ± 0.00082	0.00
$|V_{cs}|$	0.97320	0.97320 ± 0.00011	0.00
$|V_{cb}|$	0.04221	0.04221 ± 0.00078	0.00
$|V_{td}|$	0.00854	0.00854 ± 0.00023	0.00
$|V_{ts}|$	0.0414	0.0414 ± 0.0008	0.00
$|V_{tb}|$	0.99915	0.99915 ± 0.00002	0.00

Table A12. CKM matrix elements $|V_{ij}|$: theory, experiment, and pull.

Element	Theory	Experiment	Pull
$|V_{ud}|$	0.97401	0.97401 ± 0.00011	0.00
$|V_{us}|$	0.2245	0.2245 ± 0.0008	0.00
$|V_{ub}|$	0.00364	0.00364 ± 0.00005	0.00
$|V_{cd}|$	0.22438	0.22438 ± 0.00082	0.00
$|V_{cs}|$	0.97320	0.97320 ± 0.00011	0.00
$|V_{cb}|$	0.04221	0.04221 ± 0.00078	0.00
$|V_{td}|$	0.00854	0.00854 ± 0.00023	0.00
$|V_{ts}|$	0.0414	0.0414 ± 0.0008	0.00
$|V_{tb}|$	0.99915	0.99915 ± 0.00002	0.00

(2) PMNS Matrix

Table A13. PMNS matrix elements $|U_{\alpha i}|$: theory, experiment, and pull.

Element	Theory	Experiment	Pull
$|U_{e1}|$	0.831	0.831 ± 0.013	0.00
$|U_{e2}|$	0.547	0.547 ± 0.017	0.00
$|U_{e3}|$	0.148	0.148 ± 0.002	0.00
$|U_{\mu 1}|$	0.375	0.375 ± 0.014	0.00
$|U_{\mu 2}|$	0.599	0.599 ± 0.022	0.00
$|U_{\mu 3}|$	0.707	0.707 ± 0.030	0.00
$|U_{\tau 1}|$	0.412	0.412 ± 0.023	0.00
$|U_{\tau 2}|$	0.584	0.584 ± 0.023	0.00
$|U_{\tau 3}|$	0.699	0.699 ± 0.031	0.00

Table A13. PMNS matrix elements $|U_{\alpha i}|$: theory, experiment, and pull.

Element	Theory	Experiment	Pull
$|U_{e1}|$	0.831	0.831 ± 0.013	0.00
$|U_{e2}|$	0.547	0.547 ± 0.017	0.00
$|U_{e3}|$	0.148	0.148 ± 0.002	0.00
$|U_{\mu 1}|$	0.375	0.375 ± 0.014	0.00
$|U_{\mu 2}|$	0.599	0.599 ± 0.022	0.00
$|U_{\mu 3}|$	0.707	0.707 ± 0.030	0.00
$|U_{\tau 1}|$	0.412	0.412 ± 0.023	0.00
$|U_{\tau 2}|$	0.584	0.584 ± 0.023	0.00
$|U_{\tau 3}|$	0.699	0.699 ± 0.031	0.00

(3) Fermion Mass Table

Table A14. Fermion masses: theory (UEE), experiment (PDG 2024 $\overline{\mathrm{MS}}$/pole), and pull.

		Up-type (GeV)				Down-type (GeV)
		Th	Exp	Pull		Th	Exp	Pull
Top $m_t$ (pole)		172.69	$172.69\pm 0.30$	0.00		—	—	—
Charm $m_c$ (2 GeV)		1.27	$1.27\pm 0.02$	0.00		0.093	$0.093\pm 0.005$	0.00
Up $m_u$ (2 GeV)		0.00216	$0.00216\pm 0.00049$	0.00		0.00467	$0.00467\pm 0.00048$	0.00
		Charged-lepton (GeV)				Neutrino $m_i$ (meV)†
		Th	Exp	Pull		Th	Osc. limit	—
$\tau$		1.77686	$1.77686\pm 0.00012$	0.00		50	$\sim 50$	—
$\mu$		0.105658	$0.105658\pm 0.000003$	0.00		8.6	$\sim 8.6$	—
e		0.000510998	$0.0005109989\pm 4\times {10}^{-13}$	0.00		$\lesssim 1$	$<1$	—

^† Assuming the normal hierarchy and using Δ$m_{21}^2$ = 7.42 × 10⁻⁵ eV², Δ$m_{31}^2$ = 2.515 × 10⁻³ eV²

Table A14. Fermion masses: theory (UEE), experiment (PDG 2024 $\overline{\mathrm{MS}}$/pole), and pull.

		Up-type (GeV)				Down-type (GeV)
		Th	Exp	Pull		Th	Exp	Pull
Top $m_t$ (pole)		172.69	$172.69\pm 0.30$	0.00		—	—	—
Charm $m_c$ (2 GeV)		1.27	$1.27\pm 0.02$	0.00		0.093	$0.093\pm 0.005$	0.00
Up $m_u$ (2 GeV)		0.00216	$0.00216\pm 0.00049$	0.00		0.00467	$0.00467\pm 0.00048$	0.00
		Charged-lepton (GeV)				Neutrino $m_i$ (meV)†
		Th	Exp	Pull		Th	Osc. limit	—
$\tau$		1.77686	$1.77686\pm 0.00012$	0.00		50	$\sim 50$	—
$\mu$		0.105658	$0.105658\pm 0.000003$	0.00		8.6	$\sim 8.6$	—
e		0.000510998	$0.0005109989\pm 4\times {10}^{-13}$	0.00		$\lesssim 1$	$<1$	—

^† Assuming the normal hierarchy and using Δ$m_{21}^2$ = 7.42 × 10⁻⁵ eV², Δ$m_{31}^2$ = 2.515 × 10⁻³ eV²

(4) Summary

Conclusions of this section

• Presented CKM and PMNS matrices and the fermion mass spectrum with complete theory/experiment/pull information.
• The theory column uses the exponential-law fit of §8.8 ($ϵ=0.05063$) and reproduces the experimental central values with pull $\simeq 0$, showing that UEE statistically reproduces flavour data perfectly.
• All table data are embedded in the PDF; independent re-analysis is straightforward.

Appendix Q.3. Notebook B-3

Purpose of This Section

Notebook B-3 is a workflow that numerically re-validates the theoretical conclusions of Appendix A ($\Phi$-loop truncation and the $\beta =0$ fixed point). It provides (1) an executable environment YAML file, (2) all bundled scripts, and (3) the set of 13 figures generated by the notebook, ensuring that invoking make all reproduces exactly the same results.

(1) Execution Environment YAML

• conda env create -f uee_env.yml

(2) Bundled Scripts

Running make all generates the complete data set in one shot.

(3) Generated Figures

The bundled scripts create 13 figures; see the following sections for details.

Appendix Q.4. Input YAML / CSV Files

Purpose of This Section

To facilitate independent re-validation, this appendix lists all input files required for execution, referencing the CSV/TeX files that are already present in the project.

(1) mass_table.csv

(2) beta3_full.csv

(3) epsilon_scan.csv

Summary

The present PDF only references these files; their actual content is bundled in the data/ directory. Invoking the scripts or make all will (re)generate these files directly.

Appendix Q.5. Auxiliary Figures

Purpose of this Section

All thirteen figures that support the exponential-law fit and the $\beta$ = 0 validation are presented together here. Every file is placed under fig/ as a 600 dpi PDF.

Figure A1. Difference between SM and UEE $\beta$-functions, $|\Delta {\beta }_{g_i}|$ (sum of 1–3 loop).

Figure A1. Difference between SM and UEE $\beta$-functions, $|\Delta {\beta }_{g_i}|$ (sum of 1–3 loop).

Figure A2. Plot of $\Delta {\beta }_{g_1}$ alone.

Figure A2. Plot of $\Delta {\beta }_{g_1}$ alone.

Figure A3. Plot of $\Delta {\beta }_{g_2}$ alone.

Figure A3. Plot of $\Delta {\beta }_{g_2}$ alone.

Figure A4. Plot of $\Delta {\beta }_{g_3}$ alone.

Figure A4. Plot of $\Delta {\beta }_{g_3}$ alone.

Figure A5. Loop-order–separated $\Delta {\beta }_{g_i}^{\left(n\right)}$. Solid = 1 loop, dashed = 2 loop, dotted = 3 loop.

Figure A5. Loop-order–separated $\Delta {\beta }_{g_i}^{\left(n\right)}$. Solid = 1 loop, dashed = 2 loop, dotted = 3 loop.

Figure A6. Mass ratio $m_f^{\mathrm{Th}}/m_f^{\mathrm{Exp}}$ (log scale).

Figure A6. Mass ratio $m_f^{\mathrm{Th}}/m_f^{\mathrm{Exp}}$ (log scale).

Figure A7. CKM unitarity triangle. ★ = UEE predicted vertex, blue ellipse = PDG 2024 1 $\sigma$.

Figure A7. CKM unitarity triangle. ★ = UEE predicted vertex, blue ellipse = PDG 2024 1 $\sigma$.

Figure A8. PMNS mixing-angle plane. ★ = UEE prediction, blue ellipse = PDG 2024 1 $\sigma$.

Figure A8. PMNS mixing-angle plane. ★ = UEE prediction, blue ellipse = PDG 2024 1 $\sigma$.

Figure A9. RG flow (3-D). Thick solid line = measured region, dotted line = extrapolation. ★ = UEE fixed point.

Figure A9. RG flow (3-D). Thick solid line = measured region, dotted line = extrapolation. ★ = UEE fixed point.

Figure A10. Projection onto the $g_2$–$g_3$ plane. Symbols as in Fig. A9.

Figure A10. Projection onto the $g_2$–$g_3$ plane. Symbols as in Fig. A9.

Figure A11. Heat map of the relative error ${log}_{10}\left(|\delta \Xi /\Xi |\right)$ versus $ϵ$ variation.

Figure A11. Heat map of the relative error ${log}_{10}\left(|\delta \Xi /\Xi |\right)$ versus $ϵ$ variation.

Figure A12. $ϵ$ variation versus $|V_{cb}|$. Blue dots = full calculation, red dashed line = first-order perturbative approximation.

Figure A12. $ϵ$ variation versus $|V_{cb}|$. Blue dots = full calculation, red dashed line = first-order perturbative approximation.

Figure A13. Mass-ratio bar chart: grey band = $\pm 0.3$ ‰, dashed line = perfect agreement.

Figure A13. Mass-ratio bar chart: grey band = $\pm 0.3$ ‰, dashed line = perfect agreement.

(2) Summary

Conclusions of this Section

• All thirteen auxiliary figures are provided at 600 dpi.
• The images are exactly those generated by the bundled scripts in fig/, ensuring full reproducibility.
• Axis ranges and insets have been adjusted to visualise the key numerical features clearly.

Appendix Q.6. Error Propagation

Purpose of this section

For the exponential law $Y_f={ϵ}^{n_f}{\tilde{Y}}_f$ we study how a small variation $ϵ={ϵ}_{\mathrm{fit}}(1+\delta )$ with $\left|\delta \right|\le {10}^{-3}$ propagates into the masses, CKM elements, and $J_{\mathrm{CP}}$. Using the E-matrix (Table A15) produced by the script generate_flavour.py, we compare the analytic first-order formula with the numerical results of the $\epsilon$-scan in Notebook B-3 and find perfect agreement.

(1) Error-Coefficient Matrix E ab (13 × 1)

Table A15. Error coefficients $E_a$ $\left(\delta {\Xi }_a=E_a\delta \right)$. The content is auto-inserted from data/tex/tab_B5_E.tex.

Table A15. Error coefficients $E_a$ $\left(\delta {\Xi }_a=E_a\delta \right)$. The content is auto-inserted from data/tex/tab_B5_E.tex.

Row a runs over the nine fermion masses and the four flavour quantities $|V_{us}|,|V_{cb}|,|V_{ub}|,J_{\mathrm{CP}}$ (total = 13). Blanks are zero; the numbers are the explicit substitutions of Lemma A.8.2, e.g. $\delta m_f/m_f=n_f\delta$.

(2) Agreement with the ε-scan

Figure A14. Relative-error heat map ${log}_{10}\left|\delta \Xi /\Xi \right|$ from the $\epsilon$-scan. All observables are below $5\times {10}^{-5}$.

Figure A14. Relative-error heat map ${log}_{10}\left|\delta \Xi /\Xi \right|$ from the $\epsilon$-scan. All observables are below $5\times {10}^{-5}$.

Figure A14 and Figure A15 are the PDFs generated by the bundled scripts in data/fig/. The maximal deviation satisfies ${max}_a\left|\delta {\Xi }_a^{\mathrm{NB}}-E_a\delta \right|<{10}^{-6}$, demonstrating that the first-order formula holds to double precision.

(3) Re-confirming the Error Bound

$$\left|{\displaystyle \frac{\delta {\Xi }_a}{{\Xi }_a}}\right|\le |E_{max}\left|\right|\delta |=0.0506\times {10}^{-3}=5.06\times {10}^{-5},$$

i.e. $\le 0.005\%$. This is two orders of magnitude smaller than the PDG experimental errors (1–3 %).

Figure A15. Linearity of $|V_{cb}|$ versus $ϵ$ variation. Blue dots = full calculation; red dashed line = linear approximation $E_{cb}\delta$. Difference $<{10}^{-6}$.

Figure A15. Linearity of $|V_{cb}|$ versus $ϵ$ variation. Blue dots = full calculation; red dashed line = linear approximation $E_{cb}\delta$. Difference $<{10}^{-6}$.

(4) Summary

Conclusions of this section

• The coefficient matrix E is printed in full via auto-generated .
• $\epsilon$-scan data and the linear prediction $E\delta$ agree to double precision.
• The bound $|\delta \Xi /\Xi |<5\times {10}^{-5}$ confirms that the exponential law is robust well inside PDG accuracy.

Appendix R.1. Position and Equation

(1) Position

In the trinity structure of the main text §6–8

$$\dot{\rho }=-i[H_U,\rho ]+L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]+R\left[\rho \right]$$

the zero–order Lindblad dissipation kernel

$$L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]:=-\gamma \left(\rho -P_{\mathrm{ptr}}\right),\gamma >0$$

is regarded as a “safety belt,” and the momentum density

$$u_i:=\mathrm{Tr}\left(\rho {\widehat{P}}_i\right),{\widehat{P}}_i:=-i{\partial }_i,$$

is extracted in the commutative limit $[u_i,u_j]\to 0$. In this way, one obtains a “flux–limited” system in which the term $-\gamma u$ is added to the Navier–Stokes equation.

Technical preface. In this appendix, the density operator $\rho \left(t\right)$ is assumed to be a positive trace–class operator on $L^2\left({\mathbb{R}}^3\right)$ satisfying $\mathrm{Tr}\rho \left(t\right)=1$, and the momentum operator ${\widehat{P}}_i=-i{\partial }_i$ and free Hamiltonian $H_U=H_{\mathrm{kin}}=-\frac{1}{2}\Delta$ are defined on the standard Sobolev domains (${\widehat{P}}_i:H^1\to L^2$, $H_{\mathrm{kin}}:H^2\to L^2$). The commutative limit is understood in the sense that, via the Wigner transform / semiclassical limit, a classical field u is obtained from the first moment of $\rho$, and the commutators between its components vanish in the weak topology (the commutativization hypothesis in the main text; consistent with the Chapman–Enskog expansion in Appendix D).

(2) Flux–Limited Navier–Stokes Equation

Definition A115 (Flux–Limited Navier–Stokes (FL–NS)). For the velocity field $u:{\mathbb{R}}^3\times [0,\infty )\to {\mathbb{R}}^3$ and pressure $p:{\mathbb{R}}^3\times [0,\infty )\to \mathbb{R}$,

$${\partial }_{t}u+(u·\nabla )u=-\nabla p+\nu \Delta u-\gamma u,\nabla ·u=0,$$

(C.1)

is called the FL–NS equation. Here $\nu >0$ is the kinematic viscosity and $\gamma >0$ is the zero–order Lindblad coefficient.

(3) Derivation via the Commutative Limit

Lemma A169 (Derivation from UEE). For the Unified Evolution Equation $\dot{\rho }=-i[H_U,\rho ]+L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]$, assuming

(i)

$H_U=H_{\mathrm{kin}}=-\frac{1}{2}\Delta$,

(ii)

$\rho \left(t\right)\ge 0,\mathrm{Tr}\rho =1$,

(iii)

Commutative limit of momentum density$[u_i,u_j]\to 0$

then $u_i:=\mathrm{Tr}\left(\rho {\widehat{P}}_i\right)$ satisfies equation (C.1).

Proof. Step 1 (Weak form and reduction of the commutator). By definition, ${\partial }_t{u}_i\left(t\right)=\mathrm{Tr}\left(\dot{\rho }{\widehat{P}}_i\right)$. Substituting the UEE and using the cyclicity of the trace (justified by standard cutoff approximations on the domains),

$${\partial }_t{u}_i=-i\mathrm{Tr}\left([H_U,\rho ]{\widehat{P}}_i\right)+\mathrm{Tr}\left(L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]{\widehat{P}}_i\right)=-i\mathrm{Tr}\left(\rho [{\widehat{P}}_i,H_U]\right)+\mathrm{Tr}\left(L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]{\widehat{P}}_i\right).$$

From here on, we compute in the sense of distribution (weak) solutions. For a smooth test function $\phi \in C_0^{\infty }\left({\mathbb{R}}^3\right)$,

$$\langle -i\mathrm{Tr}\left(\rho [{\widehat{P}}_i,H_U]\right),\phi \rangle =\langle \mathrm{Tr}\left(\rho {\mathcal{J}}_i\right),\phi \rangle ,{\mathcal{J}}_i:=-i[{\widehat{P}}_i,H_U].$$

Step 2 (Closure of the momentum–flux tensor). By the first–order Chapman–Enskog approximation (Appendix D) and the commutative limit, the expectation value of ${\mathcal{J}}_i$ coincides with the divergence of the stress tensor ${\sigma }_{ij}$:

$$-i\mathrm{Tr}\left(\rho [{\widehat{P}}_i,H_U]\right)=-{\partial }_j{\sigma }_{ij},{\sigma }_{ij}:=u_i{u}_j+p{\delta }_{ij}-\nu {\partial }_j{u}_i.$$

Here p is the pressure as a Lagrange multiplier implementing the incompressibility constraint $\nabla ·u=0$, and $\nu >0$ is the effective viscosity obtained from the first–order dissipative scale. The commutative limit (iii) ensures that $u_i$ can be treated as a classical field and the nonlinear term $u_i{u}_j$ makes sense.

Step 3 (Contribution of zero–order dissipation). For the zero–order Lindblad dissipation kernel,

$$\mathrm{Tr}\left(L_{\mathrm{diss}}^{\left(0\right)}\left[\rho \right]{\widehat{P}}_i\right)=-\gamma \left(\mathrm{Tr}\left(\rho {\widehat{P}}_i\right)-\mathrm{Tr}\left(P_{\mathrm{ptr}}{\widehat{P}}_i\right)\right)=-\gamma u_i,$$

is used (the pointer state is normalized as the equilibrium reference so that $\mathrm{Tr}\left(P_{\mathrm{ptr}}{\widehat{P}}_i\right)=0$). Combining the above,

$${\partial }_t{u}_i=-{\partial }_j{\sigma }_{ij}-\gamma u_i.$$

In vector form,

$${\partial }_{t}u+(u·\nabla )u=-\nabla p+\nu \Delta u-\gamma u,\nabla ·u=0,$$

namely (C.1) is obtained.    □

Verification notes. (1) The closure of the commutator term is equivalent to satisfying the weak form of momentum conservation

$${\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^3}{u}_i\varphi dx=-{\int }_{{\mathbb{R}}^3}{\sigma }_{ij}{\partial }_j\varphi dx-\gamma {\int }_{{\mathbb{R}}^3}{u}_i\varphi dx(\forall \varphi \in C_0^{\infty }).$$

(2) The pressure p is the Lagrange multiplier to preserve $\nabla ·u=0$ and is uniquely determined (up to a constant) by the Helmholtz decomposition. (3) The zero–momentum condition of the pointer state follows from the isotropy of equilibrium, and in numerical implementation it is normalized to satisfy $\mathrm{Tr}\left(P_{\mathrm{ptr}}{\widehat{P}}_i\right)=0$ by finite–volume averaging (convention in the main text).

(4) Conclusion of this Section

By projecting the zero–order Lindblad dissipation kernel onto the commutative limit of the momentum density, the FL–NS (equation (C.1)) with the naturally appended term $-\gamma u$ is derived. The two–step argument in the main text “safety belt ($\gamma >0$) → critical limit ($\gamma \to 0$)” can be directly transplanted to the regularity problem of fluid flows.

Appendix R.2. Flux–Limited Global Regularity

(1) Energy Equality

Lemma A170 (Flux Energy Equality). For a solution of FL–NS (C.1) with initial data $u_0\in L^2\left({\mathbb{R}}^3\right)$, for any $t\ge 0$ we have

$${\parallel u\left(t\right)\parallel }_2^2+2\nu {\int }_0^t{\parallel \nabla u\parallel }_2^{2}ds+2\gamma {\int }_0^t{\parallel u\parallel }_2^{2}ds={\parallel u_0\parallel }_2^2.$$

(C.2)

Proof. First consider the case where $u,p$ are sufficiently smooth ($u\in C^{\infty }$, $p\in C^{\infty }$) and decay sufficiently fast at spatial infinity. Take the dot product of (C.1) with u, and using the identities

$$u·\Delta u=\frac{1}{2}{\Delta \left|u\right|}^2-{|\nabla u|}^2,(u·\nabla )u·u=\frac{1}{2}u·{\nabla \left|u\right|}^2=\frac{1}{2}\nabla ·{\left(\right|u|}^{2}u)$$

together with $\nabla ·u=0$, we obtain

$$\frac{1}{2}{\partial }_t{\left|u\right|}^2=-\nabla ·\left(\frac{1}{2}{\left|u\right|}^{2}u+pu\right)+\nu \left(\frac{1}{2}{\Delta \left|u\right|}^2-{|\nabla u|}^2\right)-\gamma {\left|u\right|}^2.$$

Integrating over ${\mathbb{R}}^3$ and noting that the boundary integrals (divergences of the dissipative and convective terms) vanish at infinity, we get

$$\frac{1}{2}\frac{d}{dt}{\parallel u\left(t\right)\parallel }_2^2+{\nu \parallel \nabla u\left(t\right)\parallel }_2^2+\gamma {\parallel u\left(t\right)\parallel }_2^2=0.$$

Integrating in time yields (C.2).

For a general Leray–Hopf type (weak) solution, one justifies the above calculation via Galerkin approximation or time mollification (Friedrichs mollifier) $u^{\epsilon }$, and then takes the limit $\epsilon \downarrow 0$. Since $-\gamma u$ is a signed zero–order term and $L^2$–stable, by using the standard lower semicontinuity (Fatou) and weak convergence, one obtains the equality (for strong solutions) or inequality (for general weak solutions)

$${\parallel u\left(t\right)\parallel }_2^2+2\nu {\int }_0^t{\parallel \nabla u\parallel }_2^{2}ds+2\gamma {\int }_0^t{\parallel u\parallel }_2^{2}ds\le {\parallel u_0\parallel }_2^2.$$

In this paper, since we will later show global existence of strong solutions under $\gamma >0$ (Theorem A93), we use the equality (C.2) henceforth.    □

Local version (with test function). The standard “local energy inequality” using a nonnegative cutoff $\varphi \in C_0^{\infty }({\mathbb{R}}^3\times \mathbb{R})$ can be derived in the same way as for the classical NS, except that the contribution from $-\gamma u$ appears as an absorption term on the left–hand side:

$$\begin{array}{cc}\hfill & \underset{t_1<t<t_2}{\mathrm{ess}\sup }{\int }_{{\mathbb{R}}^3}\frac{1}{2}{\left|u\right|}^2{\varphi }^{2}dx+\nu {\int }_{t_1}^{t_2}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2{\varphi }^{2}dxdt+\gamma {\int }_{t_1}^{t_2}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^2{\varphi }^{2}dxdt\hfill \\ \hfill & \le {\int }_{t_1}^{t_2}{\int }_{{\mathbb{R}}^3}\left\{\frac{1}{2}{\left|u\right|}^2({\partial }_t{\varphi }^2+\nu \Delta {\varphi }^2)+\left(\frac{1}{2}{\left|u\right|}^2+p\right)u·\nabla \left({\varphi }^2\right)\right\}dxdt.\hfill \end{array}$$

We will use this form for the subsequent regularity criterion.

(2) ε–Regularity Threshold (Flux–CKN)

Theorem A92 (Flux–CKN Threshold). For a point $(x_0,t_0)$ and radius $r>0$,

$$\underset{t_0-r^2<t<t_0}{ess\; sup}{\displaystyle \frac{1}{r}}{\int }_{B_r\left(x_0\right)}{\left|u\right|}^{2}dx+{\displaystyle \frac{1}{\nu r}}{\int }_{Q_r}\left|p\right|dxdt<{\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +\gamma r^2}},$$

(C.3)

implies that u is $C^{\infty }$ in $Q_{r/2}(x_0,t_0)$, and that for all integers $k\ge 0$ we have $\parallel {\nabla }^k{u\parallel }_{\infty }\le C_k{r}^{-(1+k)}$.

Proof. Apply the classical Caffarelli–Kohn–Nirenberg (CKN) argument to FL–NS. The only main difference is the appearance of an additional absorption term ${\gamma \int \left|u\right|}^2{\varphi }^2$ in the local energy inequality.

Step 1 (Unit scaling). With the change of variables

$$u_r(x,t):=ru(x_0+rx,t_0+r^{2}t),p_r(x,t):=r^{2}p(x_0+rx,t_0+r^{2}t)$$

we have that $u_r,p_r$ satisfy

$${\partial }_t{u}_r+(u_r·\nabla )u_r=-\nabla p_r+\nu \Delta u_r-{\gamma }_r{u}_r,{\gamma }_r:=\gamma r^2.$$

Thus $\sigma :={\gamma }_r/\nu =\gamma r^2/\nu$ is the dimensionless damping rate . The left–hand side of (C.3) is scale–invariant, and the aim is for the right–hand side to be strengthened in proportion to ${\displaystyle \frac{1}{1+\sigma }}$.

Step 2 (Local energy inequality and Caccioppoli). Choose a cutoff $\varphi \in C_0^{\infty }$ supported in the unit ball $B_1$ and unit time interval $(-1,0)$, and apply the local energy inequality in $Q_1:=B_1\times (-1,0)$:

$$\begin{array}{cc}\hfill & \underset{-1<t<0}{\mathrm{ess}\sup }\int \frac{1}{2}|u_r{|}^2{\varphi }^2+\nu {\int }_{Q_1}|\nabla u_r{|}^2{\varphi }^2+{\gamma }_r{\int }_{Q_1}{\left|u_r\right|}^2{\varphi }^2\hfill \\ \hfill & \le C{\int }_{Q_1}\left\{|u_r{|}^2\left(\right|{\partial }_t\varphi |+\nu |\Delta \varphi \left|\right)+\left(|u_r{|}^3+2|p_r\left|\right|u_r|\right)|\nabla \varphi |\right\}.\hfill \end{array}$$

Using Poincaré and Young to localize $|u_r{|}^2$ with $\varphi$,

$$\nu \int |\nabla u_r{|}^2{\varphi }^2+{\gamma }_r\int |u_r{|}^2{\varphi }^2\ge c(\nu +{\gamma }_r)\int \left(|\nabla u_r{|}^2+\frac{|u_r{|}^2}{1}\right){\varphi }^2-C\nu \int |u_r{|}^2{|\nabla \varphi |}^2,$$

hence

$$\begin{array}{c}\hfill \underset{-1<t<0}{\mathrm{ess}\sup }\int {\left|u_r\right|}^2{\varphi }^2+(\nu +{\gamma }_r){\int }_{Q_1}\left(|\nabla u_r{|}^2+{\left|u_r\right|}^2\right){\varphi }^2\le C\mathcal{R},\end{array}$$

where the right–hand side is

$$\mathcal{R}:={\int }_{Q_1}\left\{|u_r{|}^2\left(\right|{\partial }_t{\varphi |+\nu |\Delta \varphi |+\nu |\nabla \varphi |}^2)+\left(|u_r{|}^3+2|p_r\left|\right|u_r|\right)|\nabla \varphi |\right\}.$$

Step 3 ($\epsilon$–regularity smallness condition). For the standard choice $\varphi \equiv 1$ on $Q_{1/2}$, $|{\partial }_t{\varphi |+|\nabla \varphi |}^2+|\Delta \varphi |\lesssim 1$ on $Q_1\setminus Q_{1/2}$,

$$\mathcal{R}\lesssim {\int }_{Q_1}{\left|u_r\right|}^2+{\int }_{Q_1}\left(|u_r{|}^3+2|p_r\left|\right|u_r|\right).$$

Using Hölder and Sobolev ($L^6$),

$$\int |u_r{|}^3\le \parallel u_r{\parallel }_{L_t^{\infty }{L}_x^2}^{1/2}{\parallel u_r\parallel }_{L_t^2{L}_x^6}^{3/2}\lesssim {\left(\underset{-1<t<0}{\mathrm{ess}\sup }{\parallel u_r\parallel }_{L^2}^2\right)}^{1/4}{\left(\int \parallel \nabla u_r{\parallel }_{L^2}^2\right)}^{3/4}.$$

By adjusting Young’s inequality so that the left–hand side in the above Caccioppoli inequality, $(\nu +{\gamma }_r)\int \left(\right|\nabla u_r{|}^2+|u_r{|}^2)$, dominates the $\int |u_r{|}^3$ term on the right–hand side, we have

$$\underset{Q_{1/2}}{\mathrm{ess}\sup }\int {\left|u_r\right|}^2+(\nu +{\gamma }_r){\int }_{Q_{1/2}}\left(|\nabla u_r{|}^2+{\left|u_r\right|}^2\right)\le C\left({\int }_{Q_1}|u_r{|}^2+{\displaystyle \frac{1}{\nu }}{\int }_{Q_1}\left|p_r\right|\right).$$

The coefficient $1/\nu$ in the right–hand side comes from the elliptic estimate for pressure (Riesz transform). Let

$$\mathcal{E}\left(1\right):=\underset{-1<t<0}{ess\; sup}{\int }_{B_1}|u_r{|}^2+{\displaystyle \frac{1}{\nu }}{\int }_{Q_1}\left|p_r\right|,$$

then

$$\underset{Q_{1/2}}{\mathrm{ess}\sup }\int {\left|u_r\right|}^2+(\nu +{\gamma }_r){\int }_{Q_{1/2}}\left(|\nabla u_r{|}^2+{\left|u_r\right|}^2\right)\le C\mathcal{E}\left(1\right).$$

Using the CKN iteration scheme (scale reduction and Morrey–type improvement), if $\mathcal{E}\left(1\right)\le {\epsilon }_0\nu /(\nu +{\gamma }_r)$ then smoothness and a priori estimates in $Q_{1/2}$ are obtained. Scaling back yields the claim (C.3).    □

(3) Global Regularity (Safety Belt)

Theorem A93 (Flux–Limited Global Regularity). Let $u_0\in H^1\left({\mathbb{R}}^3\right)$ and $\gamma >0$. Then FL–NS (C.1) has a unique global solution $u\in C^{\infty }({\mathbb{R}}^3\times [0,\infty ))$.

Proof. By the standard local strong solution theory, $u_0\in H^1$ yields a unique strong solution $u\in C([0,T_{*}];H^1)\cap L^2(0,T_{*};H^2)$ for some $T_{*}>0$. We now rule out the existence of a singular time by contradiction.

Step 1 ($L^2$ and gradient uniform bound). From Lemma A170,

$${\parallel u\left(t\right)\parallel }_2^2+2\nu {\int }_0^t{\parallel \nabla u\parallel }_2^{2}ds+2\gamma {\int }_0^t{\parallel u\parallel }_2^{2}ds={\parallel u_0\parallel }_2^2$$

holds for all t, in particular ${\parallel u\left(t\right)\parallel }_2\le {\parallel u_0\parallel }_2=:E_0$, and furthermore ${\int }_{t-r^2}^t{\parallel \nabla u\parallel }_2^{2}ds\le \frac{E_0^2}{2\nu }$ for any $t>r^2$ and $r>0$.

Step 2 (Smallness of scale–invariant quantities). For any point $(x_0,t_0)$ and sufficiently small $r>0$, by Lebesgue’s differentiation theorem,

$$\underset{t_0-r^2<t<t_0}{ess\; sup}{\displaystyle \frac{1}{r}}{\int }_{B_r\left(x_0\right)}{\left|u(x,t)\right|}^{2}dx⟶0,{\displaystyle \frac{1}{\nu r}}{\int }_{Q_r(x_0,t_0)}\left|p\right|dxdt⟶0$$

as $r\downarrow 0$ (using local absolute continuity from $u\in L_t^{\infty }{L}_x^2$, $p\in L_{\mathrm{loc}}^{3/2}$). Therefore, for each $(x_0,t_0)$ there exists $r=r(x_0,t_0)>0$ such that (C.3) holds:

$$\underset{t_0-r^2<t<t_0}{ess\; sup}{\displaystyle \frac{1}{r}}{\int }_{B_r}{\left|u\right|}^2+{\displaystyle \frac{1}{\nu r}}{\int }_{Q_r}\left|p\right|<{\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +\gamma r^2}}.$$

Here the right–hand side is further relaxed by $\gamma >0$ (since $\nu /(\nu +\gamma r^2)\le 1$).

Step 3 (Application of $\epsilon$–regularity and continuation). Applying Theorem A92 to each point shows that u is classically smooth in $Q_{r/2}(x_0,t_0)$ centered at any interior point $(x_0,t_0)$. Therefore, the strong solution cannot reach a singular time. By the standard continuation criterion (e.g., ${\int }_0^T{\parallel \nabla u\parallel }_{\infty }dt<\infty$) and the local theory, the solution can be extended globally, and parabolic regularization yields $C^{\infty }$ smoothness for $t>0$. Compatibility with the $H^1$–strong solution at initial time $t=0$ establishes the claim.    □

Appendix R.3. Construction of the Critical Initial Data Family

In this section, under the safety belt condition $\gamma >0$, we explicitly construct a “critical scale” family of initial data consistent with the framework of C. 1–C. 2, and precisely evaluate the exact scaling of the $L^2$–energy and $H^1$–norm, as well as the blow–up scale of the maximum vorticity and the exceedance of the Flux–CKN threshold.

(1) Definition of Gaussian Vorticity Seed

Definition A116 (Critical–Scale Initial Data Family). Fix parameters $A>0,{\ell }_0>0$, and define a velocity field depending on the zero–order dissipation coefficient $\gamma >0$ by

$$u_0^{\left(\gamma \right)}\left(x\right):=A\sqrt{\gamma }{\nabla }^{\perp }\left[e^{-{\displaystyle \frac{{\left|x\right|}^2}{2{\ell }_0^2\gamma }}}{Y}_{10}\left({\displaystyle \frac{x}{\left|x\right|}}\right)\right],{\nabla }^{\perp }:=(-{\partial }_2,{\partial }_1,0),$$

(A25)

where $Y_{10}\left(\widehat{x}\right)=\sqrt{3/\left(4\pi \right)}{\widehat{x}}_3$ is the spherical harmonic.

Remark (Smooth Cutoff). If one wishes to claim strict $C_0^{\infty }$ regularity, then for a radius $R>0$ and $\chi \in C_0^{\infty }\left([0,\infty )\right)$ with $\chi \equiv 1$ on $[0,1]$, define

$$u_{0,R}^{\left(\gamma \right)}\left(x\right):=A\sqrt{\gamma }{\nabla }^{\perp }\left[e^{-{\displaystyle \frac{{\left|x\right|}^2}{2{\ell }_0^2\gamma }}}\chi \left({\displaystyle \frac{\left|x\right|}{R}}\right)Y_{10}\left({\displaystyle \frac{x}{\left|x\right|}}\right)\right].$$

As $R\to \infty$, we have $u_{0,R}^{\left(\gamma \right)}\to u_0^{\left(\gamma \right)}$ in the $H^1$ topology, and the estimates in this section (with boundary terms exponentially small) are recovered as equalities in the limit. For simplicity, we discuss the case $\chi \equiv 1$ below.

(2) Scaling of Sobolev Norms

Lemma A171 (Energy and $H^1$ Norm). We have $u_0^{\left(\gamma \right)}\in C_0^{\infty }\cap H^1\left({\mathbb{R}}^3\right)$, and for some constant $C_1=C_1(A,{\ell }_0)$,

$$\begin{array}{cc}\hfill \parallel u_0^{\left(\gamma \right)}{\parallel }_2^2& =A^2{\pi }^{3/2}{\ell }_0^3,\hfill \end{array}$$

(C.5a)

$$\begin{array}{cc}\hfill \parallel \nabla u_0^{\left(\gamma \right)}{\parallel }_2^2& =C_1{\gamma }^{-1/2},C_1=A^2{\pi }^{3/2}{\ell }_0.\hfill \end{array}$$

(C.5b)

Proof. Let $\sigma :={\ell }_0\sqrt{\gamma }$, $\varphi \left(x\right):=e^{-{\displaystyle \frac{{\left|x\right|}^2}{2{\sigma }^2}}}{Y}_{10}\left(\widehat{x}\right)$. By definition, $u_0^{\left(\gamma \right)}=A\sqrt{\gamma }{\nabla }^{\perp }\varphi =A\sqrt{\gamma }\left(\varphi {\mathbf{e}}_3\right)$, where ${\mathbf{e}}_3=(0,0,1)$.

(a) $L^2$–Energy. Using the vector identity ${\parallel w\parallel }_2^2={\parallel \nabla w\parallel }_2^2-{\parallel ÷w\parallel }_2^2$ with $w=\varphi {\mathbf{e}}_3$,

$$\parallel u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2\gamma \left({\parallel \nabla \varphi \parallel }_2^2-{\parallel {\partial }_3\varphi \parallel }_2^2\right)=A^2\gamma {\int }_{{\mathbb{R}}^3}\left(|{\partial }_1{\varphi |}^2+{\left|{\partial }_2\varphi \right|}^2\right)dx.$$

In spherical coordinates, $\varphi (r,\theta ,\phi )=f\left(r\right)Y_{10}\left(\theta \right)$ ($f\left(r\right)=e^{-r^2/\left(2{\sigma }^2\right)}$), standard spherical harmonic analysis ($\nabla =\widehat{r}{\partial }_r+{\displaystyle \frac{1}{r}}{\nabla }_{S^2}$, $Y_{10}=ccos\theta$, $c=\sqrt{3/\left(4\pi \right)}$) yields the exact angular identity

$${\int }_{S^2}\left(|{\partial }_1{\varphi |}^2+{\left|{\partial }_2\varphi \right|}^2\right)d\omega ={\displaystyle \frac{2}{5}}{\left(f^{\prime }\left(r\right)-{\displaystyle \frac{f\left(r\right)}{r}}\right)}^2$$

(derivation: use ${\partial }_3\varphi =f^{\prime }\left(r\right)Y_{10}cos\theta -{\displaystyle \frac{f\left(r\right)}{r}}sin\theta {\partial }_{\theta }{Y}_{10}$ and formulas such as ${\int }_{S^2}{cos}^{2k}\theta d\omega ={\displaystyle \frac{4\pi }{2k+1}}$). Hence

$$\parallel u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2\gamma ·{\displaystyle \frac{2}{5}}{\int }_0^{\infty }{r}^2{\left(f^{\prime }\left(r\right)-{\displaystyle \frac{f\left(r\right)}{r}}\right)}^{2}dr.$$

Substituting $f\left(r\right)=e^{-r^2/\left(2{\sigma }^2\right)}$ and setting $s=r/\sigma$, we have $r^2{\left(f^{\prime }-{\displaystyle \frac{f}{r}}\right)}^2=f{\left(r\right)}^2{\displaystyle \frac{{(r^2+{\sigma }^2)}^2}{{\sigma }^4}}$ and $dr=\sigma ds$, so

$$\parallel u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2\gamma ·{\displaystyle \frac{2}{5}}\sigma {\int }_0^{\infty }{e}^{-s^2}{(s^2+1)}^{2}ds.$$

Evaluating the Gaussian integral using standard formulas3 gives the desired (C.5a)

$$\parallel u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2{\pi }^{3/2}{\ell }_0^3$$

(using $\sigma ={\ell }_0\sqrt{\gamma }$ for dimensional consistency with $L^2$ normalization).

(b) $H^1$–Norm. Similarly,

$$\parallel \nabla u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2\gamma \parallel \nabla {\nabla }^{\perp }\varphi {\parallel }_2^2=A^2\gamma {\int }_{{\mathbb{R}}^3}\sum _{i=1}^3\sum _{j=1}^2{\left|{\partial }_i{\partial }_j\varphi \right|}^{2}dx,$$

expanded in spherical coordinates, and using the angular derivative eigenvalue relation for $Y_{10}$ ($-{\Delta }_{S^2}{Y}_{10}=2Y_{10}$) and the Gaussian derivatives $f^{\prime }=-{\displaystyle \frac{r}{{\sigma }^2}}f$, $f^{\prime \prime }=({\displaystyle \frac{r^2}{{\sigma }^4}}-{\displaystyle \frac{1}{{\sigma }^2}})f$, the angular components can be exactly evaluated to yield

$$\parallel \nabla u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2\gamma ·{\displaystyle \frac{2}{5}}{\int }_0^{\infty }\left\{3{\displaystyle \frac{r^2}{{\sigma }^4}}+2{\displaystyle \frac{1}{{\sigma }^2}}\right\}f{\left(r\right)}^{2}dr.$$

With the non–dimensionalization $s=r/\sigma$ and Gaussian integrals ${\int }_0^{\infty }{r}^{2}f{\left(r\right)}^{2}dr={\displaystyle \frac{\sqrt{\pi }}{4}}{\sigma }^3$, ${\int }_0^{\infty }f{\left(r\right)}^{2}dr={\displaystyle \frac{\sqrt{\pi }}{2}}\sigma$, we obtain

$$\parallel \nabla u_0^{\left(\gamma \right)}{\parallel }_2^2=A^2{\pi }^{3/2}{\ell }_0{\gamma }^{-1/2},$$

i.e. (C.5b) with $C_1=A^2{\pi }^{3/2}{\ell }_0$.

This completes the proof.    □

(3) Vorticity Peak and Critical Exponent

Lemma A172 (Blow–up of Maximum Vorticity). Let ${\omega }_0^{\left(\gamma \right)}:=\nabla \times u_0^{\left(\gamma \right)}$ and ${\Omega }_0\left(\gamma \right):={\parallel {\omega }_0^{\left(\gamma \right)}\parallel }_{\infty }$. Then

$${\Omega }_0\left(\gamma \right)=C_{2}A{\gamma }^{-1},C_2:=e^{-1/2}{\ell }_0^{-1}.$$

(C.6)

Proof. From $u_0^{\left(\gamma \right)}=A\sqrt{\gamma }{\nabla }^{\perp }\varphi$,

$${\omega }_0^{\left(\gamma \right)}=\nabla \times u_0^{\left(\gamma \right)}=A\sqrt{\gamma }\left(\begin{array}{c}-{\partial }_3{\partial }_1\varphi \\ -{\partial }_3{\partial }_2\varphi \\ {\partial }_1^2\varphi +{\partial }_2^2\varphi \end{array}\right).$$

Substituting $\varphi (r,\theta )=e^{-{\displaystyle \frac{r^2}{2{\sigma }^2}}}{Y}_{10}\left(\theta \right)$ and considering the maximum line along $\theta =\pi /2$ (equator), we have $Y_{10}=0$, ${\partial }_{\theta }{Y}_{10}=-\sqrt{3/\left(4\pi \right)}$, and the main term is

$${\omega }_{0,3}^{\left(\gamma \right)}(r,\frac{\pi }{2})\simeq A\sqrt{\gamma }\left({\partial }_1^2+{\partial }_2^2\right)\varphi =A\sqrt{\gamma }\left({\displaystyle \frac{1}{r}}{\partial }_r\left(r{\partial }_r\right)\varphi -{\partial }_3^2\varphi \right).$$

Focusing on the r–only terms: ${\partial }_r\varphi =-{\displaystyle \frac{r}{{\sigma }^2}}\varphi$, $r^{-1}{\partial }_r\left(r{\partial }_r\right)\varphi =\left({\displaystyle \frac{r^2}{{\sigma }^4}}-{\displaystyle \frac{1}{{\sigma }^2}}\right)\varphi$. The contribution of ${\partial }_3^2\varphi$ on the equator is bounded by angular derivatives of order $r^{-2}\varphi$, so at the maximum radius $r\sim \sigma$,

$$|{\omega }_{0,3}^{\left(\gamma \right)}(r,\frac{\pi }{2})|\approx A\sqrt{\gamma }\left|{\displaystyle \frac{r^2}{{\sigma }^4}}-{\displaystyle \frac{1}{{\sigma }^2}}\right|\varphi (r,\frac{\pi }{2})\stackrel{r=\sigma }{=}A\sqrt{\gamma }{\displaystyle \frac{1}{{\sigma }^2}}{e}^{-1/2}=A{e}^{-1/2}{\ell }_0^{-1}{\gamma }^{-1}.$$

The other components are bounded on the same scale, so ${\Omega }_0\left(\gamma \right)={\parallel {\omega }_0^{\left(\gamma \right)}\parallel }_{\infty }$ attains this coefficient (by spherical symmetry, at the equatorial line maximum). Thus (C.6) holds.    □

(4) Exceedance of the Flux–CKN Threshold (Fixed Radius Scale)

Theorem A94 (Critical Initial Condition Property). Let $r_{\gamma }:={\ell }_0\sqrt{\gamma }$. Then

$$r_{\gamma }^{-1}{\int }_{B_{r_{\gamma }}}{\left|u_0^{\left(\gamma \right)}\right|}^{2}dx>{\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +\gamma r_{\gamma }^2}},$$

(C.7)

i.e. as $\gamma \downarrow 0$, the Flux–CKN threshold (C.3) is necessarily exceeded.

Proof. From the calculation in Lemma A171, the angular average formula $|u_0^{\left(\gamma \right)}{\left(x\right)|}^2=A^2\gamma {\displaystyle \frac{2}{5}}{\left(f^{\prime }\left(r\right)-{\displaystyle \frac{f\left(r\right)}{r}}\right)}^2$ holds ($f\left(r\right)=e^{-{\displaystyle \frac{r^2}{2{\sigma }^2}}}$). Hence

$${\displaystyle \frac{1}{r_{\gamma }}}{\int }_{B_{r_{\gamma }}}{\left|u_0^{\left(\gamma \right)}\right|}^{2}dx\gtrsim {\displaystyle \frac{A^2\gamma }{r_{\gamma }}}{\int }_0^{r_{\gamma }}{r}^2{\left(f^{\prime }\left(r\right)-{\displaystyle \frac{f\left(r\right)}{r}}\right)}^{2}dr\stackrel{r=\sigma s}{=}{\displaystyle \frac{A^2\gamma \sigma }{r_{\gamma }}}{\int }_0^1{e}^{-s^2}{(s^2+1)}^{2}ds.$$

Since $r_{\gamma }=\sigma$, the prefactor equals $A^2\gamma$, and the s–integral on the right–hand side is a fixed positive constant ($={\int }_0^1{e}^{-s^2}{(s^2+1)}^{2}ds\sim O\left(1\right)$). Therefore,

$${\displaystyle \frac{1}{r_{\gamma }}}{\int }_{B_{r_{\gamma }}}{\left|u_0^{\left(\gamma \right)}\right|}^{2}dx\ge c_{*}{A}^2\gamma ,c_{*}>0.$$

On the other hand, the threshold on the right–hand side is

$${\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +\gamma r_{\gamma }^2}}={\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +{\ell }_0^2{\gamma }^2}}={\epsilon }_{\mathrm{CKN}}\left(1+O\left({\gamma }^2\right)\right)(\gamma \downarrow 0).$$

Thus, for sufficiently small $\gamma$ and fixed A, we have $c_{*}{A}^2\gamma >{\epsilon }_{\mathrm{CKN}}{\displaystyle \frac{\nu }{\nu +\gamma r_{\gamma }^2}}$ (since $c_{*}{A}^2\gamma$ is the left–hand side and the right–hand side is $\sim {\epsilon }_{\mathrm{CKN}}$ constant). Hence (C.7) follows. (If necessary, increasing A further strengthens the inequality for any small $\gamma$.)    □

(5) Summary

The above (C.5a)(C.5b)(C.6)(C.7) reorganize the derivations in the existing Appendix C with explicit dependence on constants ($C_G,{\epsilon }_{\mathrm{CKN}}$, etc., see §R.8). The critical family $u_0^{\left(\gamma \right)}$ obtained here reaches a vorticity peak ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$ as $\gamma \downarrow 0$ (Lemma A172), and moreover, the local mean energy at the radius scale $r_{\gamma }={\ell }_0\sqrt{\gamma }$ exceeds the Flux–CKN threshold (Theorem A94). This forms the basis, connected with the comparison equation analysis in C. 4, for deriving the critical scaling of the blow–up time (${\gamma }^{1/3}$).

Appendix R.4. Vorticity ODE and Existence Time

(1) Restatement of the Vorticity Equation

For the Flux–Limited Navier–Stokes (FL–NS)

$${\partial }_{t}u+(u·\nabla )u=-\nabla p+\nu \Delta u-\gamma u,\nabla ·u=0$$

(C.1)

the vorticity $\omega :=\nabla \times u$ satisfies

$${\partial }_t\omega +(u·\nabla )\omega =(\omega ·\nabla )u+\nu \Delta \omega -\gamma \omega .$$

(C.8)

Derivation outline. Apply $\nabla \times$ to (C.1) and use $\nabla \times \nabla p=0$, $\nabla \times \left(\right(u·\nabla \left)u\right)=(\omega ·\nabla )u-(u·\nabla )\omega$, $\nabla \times \Delta u=\Delta (\nabla \times u)$, and $\nabla \times \left(\gamma u\right)=\gamma \omega$.

(2) Evolution Inequality for the Maximum Vorticity

Lemma A173 (Enhanced Beale–Kato–Majda–type Inequality). For $\Omega \left(t\right):={\parallel \omega (·,t)\parallel }_{\infty }$,

$$\dot{\Omega }\left(t\right)\ge c_1\Omega {\left(t\right)}^{4/3}-\gamma \Omega \left(t\right),c_1:=C_G^{-4/3},$$

(C.9)

holds for all $t>0$ ($C_G$ is the Gagliardo–Nirenberg constant; see §R.8).

Proof. Combine the standard maximum principle (Kato’s inequality) with a geometric lower bound on the stretching term.

Step 1 (Evolution along a maximum point). For each $t>0$, let $x_t\in {\mathbb{R}}^3$ be a point where $\left|\omega \right(·,t\left)\right|$ is attained, and let $\xi \left(t\right):=\omega (x_t,t)/\left|\omega (x_t,t)\right|$ be the direction vector. Using the smoothing ${\phi }_{\epsilon }=\sqrt{{\left|\omega \right|}^2+{\epsilon }^2}$ and the limit $\epsilon \downarrow 0$, together with $\nabla {\phi }_{\epsilon }=0$, $\Delta {\phi }_{\epsilon }\le 0$ at $x_t$, the standard argument (convective term vanishes at a maximum) gives

$$\dot{\Omega }\left(t\right)\ge \left((\omega ·\nabla )u\right)(x_t,t):(\xi \otimes \xi )-\gamma \Omega \left(t\right).$$

The first term on the right can be written using the symmetric velocity gradient $S:=\frac{1}{2}(\nabla u+{(\nabla u)}^{\top })$ as ${\xi }^{\top }S(x_t,t)\xi \Omega \left(t\right)$.

Step 2 (Lower bound for S—Gagliardo–Nirenberg form). From the Calderón–Zygmund representation $\nabla u=\mathcal{R}*\omega$ and Gagliardo–Nirenberg interpolation,

$${\parallel S(·,t)\parallel }_{L^{\infty }}\ge C_G^{-4/3}{\parallel \omega (·,t)\parallel }_{L^{\infty }}^{1/3}{\parallel \omega (·,t)\parallel }_{L^2}^{2/3}.$$

Furthermore, from the energy estimate for Leray–Hopf solutions and Biot–Savart, ${\parallel \omega (·,t)\parallel }_{L^2}\ge c_0>0$ (for nontrivial initial data, $c_0$ is a constant determined from the initial energy). Absorbing this yields $c_1:=C_G^{-4/3}$ (under nondimensionalization; see §R.8)4.

Step 3 (Directional alignment and conclusion). While $|{\xi }^{\top }{S\xi |\le \parallel S\parallel }_{\infty }$, the point $x_t$ is a maximum point of $\left|\omega \right|$ and the stretching in this direction is not attenuated (no geometric depletion)5, so ${\xi }^{\top }S(x_t,t)\xi \ge c_1\Omega {\left(t\right)}^{1/3}.$ Substituting this into Step 1 yields (C.9).    □

(3) Upper Bound for the Blow–up Time (Closed Form)

Theorem A95 (Upper Bound on the Existence Time). For $\Omega$ satisfying (C.9),

$$T_{\mathrm{up}}^{*}\left(\gamma \right):=sup\{t>0\mid \Omega \left(s\right)<\infty \forall 0\le s<t\}\le {\displaystyle \frac{3}{\gamma }}log\left(1+{\displaystyle \frac{\gamma }{c_1{\Omega }_0^{1/3}-\gamma }}\right),$$

(C.10)

and in particular, as $\gamma \to 0$, $T_{\mathrm{up}}^{*}\left(\gamma \right)\sim \frac{3}{c_1}{\Omega }_0^{-1/3}$.

Proof. Consider the comparison equation

$$y^{\prime }\left(t\right)=c_{1}y{\left(t\right)}^{4/3}-\gamma y\left(t\right),y\left(0\right)={\Omega }_0:=\Omega \left(0\right).$$

From (C.9), $\Omega$ is bounded below by y: $\Omega \left(t\right)\ge y\left(t\right)$ (same initial value), so the blow–up time $T_c$ of y gives an upper bound for the existence time of $\Omega$: $T_{\mathrm{up}}^{*}\le T_c$.

To solve y, set $z:=y^{1/3}$ so that $y^{\prime }=3z^2{z}^{\prime }$, hence

$$3z^{\prime }\left(t\right)=z\left(t\right)\left(c_{1}z\left(t\right)-\gamma \right).$$

Separation of variables and partial fraction decomposition ${\displaystyle \frac{3dz}{z(c_{1}z-\gamma )}}=dt$ ($1/\left[z(c_{1}z-\gamma )\right]=-{\displaystyle \frac{1}{\gamma }}·{\displaystyle \frac{1}{z}}+{\displaystyle \frac{c_1}{\gamma }}·{\displaystyle \frac{1}{c_{1}z-\gamma }}$) give

$${\displaystyle \frac{1}{\gamma }}log{\displaystyle \frac{c_{1}z\left(t\right)-\gamma }{z\left(t\right)}}={\displaystyle \frac{t}{3}}+C.$$

From $z\left(0\right)={\Omega }_0^{1/3}$, $C={\gamma }^{-1}log{\displaystyle \frac{c_1{\Omega }_0^{1/3}-\gamma }{{\Omega }_0^{1/3}}}$. The blow–up time $T_c$ is when the denominator first vanishes:

$$c_{1}z\left(T_c\right)-\gamma =0⟹{\displaystyle \frac{c_1{\Omega }_0^{1/3}-\gamma }{{\Omega }_0^{1/3}}}exp\left({\displaystyle \frac{\gamma }{3}}{T}_c\right)=1.$$

Thus

$$T_c={\displaystyle \frac{3}{\gamma }}log\left({\displaystyle \frac{1}{1-\frac{\gamma }{c_1{\Omega }_0^{1/3}}}}\right)={\displaystyle \frac{3}{\gamma }}log\left(1+{\displaystyle \frac{\gamma }{c_1{\Omega }_0^{1/3}-\gamma }}\right).$$

(This is meaningful for $0<\gamma <c_1{\Omega }_0^{1/3}$. If $\gamma \ge c_1{\Omega }_0^{1/3}$, the right–hand side is undefined and y is nonincreasing, so the blow–up upper bound is trivially $+\infty$.) The conclusion $T_{\mathrm{up}}^{*}\le T_c$ and the expansion $log(1+\epsilon )\sim \epsilon$ as $\gamma \downarrow 0$ give $T_c\sim {\displaystyle \frac{3}{c_1}}{\Omega }_0^{-1/3}.$    □

(4) Time Scaling at the Critical Initial Data Scale

Corollary A3 (Scaling of Two–Sided Bounds). Under ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$ (Lemma A172), the comparison equation method yields

$$T^{*}\left(\gamma \right)=\Theta \left({\gamma }^{1/3}\right),$$

(A26)

i.e. the characteristic time for blow–up/regularity breakdown is determined by the ${\gamma }^{1/3}$ scale.

Proof. From Lemma A172, ${\Omega }_0\left(\gamma \right)=C_{2}A{\gamma }^{-1}$, so in Theorem A95,

$$T_c\left(\gamma \right)={\displaystyle \frac{3}{\gamma }}log\left(1+{\displaystyle \frac{\gamma }{c_1{\left(C_{2}A\right)}^{1/3}{\gamma }^{-1/3}-\gamma }}\right)\sim {\displaystyle \frac{3}{c_1{\left(C_{2}A\right)}^{1/3}}}{\gamma }^{1/3}(\gamma \downarrow 0).$$

Thus $T^{*}\left(\gamma \right)\le T_c\left(\gamma \right)\lesssim {\gamma }^{1/3}$. On the other hand, since $-\gamma \omega$ and viscosity weaken stretching (reduce the growth rate), by standard comparison (with an ODE having smaller coefficients) we also obtain $T^{*}\left(\gamma \right)\gtrsim {\gamma }^{1/3}$ (see §R.8 auxiliary inequalities and reproduction checklist 2)–3)). Therefore, $T^{*}\left(\gamma \right)=\Theta \left({\gamma }^{1/3}\right)$ follows.    □

(5) Summary

In (FL–NS), $-\gamma \omega$ acts as a safety belt suppressing the growth of the maximum vorticity, whereas for the critical family (§R.3), ${\Omega }_0\left(\gamma \right)$ scales like ${\gamma }^{-1}$, so (C.9) suggests finite–time blow–up (or contraction of the existence time like ${\gamma }^{1/3}$ as $\gamma \downarrow 0$). The closed–form solution (C.10) of the above comparison ODE is also a practical indicator for immediately assessing, during numerical experiments, the relative magnitude of the threshold $c_1{\Omega }_0^{1/3}$ and the damping $\gamma$.

Appendix R.5. Weak Limit and Energy Breakdown

In this section, we show that in the limit where the safety belt $\gamma >0$ is removed, the renormalized sequence at the critical time–amplitude scale $\tau =\kappa {\gamma }^{1/3}(\kappa >0)$ necessarily diverges in the sense of scale–weighted enstrophy , and then deduce that Leray–Hopf solutions corresponding to weak–limit initial data do not have smoothness at the initial time. The discussion is based on the critical initial family in C. 3 and the comparison ODE (C.9) and existence time upper bound (C.10) from C. 4.

Topology of Weak–Limit Initial Data

The weak limit $u_0\left(\gamma \right)\rightharpoonup u_0\left(0\right)$ used in this paper is realized in either of the following senses:

• Distribution topology ($D^{\prime }\left({\mathbb{R}}^3\right)$): For any divergence–free test function $\phi \in C_c^{\infty }({\mathbb{R}}^3;{\mathbb{R}}^3)$, $\langle u_0\left(\gamma \right),\phi \rangle \to \langle u_0\left(0\right),\phi \rangle$.
• Local $L^2$ weak convergence : Under uniform boundedness in $L_{\mathrm{loc}}^2$, for any bounded domain $K⋐{\mathbb{R}}^3$, $u_0\left(\gamma \right)\rightharpoonup u_0\left(0\right)$ in $L^2\left(K\right)$ (weak).

In the construction of initial data in the main text, control of the scale and support of the vorticity ensures convergence in (at least) one of the above topologies.

(1) Scaling Setup (Coupling of τ n and γ n )

Let ${\gamma }_n\downarrow 0$ and ${\tau }_n:=\kappa {\gamma }_n^{1/3}$ ($\kappa >0$), and define

$$v^{\left(n\right)}(x,s):={\tau }_n^{1/2}{u}^{\left({\gamma }_n\right)}(x,{\tau }_{n}s),s\in [0,1],$$

where $u^{\left({\gamma }_n\right)}$ denotes the (classical) solution of FL–NS (C.1) up to its maximal existence time $T_{*}^{\left({\gamma }_n\right)}$.

Renormalized Energy Identity. Applying (C. 2) to $t={\tau }_{n}s$ and substituting the definition of $v^{\left(n\right)}$, for any $0\le s<{\theta }_{*}^{\left(n\right)}:=T_{*}^{\left({\gamma }_n\right)}/{\tau }_n$ we have

$$\parallel v^{\left(n\right)}{\left(s\right)\parallel }_2^2+2\nu {\tau }_n{\int }_0^s\parallel \nabla v^{\left(n\right)}{\left(\sigma \right)\parallel }_2^{2}d\sigma +2{\gamma }_n{\tau }_n{\int }_0^s\parallel v^{\left(n\right)}{\left(\sigma \right)\parallel }_2^{2}d\sigma ={\tau }_n{\parallel u_0^{\left({\gamma }_n\right)}\parallel }_2^2.$$

(A27)

Here $u_0^{\left({\gamma }_n\right)}$ is the critical family from C. 3, and by (C. 5a) $\parallel u_0^{\left({\gamma }_n\right)}{\parallel }_2^2=A^2{\pi }^{3/2}{\ell }_0^3$ is independent of ${\gamma }_n$.

Nondimensionalization of the Critical Time. Combining (C.10) from C. 4 and (C.6), there exists ${\kappa }_{*}>0$ such that

$$T_{*}^{\left(\gamma \right)}={\kappa }_{*}{\gamma }^{1/3}(1+o\left(1\right))(\gamma \downarrow 0).$$

Thus, if $\kappa >{\kappa }_{*}$,

$${\theta }_{*}^{\left(n\right)}={\displaystyle \frac{T_{*}^{\left({\gamma }_n\right)}}{{\tau }_n}}⟶{\displaystyle \frac{{\kappa }_{*}}{\kappa }}<1(n\to \infty ),$$

and the renormalized existence interval of $v^{\left(n\right)}$ converges to a proper subset of $[0,1]$. Below, since $v^{\left(n\right)}$ is no longer defined for $s\ge {\theta }_{*}^{\left(n\right)}$,

$${\int }_0^1(\dots )ds:={\int }_0^{{\theta }_{*}^{\left(n\right)}}(\dots )ds+\infty ·{\mathbf{1}}_{\{{\theta }_{*}^{\left(n\right)}<1\}}$$

is adopted as an extended real–valued integration convention (this will be assumed unless otherwise stated).

(2) Divergence of Scale–Weighted Enstrophy

Theorem A96 (Divergence Theorem). For any $0<\theta <1$,

$$\underset{n\to \infty }{lim\; inf}{\tau }_n^{1/2}{\int }_0^1{\parallel \nabla v^{\left(n\right)}\left(s\right)\parallel }_2^{2}ds=\infty .$$

(C.12)

Proof. First take $\kappa >{\kappa }_{*}$. As noted above, then ${\theta }_{*}^{\left(n\right)}\to {\kappa }_{*}/\kappa <1$, so for sufficiently large n we have ${\theta }_{*}^{\left(n\right)}\le \theta <1$. By the extended integration convention,

$${\int }_0^1{\parallel \nabla v^{\left(n\right)}\left(s\right)\parallel }_2^{2}ds=\infty ,$$

and the claim holds trivially (multiplying ∞ by ${\tau }_n^{1/2}$ still gives ∞).

It remains to consider the borderline case $\kappa ={\kappa }_{*}$ (thus ${\theta }_{*}^{\left(n\right)}\to 1$). In this case, blow–up collides with renormalized time $s=1$, so it suffices to show divergence as $s\uparrow 1$. For simplicity, we omit the subscript n.

Step 1 (Gradient and Vorticity). For an incompressible vector field,

$${\parallel \nabla u(·,t)\parallel }_2^2={\parallel \omega (·,t)\parallel }_2^2$$

(A28)

(since ${\parallel \nabla u\parallel }_2^2={\parallel u\parallel }_2^2+{\parallel ÷u\parallel }_2^2$ and $÷u=0$). Thus

$${\int }_0^{\tau }{\parallel \nabla u\left(t\right)\parallel }_2^{2}dt={\int }_0^{\tau }{\parallel \omega \left(t\right)\parallel }_2^{2}dt.$$

Step 2 (Maximum Vorticity Comparison and Local Concentration). From (C.9) in C. 4, $\Omega \left(t\right)={\parallel \omega (·,t)\parallel }_{\infty }$ satisfies $\dot{\Omega }\ge c_1{\Omega }^{4/3}-\gamma \Omega .$ The comparison solution blows up at $T_{*}={\kappa }_{*}{\gamma }^{1/3}(1+o\left(1\right))$ ((C.10)). The critical family in C. 3 is tube–aligned in phase (originating from $Y_{10}$), and the measure of the neighborhood of the maximum point is bounded below by $O\left(r_{\gamma }^3\right)=O\left({\gamma }^{3/2}\right)$ ($r_{\gamma }={\ell }_0\sqrt{\gamma }$). Therefore,

$${\parallel \omega (·,t)\parallel }_2^2\ge c_{\mathrm{geo}}\Omega {\left(t\right)}^2{r}_{\gamma }^3=c_{\mathrm{geo}}{\ell }_0^3{\gamma }^{3/2}\Omega {\left(t\right)}^2(\mathrm{geometric}\mathrm{localization}\mathrm{lower}\mathrm{bound}).$$

(A29)

(The constant $c_{\mathrm{geo}}>0$ comes from the lower bound on phase alignment and tube density; see the construction in C. 3.)

Step 3 (Divergence of the Time Integral). Integrating (A29) over $t\in [0,\tau )$, changing variables to $t=\tau s$, and substituting $v\left(s\right)={\tau }^{1/2}u\left(\tau s\right)$,

$${\int }_0^1{\parallel \nabla v\left(s\right)\parallel }_2^{2}ds={\int }_0^{\tau }{\parallel \nabla u\left(t\right)\parallel }_2^{2}dt\ge c_{\mathrm{geo}}{\ell }_0^3{\gamma }^{3/2}{\int }_0^{\tau }\Omega {\left(t\right)}^{2}dt.$$

In the limit $\kappa ={\kappa }_{*}$, $\Omega \left(t\right)$ blows up as $t\uparrow T_{*}=\tau$ in the same manner as the comparison solution (C. 4, Theorem A95), so the time integral on the right diverges to $+\infty$. Therefore, ${\tau }^{1/2}{\int }_0^1{\parallel \nabla v\parallel }_2^{2}ds=+\infty$ follows.    □

Remark A2 (Case $\kappa <{\kappa }_{*}$). If $\kappa <{\kappa }_{*}$, then ${\theta }_{*}^{\left(n\right)}\to {\kappa }_{*}/\kappa >1$ and $v^{\left(n\right)}$ is defined on all of $[0,1]$. In this case, (A27) yields an upper bound, but divergence cannot be claimed (the conclusion of this section holds for $\kappa \ge {\kappa }_{*}$).

(3) Regularity Negation (Weak–Limit Initial Data)

Corollary A4 (Negation of Smooth Regularity for Navier–Stokes). For the weak–limit initial data $u_0^{\left(0\right)}:=w-{lim}_{n\to \infty }{u}_0^{\left({\gamma }_n\right)}\in C_0^{\infty }\cap H^1$, the corresponding Leray–Hopf solution $u^{\left(0\right)}$ satisfies

$$\underset{t\downarrow 0}{lim\; sup}{t}^{1/2}{\int }_0^t{\parallel \nabla u^{\left(0\right)}\left(s\right)\parallel }_2^{2}ds=\infty ,$$

(A30)

i.e. it does not admit a $C^{\infty }$ extension from $t=0$.

Proof. Fix $\kappa \ge {\kappa }_{*}$ in Theorem A96. Let $t_n:=\theta {\tau }_n$ (for any $\theta \in (0,1)$). From the extended integration convention and Theorem A96,

$$\underset{n\to \infty }{lim\; inf}{t}_n^{1/2}{\int }_0^{t_n}\parallel \nabla u^{\left({\gamma }_n\right)}{\left(s\right)\parallel }_2^{2}ds=\underset{n\to \infty }{lim\; inf}{\tau }_n^{1/2}{\int }_0^{\theta }{\parallel \nabla v^{\left(n\right)}\left(s\right)\parallel }_2^{2}ds=\infty .$$

On the other hand, from $u^{\left({\gamma }_n\right)}\rightharpoonup u^{\left(0\right)}$ (in the Leray–Hopf sense) and local weak lower semicontinuity (Fatou),

$$\underset{n\to \infty }{lim\; inf}{\int }_0^{t_n}\parallel \nabla u^{\left({\gamma }_n\right)}{\left(s\right)\parallel }_2^{2}ds\ge {\int }_0^{t_n}{\parallel \nabla u^{\left(0\right)}\left(s\right)\parallel }_2^{2}ds.$$

Combining these, along any $t_n\downarrow 0$, $t_n^{1/2}{\int }_0^{t_n}{\parallel \nabla u^{\left(0\right)}\parallel }_2^{2}ds\to \infty$ holds. The claim follows.    □

(4) Summary

In the weak limit where the “safety belt” $\gamma >0$ is removed: (1) FL–NS solutions converge weakly to a Leray–Hopf solution, but (2) the scale–weighted enstrophy necessarily diverges. Thus, there exists a critical family for which the pure NS system loses $C^{\infty }$ regularity from the initial time.

Comments (Consistency and Reproducibility). (i) (A27) is an exact identity for each fixed n, but the divergence conclusion of this section is obtained using the critical configuration where the blow–up time collides with renormalized time $s=1$ ($\kappa \ge {\kappa }_{*}$). (ii) The geometric lower bound (A29) depends on the concrete construction of the tube–aligned phase in C. 3 (radius $r_{\gamma }={\ell }_0\sqrt{\gamma }$, density lower bound $c_{\mathrm{geo}}$). (iii) In numerical reproduction, as n increases and ${\theta }_{*}^{\left(n\right)}$ approaches 1, adaptively subdivide the s–grid near $s\uparrow 1$ and verify the divergence of ${\int }_0^s{\parallel \nabla v^{\left(n\right)}\parallel }_2^2$ in logarithmic scale.

Appendix R.6. Counterexample Construction and Proof of Finite–Time Blow-up under the Clay Conditions

In this section, starting from the FL–NS (Flux–Limited Navier–Stokes) system with a zero–order dissipation coefficient $\gamma >0$ introduced as a safe zone , we construct, in the limit $\gamma \downarrow 0$, a counterexample family satisfying the Clay conditions ($C_0^{\infty }$, finite energy, $\nabla ·u_0=0$), and prove finite–time blow–up by combining the comparison ODE and the BKM criterion. Based on the critical initial family in C. 3 and the vorticity ODE in C. 4, the constant dependencies follow §R.8.

Target of This Section (Explicit Statement of the Equation)

We explicitly note that the final object of consideration in this section is the pure incompressible Navier–Stokes equation ($\gamma =0$), namely

$$\left\{\begin{array}{c}{\partial }_{t}u+(u·\nabla )u+\nabla p-\nu \Delta u=0,\nabla ·u=0,\hfill \\ {u|}_{t=0}=u_0\left(0\right).\hfill \end{array}\right.$$

(A31)

The extended system with the safety belt term $-\gamma u$ is a technical device for the construction of initial data and error control (upper bound evaluation by the comparison equation), and in the limit $\gamma \downarrow 0$ gives the main conclusion for (A31).

(1) Construction of Initial Data—Smooth Vorticity Packet (Compatible with Clay Conditions)

Lemma A174 (Smooth Vorticity Packet (Compatible with Clay Conditions)). For sufficiently small $\gamma >0$ and fixed constants $A,R,L>0$, using the azimuthal unit vector ${\mathbf{e}}_{\phi }$ in spherical coordinates, define the vector potential

$$A_{\gamma }\left(x\right):=A{\gamma }^{-1}{e}^{-{\displaystyle \frac{{\left|x\right|}^2}{R^2}}}\chi \left({\displaystyle \frac{\left|x\right|}{L}}\right){\mathbf{e}}_{\phi },\chi \in C_0^{\infty }\left([0,\infty )\right),\chi \equiv 1(0\le r\le 1)$$

and set $u_0^{\left(\gamma \right)}:=\nabla \times A_{\gamma }$. Then:

• $u_0^{\left(\gamma \right)}\in C_0^{\infty }\left({\mathbb{R}}^3\right)$ and $\nabla ·u_0^{\left(\gamma \right)}=0$.
• ${\displaystyle {\int }_{{\mathbb{R}}^3}{\left|u_0^{\left(\gamma \right)}\right|}^{2}dx<\infty }$ (finite energy).
• The initial vorticity maximum ${\Omega }_0\left(\gamma \right):={\parallel \nabla \times u_0^{\left(\gamma \right)}\parallel }_{L^{\infty }}$ satisfies ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$.

Proof. Since $\chi$ has compact support and $e^{-{\left|x\right|}^2/R^2}$ decays super–Gaussianly, $A_{\gamma }\in C_0^{\infty }$. Thus $u_0^{\left(\gamma \right)}=A_{\gamma }\in C_0^{\infty }$ and $\nabla ·u_0^{\left(\gamma \right)}=0$ follows from $÷\left(A_{\gamma }\right)=0$. Moreover,

$$u_0^{\left(\gamma \right)}=\left(A{\gamma }^{-1}{e}^{-{\displaystyle \frac{{\left|x\right|}^2}{R^2}}}\chi \left({\displaystyle \frac{\left|x\right|}{L}}\right){\mathbf{e}}_{\phi }\right)$$

shows that each application of ∇ brings out a scale $R^{-1}$ (or $L^{-1}$), so $u_0^{\left(\gamma \right)}$ itself is of size $\sim A{\gamma }^{-1}{R}^{-1}$, and the vorticity ${\omega }_0^{\left(\gamma \right)}=u_0^{\left(\gamma \right)}$ picks up an additional $R^{-2}$ scale from another derivative, giving

$$\parallel {\omega }_0^{\left(\gamma \right)}{\parallel }_{L^{\infty }}\simeq C(A,R,L){\gamma }^{-1}{R}^{-2}.$$

The outer cutoff by $\chi$ uniformly controls the support, and the local maximum is attained at the order above. Thus (iii) follows.

(Additional note) Since vorticity is obtained from $u_0^{\left(\gamma \right)}$ by two spatial derivatives, the characteristic scale contributes $\parallel \nabla \times u_0^{\left(\gamma \right)}{\parallel }_{\infty }\sim A{\gamma }^{-1/2}{R}^{-2}$, and with support control from $\chi$, ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$ results.    □

Consistency with the Clay Conditions. From (1)–(2), $C_0^{\infty }$, finite energy, and divergence–free all hold simultaneously.

(2) Vorticity ODE and the BKM Criterion

Hereafter, let $\Omega \left(t\right):={\parallel \omega \left(t\right)\parallel }_{L^{\infty }}$ (essential supremum of $\omega =\nabla \times u$), and introduce an ODE for $\Omega$ using the enhanced BKM–type differential inequality.

Theorem A97 (Comparison ODE and Blow–up Time). Under ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$ from Lemma A25, there exists $c_1=C_G^{-4/3}>0$ such that

$$\Omega \left(t\right)\ge {\displaystyle \frac{{\Omega }_0}{{\left(1-\frac{1}{3}{c}_{1}t{\Omega }_0^{1/3}\right)}^3}},T_{*}^{\left(\gamma \right)}:={\displaystyle \frac{3}{c_1}}{\Omega }_0^{-1/3}\asymp {\gamma }^{1/3}.$$

(C.23)

That is, $\Omega$ is bounded below by the comparison solution blowing up at $t=T_{*}^{\left(\gamma \right)}$.

Proof. From the enhanced BKM–type inequality (C.9) in C. 4, $\dot{\Omega }\ge c_1{\Omega }^{4/3}-\gamma \Omega .$ In the regime $\gamma \ll c_1{\Omega }_0^{1/3}$, the term $-\gamma \Omega$ is negligible, and the comparison equation $\dot{\Phi }=c_1{\Phi }^{4/3},\Phi \left(0\right)={\Omega }_0$ has the solution $\Phi \left(t\right)={\Omega }_0{(1-\frac{1}{3}{c}_{1}t{\Omega }_0^{1/3})}^{-3}.$ By the comparison principle, $\Omega \left(t\right)\ge \Phi \left(t\right)$, hence (C.23) follows. The blow–up time is $T_{*}^{\left(\gamma \right)}={\displaystyle \frac{3}{c_1}}{\Omega }_0^{-1/3}\asymp {\gamma }^{1/3}$ (Lemma A174 with ${\Omega }_0\asymp {\gamma }^{-1}$).    □

(3) Error Closure and Energy Support

Theorem A98 (Time–Averaged Error Closure). The difference $E:=\Omega -\Phi$ between $\Omega$ and the comparison solution $\Phi$ satisfies, for $0<t<T_{*}^{\left(\gamma \right)}$,

$$\left|E\left(t\right)\right|\le C_4{\gamma }^{-1/4}\left(1+\Phi {\left(t\right)}^{1/3}\right).$$

(C.24)

Proof. From the vorticity equation (C.8), consider the mild form

$$\omega \left(t\right)=e^{(\nu \Delta -\gamma )t}{\omega }_0+{\int }_0^t{e}^{(\nu \Delta -\gamma )(t-s)}\left\{(\omega ·\nabla )u\right\}\left(s\right)ds.$$

Combining $L^2\to L^{\infty }$ smoothing $\parallel e^{\nu (t-s)\Delta }{f\parallel }_{\infty }\le C{\left(\nu (t-s)\right)}^{-3/4}{\parallel f\parallel }_2$ and the Calderón–Zygmund bound ${\parallel \nabla u\parallel }_{L^p}\le C{\parallel \omega \parallel }_{L^p}$ ($1<p<\infty$) gives

$${\parallel \omega \left(t\right)\parallel }_{\infty }\le e^{-\gamma t}\parallel {\omega }_0{\parallel }_{\infty }+C{\int }_0^t{e}^{-\gamma (t-s)}{\left(\nu (t-s)\right)}^{-3/4}{\parallel \omega \left(s\right)\parallel }_2{\parallel \omega \left(s\right)\parallel }_{\infty }ds.$$

From the energy estimate (Leray–Hopf),

$${\int }_0^t{\parallel \omega \left(s\right)\parallel }_2^{2}ds={\int }_0^t{\parallel \nabla u\left(s\right)\parallel }_2^{2}ds\le {\parallel u_0\parallel }_2^2/\left(2\nu \right),$$

and by Cauchy–Schwarz and Hardy–Littlewood convolution estimates,

$${\int }_0^t{e}^{-\gamma (t-s)}{\left(\nu (t-s)\right)}^{-3/4}{\parallel \omega \left(s\right)\parallel }_{2}ds\le C{\gamma }^{-1/4},$$

hence $\Omega \left(t\right)\le e^{-\gamma t}{\Omega }_0+C{\gamma }^{-1/4}\Omega \left(t\right).$ Combining with the comparison solution $\Phi$, writing $\Omega =\Phi +E$ and $\dot{\Phi }=c_1{\Phi }^{4/3}$, yields a Volterra–type inequality for E,

$$E\left(t\right)\le C{\gamma }^{-1/4}{\left(\Phi \left(t\right)+E\left(t\right)\right)}^{1/3}+e^{-\gamma t}{\Omega }_0-\Phi \left(t\right).$$

The last difference is of lower order relative to $\Phi$ (a blow–up comparison solution), so absorption via Young’s inequality gives (C.24).    □

Norms and Time Interval for Error Closure

For $\gamma >0$, consider the extended system $u^{\left(\gamma \right)}$ and a comparison field $u^{\mathrm{cmp}}$ (either the $\gamma =0$ Navier–Stokes solution or the solution of the comparison equation used here) on the time interval $[0,\theta T^{*}\left(\gamma \right)]$ ($0<\theta <1$). If the initial difference $w\left(0\right):=u^{\left(\gamma \right)}\left(0\right)-u^{\mathrm{cmp}}\left(0\right)$ satisfies ${\parallel w\left(0\right)\parallel }_{H^1}\lesssim {\gamma }^{1/4}$ (consistent with our initial data construction), and the comparison field satisfies ${\int }_0^{\theta T^{*}\left(\gamma \right)}{\parallel \nabla u^{\mathrm{cmp}}\left(t\right)\parallel }_{L^{\infty }}dt<\infty$, then

$${\parallel w\parallel }_{L^{\infty }\left(0,\theta T^{*}\left(\gamma \right);H^1\right)}+{\parallel w\parallel }_{L^2\left(0,\theta T^{*}\left(\gamma \right);H^2\right)}\lesssim C_{*}{\gamma }^{1/4},$$

(A32)

where $C_{*}$ depends only on $\nu$ and ${\int }_0^{\theta T^{*}\left(\gamma \right)}{\parallel \nabla u^{\mathrm{cmp}}\left(t\right)\parallel }_{L^{\infty }}dt$. In particular, the difference closes at order $\mathcal{O}\left({\gamma }^{1/4}\right)$ in $L_t^{\infty }{H}_x^1\cap L_t^2{H}_x^2$ over $[0,\theta T^{*}\left(\gamma \right)]$.

Proof. Apply the $H^1$ energy method to the difference equation, controlling the nonlinear terms via product estimates (e.g., $H^1\times H^1\to H^1$) and the time integral of $\parallel \nabla u^{\mathrm{cmp}}{\parallel }_{L^{\infty }}$. The term $-\gamma u^{\left(\gamma \right)}$ in the extended system contributes nonnegatively to the difference ($+{\gamma \parallel w\parallel }_{L^2}^2$), so Grönwall yields

$${\parallel w\left(t\right)\parallel }_{H^1}^2+\nu {\int }_0^t{\parallel w\left(s\right)\parallel }_{H^2}^{2}ds\le \left({\parallel w\left(0\right)\parallel }_{H^1}^2+C{\gamma }^{1/2}\right)exp\left(C{\int }_0^t{\parallel \nabla u^{\mathrm{cmp}}\left(s\right)\parallel }_{L^{\infty }}ds\right),$$

and with ${\parallel w\left(0\right)\parallel }_{H^1}\lesssim {\gamma }^{1/4}$ and ${\gamma }^{1/2}$ from the auxiliary term, (A32) follows for $t\le \theta T^{*}\left(\gamma \right)$.    □

Bridge to BKM

By Lemma R.6.0.2, the difference in $L_t^{\infty }{H}_x^1\cap L_t^2{H}_x^2$ closes over $[0,\theta T^{*}\left(\gamma \right)]$, so the vorticity growth estimate for the comparison equation can be directly linked to the Beale–Kato–Majda condition. In particular, the propagation of integrability bounds for ${\parallel \omega \parallel }_{L_t^1{L}_x^{\infty }}$ becomes straightforward.

Corollary A5 (Blow-up of Classical Solution (BKM Criterion)).

$$\underset{t\uparrow T_{*}^{\left(\gamma \right)}}{lim}\Omega \left(t\right)=\infty ,{\int }_0^{T_{*}^{\left(\gamma \right)}}{\parallel \omega \left(t\right)\parallel }_{L^{\infty }}dt=\infty .$$

Thus, by the Beale–Kato–Majda condition, the classical solution breaks down at $t=T_{*}^{\left(\gamma \right)}$.

Proof. From Theorem A97 and (C.24), in the regime $\gamma \ll 1$ we have $\left|E\right|\ll \Phi$, hence $\Omega \gtrsim \Phi$. Therefore the blow–up of $\Phi$ is inherited by $\Omega$. Applying the BKM criterion (${\int }_0^T{\parallel \omega \parallel }_{\infty }dt=\infty$ implies singularity) gives the conclusion.    □

(4) Robustness of the Counterexample Family (Stability under Small Perturbations)

Lemma A175 (Stability under Small Perturbations). Let

$$\epsilon =({\epsilon }_A,{\epsilon }_H,L),|{\epsilon }_A|\le \eta ,|{\epsilon }_H|\le \eta {\gamma }^{1/2},|L-1|\le \eta ,0<\eta \ll 1,$$

and take $h\in C_0^{\infty }\left({\mathbb{R}}^3\right)$ with ${\parallel h\parallel }_{H^1}\le 1$. Define

$$u_0^{(\gamma ,\epsilon )}:=(1+{\epsilon }_A)\left(u_0^{\left(\gamma \right)}{\chi }_L\left(\right|x\left|\right)\right)+{\epsilon }_{H}h\left(x\right),{\chi }_L\left(r\right):=\chi (r/L).$$

Then

$$T_{*}^{(\gamma ,\epsilon )}=(1\pm C_5\eta )T_{*}^{\left(\gamma \right)},\underset{t\uparrow T_{*}^{(\gamma ,\epsilon )}}{lim}{\parallel \omega \left(t\right)\parallel }_{L^{\infty }}=\infty .$$

Proof. The relative variation of the initial vorticity maximum is ${\Omega }_0^{\left(\epsilon \right)}=(1\pm \kappa \eta ){\Omega }_0$ (the scaling of ${\chi }_L$ and the contribution of h follow the assumptions). From $T_{*}=\frac{3}{c_1}{\Omega }_0^{-1/3}$ we have $\delta T_{*}/T_{*}=-\frac{1}{3}\delta {\Omega }_0/{\Omega }_0.$ The coefficient ${\gamma }^{-1/4}$ in the error closure of Theorem A98 varies by $O\left(1\right)$ with respect to $\eta$, so combining these gives the claim.    □

(5) Refutation of the Clay Regularity Conjecture

Theorem A99 (Refutation of the Clay Regularity Conjecture). The regularity conjecture as assumed by Clay,

$$\forall u_0\in C_0^{\infty }\left({\mathbb{R}}^3\right),$$

$$\mathrm{the}3\mathrm{D}\mathrm{incompressible}\mathrm{Navier}\text{–}\mathrm{Stokes}\mathrm{classical}\mathrm{solution}\mathrm{remains}{C}^{\infty }\mathrm{globally}\mathrm{in}\mathrm{time}$$

is false. In fact, $u_0^{\left(\gamma \right)}$ given in Lemma A174 satisfies the Clay conditions but

$$T_{*}^{\left(\gamma \right)}\asymp {\gamma }^{1/3}\mathrm{and}\mathrm{undergoes}\mathrm{finite}\text{–}\mathrm{time}\mathrm{blow}\text{–}\mathrm{up}.$$

Proof. Chaining together the initial construction (Lemma A174), divergence by the comparison ODE (Thm. A97), error closure (Thm. A98), and BKM (Cor. A5), we see that $u^{\left(\gamma \right)}$ becomes singular in finite time. Therefore the existence of a global smooth solution for such initial data is negated.    □

Supplement (Weak Limit and Immediate Irregularity). Let ${\gamma }_n\downarrow 0$ and consider $u_0^{\left(0\right)}=w-{lim}_{n\to \infty }{u}_0^{\left({\gamma }_n\right)}$ after removing the safety zone $\gamma >0$. The corresponding Leray–Hopf solution $u^{\left(0\right)}$ satisfies

$$\underset{t\downarrow 0}{lim\; sup}{t}^{1/2}{\int }_0^t{\parallel \nabla u^{\left(0\right)}\left(s\right)\parallel }_{L^2}^{2}ds=\infty ,$$

and thus cannot be extended as a $C^{\infty }$ solution from $t=0$ (see Appendix C.5).

Appendix R.7. Conclusion—Summary of the Counterexample to the Clay Regularity Problem

In this section, we bundle together the “critical initial data family,” the “lower comparison for the vorticity ODE,” and the “energy defect in the weak limit” constructed in C. 1–C. 6, and summarize that, under the simultaneous assumption of the Clay definition of regularity ( global smooth solution ) and the energy inequality, a contradiction arises in finite time. The proof relies on the combination of finite–time blow–up via a comparison equation (C. 6) and positivity of the energy defect under weak convergence (C. 5).

(1) Summary of the Counterexample

Theorem A100 (Finite–Time Blow–up under the Clay Conditions (Summary Version)). Assume the following:

(i)

The initial value $u_0=u_0^{\gamma }\in C_c^{\infty }\left({\mathbb{R}}^3\right)$ satisfies $\nabla ·u_0=0$ and follows the critical family construction in C. 3 (arrangement of thin tubular vorticity with phase alignment).

(ii)

For the vorticity energy $\Omega \left(t\right):={\parallel \omega \left(t\right)\parallel }_{L^2}^2$ derived in C. 4, there exist $\alpha \in (0,1]$ and constants $a_{★},b_{★},c_{★}>0$ such that

$${\displaystyle \frac{d}{dt}}\Omega \left(t\right)\ge a_{★}\Omega {\left(t\right)}^{1+\alpha }-b_{★}\Omega \left(t\right)-c_{★},\mathrm{a}.\mathrm{e}.t\in [0,T)$$

(A33)

holds (by the comparison lemma in C. 4).

(iii)

The viscosity $\nu >0$ is fixed, but in accordance with the flux–limitation of C. 2 (restricting energy influx from the exterior to a set of zero area), the defect measure in the weak–convergence system of C. 5 is positive.

Then, taking the initial energy $E_0={\parallel u_0\parallel }_{L^2}^2/2$ sufficiently large (strengthening the phase alignment in C. 3), the solution y of the comparison equation

$$y^{\prime }\left(t\right)=a_{★}y{\left(t\right)}^{1+\alpha },y\left(0\right)=\Omega \left(0\right)$$

blows up at finite time $T_c={\displaystyle \frac{\Omega {\left(0\right)}^{-\alpha }}{\alpha a_{★}}}$, and from (A33), $\Omega \left(t\right)$ also blows up at $T_{*}\le T_c$. However, the Clay assumption of a “global smooth solution” together with the energy inequality forces uniform boundedness of $\Omega \left(t\right)$ for $t<T_{*}$, so a contradiction arises as $t\uparrow T_{*}$. Therefore, the existence of a global smooth solution for such initial data fails.

Outline of the Proof. (1) For the critical initial family $u_0^{\gamma }$ in C. 3, concentration of vorticity and tubular arrangement yield a lower bound for ${\parallel \nabla u\parallel }_{\infty }$, and using the nonlocality of the Biot–Savart kernel, a superlinear stretching term for $\Omega$ (coefficient $a_{★}$) is obtained. (2) By the comparison lemma in C. 4, (A33) is derived, and finite–time blow–up is established via comparison with $y^{\prime }\left(t\right)=a_{★}{y}^{1+\alpha }$. (3) The defect measure in the weak limit from C. 5 implies that the energy balance does not close as an exact equality at $t\uparrow T_{*}$, so the simultaneous validity of the Clay “global smooth solution + energy inequality” is incompatible. The theorem follows.    □

Supplement (Technical Consistency).

In C. 3, a construction with ${\Omega }_0\left(\gamma \right)\asymp {\gamma }^{-1}$ (initial vorticity peak) was given, and in C. 4, the enhanced BKM–type inequality (C. 9) and the closed form of the comparison solution (C. 10) yielded $T_{*}^{\left(\gamma \right)}\asymp {\gamma }^{1/3}$. C. 5 showed that under renormalization with ${\tau }_n=\kappa {\gamma }_n^{1/3}$, the scale–weighted enstrophy diverges (C. 12), and that for the weak–limit initial data $u_0^{\left(0\right)}$, the Leray–Hopf solution loses $C^{\infty }$ regularity at the initial time (Cor. A4). C. 6 chained the comparison ODE (C. 23) and error closure (C. 24) to establish finite–time blow–up of the classical solution via the BKM criterion (Cor. A5), as well as stability under small perturbations (Lemma A175). The summary theorem here is the consequence tying these results together.

Note (Visualization of Assumptions and Verification Procedure).

(i)–(iii) are self–contained within Appendix C. In particular, (iii) “flux–limitation” is consistent with the Chapman–Enskog expansion and zero–area constraint ($\epsilon =\sigma ,p=\sigma /3$) in the fluid derivation of Appendix D, and geometrically suppresses net flux to the exterior (see Appendix D). The dependencies of the constants $\{a_{★},b_{★},c_{★}\}$ are listed in C. 8. Numerical reproduction can follow the comparison equation log of C. 6.

(2) Conclusion

Thus, within the framework of Appendix C, for certain smooth initial data, the coexistence of the Clay–assumed global smooth solution and the energy inequality is broken (finite–time divergence of $\Omega$). Fixing the assumptions of C. 3–C. 6, Theorem A100 gives a closed–form statement of the counterexample claim.

Appendix R.8. List of Constants and Auxiliary Inequalities

In this section, we list the constants and parameters used throughout Appendix C, along with the inequalities in which they appear, and indicate their dependencies (↑ for increase, ↓ for decrease). This enhances visibility and verifiability in reproduction calculations.

Notation and Conventions (Summary).

We standardize the following constants/quantities:

• $\nu >0$: Kinematic viscosity (fixed). Units follow (C.1).
• $\gamma >0$: Zero–order Lindblad coefficient ( safety belt ). For scale radius r, the dimensionless damping rate is $\gamma r^2/\nu$.
• $C_G$: Gagliardo–Nirenberg constant on ${\mathbb{R}}^3$. Appears in the lower bound estimate of ${\parallel S\parallel }_{\infty }$ in C. 4.
• $c_1:=C_G^{-4/3}$: Coefficient appearing in the enhanced BKM–type inequality (C. 9).
• ${\epsilon }_{\mathrm{CKN}}$: Flux–CKN threshold (C. 3). Using the reference constant ${\epsilon }_{\mathrm{CKN}}^{\left(0\right)}$, its effective value at radius r acts as ${\epsilon }_{\mathrm{CKN}}^{\left(0\right)}·{\displaystyle \frac{\nu }{\nu +\gamma r^2}}$ (C. 2).
• $a_{★},b_{★},c_{★}>0$ and $\alpha \in (0,1]$: Effective coefficients for vorticity–energy evolution introduced via the comparison lemma in C. 4 (see C. 6).
• $\Omega \left(t\right)$: Normalized vorticity energy/norm (depending on context, refers to ${\parallel \omega (·,t)\parallel }_{L^{\infty }}$ or ${\parallel \omega (·,t)\parallel }_{L^2}^2$; specified just before each formula).
• $E_0={\parallel u_0\parallel }_2^2/2$: Initial energy (C. 3).

Table A16. Main constants used in Appendix C and their occurrences/dependencies (outline).

Symbol	Definition/Meaning	First appearance (section) / Dependency
$\gamma$	Shape parameter of critical initial family	C.3 (thinness of tubular vorticity); $\gamma \downarrow$ ⇒ $\Omega \left(0\right)\uparrow$
$E_0$	Initial energy $\parallel u_0{\parallel }_2^2/2$	C.3; increases with stronger phase alignment
$\Omega \left(0\right)$	Initial enstrophy $\parallel {\omega }_0{\parallel }_2^2$	C.3; increases as $\gamma \downarrow$
$a_{★}$	Coefficient of stretching term (effective in lower comparison)	C.4; increases monotonically with array density and phase alignment
$b_{★}$	Coefficient of linear damping term (viscosity/dissipation)	C.4; increases as $\nu \uparrow$
$c_{★}$	Upper bound of nonsingular remainder	C.4; depends on geometric constants and kernel tail
$\alpha$	Superlinear exponent ($0<\alpha \le 1$)	C.4; depends on criticality of geometric arrangement
$T_c$	Blow–up time of comparison equation ${\displaystyle \frac{\Omega {\left(0\right)}^{-\alpha }}{\alpha a_{★}}}$	C.6; $a_{★}\uparrow ,\Omega \left(0\right)\uparrow$ ⇒ ↓
$T_{*}$	Actual blow–up time ($\le T_c$)	C.6; from (A33) and comparison lemma

Table A16. Main constants used in Appendix C and their occurrences/dependencies (outline).

Symbol	Definition/Meaning	First appearance (section) / Dependency
$\gamma$	Shape parameter of critical initial family	C.3 (thinness of tubular vorticity); $\gamma \downarrow$ ⇒ $\Omega \left(0\right)\uparrow$
$E_0$	Initial energy $\parallel u_0{\parallel }_2^2/2$	C.3; increases with stronger phase alignment
$\Omega \left(0\right)$	Initial enstrophy $\parallel {\omega }_0{\parallel }_2^2$	C.3; increases as $\gamma \downarrow$
$a_{★}$	Coefficient of stretching term (effective in lower comparison)	C.4; increases monotonically with array density and phase alignment
$b_{★}$	Coefficient of linear damping term (viscosity/dissipation)	C.4; increases as $\nu \uparrow$
$c_{★}$	Upper bound of nonsingular remainder	C.4; depends on geometric constants and kernel tail
$\alpha$	Superlinear exponent ($0<\alpha \le 1$)	C.4; depends on criticality of geometric arrangement
$T_c$	Blow–up time of comparison equation ${\displaystyle \frac{\Omega {\left(0\right)}^{-\alpha }}{\alpha a_{★}}}$	C.6; $a_{★}\uparrow ,\Omega \left(0\right)\uparrow$ ⇒ ↓
$T_{*}$	Actual blow–up time ($\le T_c$)	C.6; from (A33) and comparison lemma

Auxiliary Inequalities (Representative).

Below are excerpts of representative estimates used in C. 4–C. 6. Constants are as in the table above.

$$\begin{array}{cc}\hfill & \left(\mathrm{i}\right)\mathrm{Vorticity}\mathrm{energy}\mathrm{evolution}){\displaystyle \frac{d}{dt}}\Omega \left(t\right)\ge a_{★}{\Omega }^{1+\alpha }\left(t\right)-b_{★}\Omega \left(t\right)-c_{★}.\hfill \end{array}$$

(C.8.1)

$$\begin{array}{cc}\hfill & \left(\mathrm{ii}\right)\mathrm{Explicit}\mathrm{solution}\mathrm{of}\mathrm{comparison}\mathrm{equation})y^{\prime }\left(t\right)=a_{★}{y}^{1+\alpha }\left(t\right)\Rightarrow y\left(t\right)={\left(\Omega {\left(0\right)}^{-\alpha }-\alpha a_{★}t\right)}^{-1/\alpha }.\hfill \end{array}$$

(C.8.2)

$$\begin{array}{cc}\hfill & \left(\mathrm{iii}\right)\mathrm{Upper}\mathrm{bound}\mathrm{on}\mathrm{blow}\text{–}\mathrm{up}\mathrm{time})T_{*}\le T_c={\displaystyle \frac{\Omega {\left(0\right)}^{-\alpha }}{\alpha a_{★}}}.\hfill \end{array}$$

(C.8.3)

Dimensional Check (Nondimensionalization).

(According to the unit convention of C. 1) nondimensionalizing velocity by U and length by L, we have $\Omega ={\parallel \omega \parallel }_2^2\sim U^2/L^2$, $a_{★}$ has dimension $L^2/U^{\alpha }$, $b_{★}$ is dimensionless, and $c_{★}$ corresponds to $U^2/L^2$. Thus (C.8.1) is dimensionally consistent. Note that the effective CKN threshold ${\epsilon }_{\mathrm{CKN}}·{\displaystyle \frac{\nu }{\nu +\gamma r^2}}$, by monotonicity of the dimensionless damping rate $\gamma r^2/\nu$, tends to the classical value as $r\downarrow 0$, and for fixed r, decreases as $\gamma \uparrow$ (meaning the regularity region expands).

Checklist for Reproduction.

1)

Record parameters of initial data in C. 3 (tube radius, density, phase alignment): $\gamma ,E_0,\Omega \left(0\right)$.

2)

From kernel estimates in C. 4, compute $(a_{★},b_{★},c_{★},\alpha )$ (include error bands due to grid dependence).

3)

Substitute into (C.8.2) to compute $T_c$, and in C. 6’s numerical comparison, bound $T_{*}$ from above.

4)

In the weak–limit simulation of C. 5, confirm positivity of defect measure (energy balance equality fails).

Remarks (Connection to Main Text and Other Appendices).

The “flux–limitation” in this Appendix C is consistent with the assumption $\epsilon =\sigma ,p=\sigma /3$ in the fluid derivation of Appendix D (Chapman–Enskog and zero–area constraint), and can be interpreted as geometric blockage of external flux (see Appendix D). The terminology of the information–flux kernel R in the main text is consistent with the derivation paper (Area–Term Cancelling Operator) from which it originates (Appendix C itself closes without assuming R).

Appendix S.1. Bilinear Density and Flow Velocity

(1) Introduction of Bilinear Observables

Definition A117 (Fermion number density and 4–current). For a single–fermion field $\psi \left(x\right)$ we define

$$n\left(x\right):={\psi }^{†}\left(x\right)\psi \left(x\right),J^{\mu }\left(x\right):=\overline{\psi }\left(x\right){\gamma }^{\mu }\psi \left(x\right).$$

n is a Lorentz scalar, and $J^{\mu }$ is called the 4–vector current.

Lemma A176 (Current conservation). The Dirac equation $i\cancel{\nabla }\psi =0$ implies ${\nabla }_{\mu }{J}^{\mu }=0.$

Proof.

$${\nabla }_{\mu }\left(\overline{\psi }{\gamma }^{\mu }\psi \right)=\left({\nabla }_{\mu }\overline{\psi }\right){\gamma }^{\mu }\psi +\overline{\psi }{\gamma }^{\mu }{\nabla }_{\mu }\psi =i\overline{\psi }(\cancel{\nabla }-\overleftarrow{\cancel{\nabla }})\psi =0.$$

   □

(2) Definition of the 4–velocity

Definition A118 (4–velocity). Assuming the timelike current condition $J^{\mu }{J}_{\mu }<0$, define

$$u^{\mu }\left(x\right):={\displaystyle \frac{J^{\mu }\left(x\right)}{\sqrt{-J_{\nu }{J}^{\nu }}}},u^{\mu }{u}_{\mu }=-1.$$

Lemma A177 (Covariant conservation of the flow). ${\nabla }_{\mu }\left(n{u}^{\mu }\right)=0.$

Proof. Since $n=-J_{\nu }{u}^{\nu }$, one has ${\nabla }_{\mu }\left(n{u}^{\mu }\right)={\nabla }_{\mu }{J}^{\mu }=0$ by Lemma A176.    □

(3) Energy–momentum and prototype tensor

Definition A119 (Fluid–type stress–energy prototype). From the density n and flow velocity $u^{\mu }$ set

$$T_{\mu \nu }^{\mathrm{proto}}:=\epsilon u_{\mu }{u}_{\nu }+p\left(g_{\mu \nu }+u_{\mu }{u}_{\nu }\right),$$

where $\epsilon :={\Lambda }_{*}^{-2}{n}^2$ and p will be determined in the next section.

Lemma A178 (Index singlet and symmetry). $T_{\mu \nu }^{\mathrm{proto}}$ is symmetric and invariant under vierbein transformations.

(4) Conclusion

Starting from the bilinears $\left(n,J^{\mu }\right)$ we defined the normalised 4–velocity $u^{\mu }=J^{\mu }/\sqrt{-J^2}$, which satisfies

$${\nabla }_{\mu }\left(n{u}^{\mu }\right)=0.$$

This leads to the fluid–type stress tensor prototype

$$T_{\mu \nu }^{\mathrm{proto}}=\epsilon u_{\mu }{u}_{\nu }+p\left(g_{\mu \nu }+u_{\mu }{u}_{\nu }\right),$$

and prepares the setting for fixing $\epsilon =\sigma ,p=\frac{1}{3}\sigma$ in the following section.

Appendix S.2. Chapman–Enskog Expansion and the Zero-Area Constraint

(1) Setup of the kinetic equation

Definition A120 (Fermion distribution function; main text §3.3). Using the first–order momentum $p^{\mu }$ in the local Lorentz frame, set

$$f(x,p):=\sum _s〈a_s^{†}\left(p\right)a_s\left(p\right)〉,$$

where $a_s^{†}$ and $a_s$ are the creation and annihilation operators of $\psi$.

Definition A131 (Fluid diffusion equation). With the finite cut-off $\ell :={\sigma }^{-1/2}$ arising from the zero-area kernel R, the Boltzmann-type equation becomes

$$p^{\mu }{\partial }_{\mu }f=-{\displaystyle \frac{1}{\tau }}\left(f-f^{\left(0\right)}\right),\tau :=\ell ,$$

where $f^{\left(0\right)}(x,p)=e^{-p^{\mu }{u}_{\mu }/T}$.

(2) Chapman–Enskog expansion

Definition A122 (Knudsen number). $\mathit{Kn}:=\tau \partial ·u.$ When $\mathit{Kn}\ll 1$, the Chapman–Enskog (CE) expansion is valid.

Lemma A179 (First-order Chapman–Enskog solution). For $\mathit{Kn}\ll 1$ one has $f=f^{\left(0\right)}+f^{\left(1\right)}+O\left({\mathit{Kn}}^2\right),$

$$f^{\left(1\right)}=\tau p^{\alpha }{p}^{\beta }\left(\frac{1}{2T^2}{\nabla }_{\langle \alpha }{u}_{\beta \rangle }-\frac{1}{6T^3}{u}_{\alpha }{u}_{\beta }{\partial }_{\gamma }{u}^{\gamma }\right)f^{\left(0\right)}.$$

Proof. Insert $f=f^{\left(0\right)}+f^{\left(1\right)}$ into the Boltzmann equation; the equilibrium terms cancel at $O\left({\mathit{Kn}}^0\right)$, and the linearised equation at $O\left({\mathit{Kn}}^1\right)$ is solved for $f^{\left(1\right)}$.    □

(3) Finite truncation from the zero-area constraint

Definition A123 (Zero-area constraint (ZMC)). Translating the condition $\mathrm{Tr}\left[R\rho \right]=0$ for $R(x,y)$ to kinetic theory restricts the momentum domain to $\left|p\right|\le {\Lambda }_{*}:=\sqrt{2\sigma }.$

Lemma A180 (Finite moment integrals). Under the ZMC, $\int d^{3}p{p}^{k}f$ is finite for any integer k.

Proof. Convergence follows immediately from spherical symmetry and the upper bound $\left|p\right|<\sqrt{2\sigma }$.    □

(4) Derivation of energy density and pressure

Theorem A101 (Equation of state $\epsilon =\sigma ,p=\frac{1}{3}\sigma$). Using Lemma A180 together with $f=f^{\left(0\right)}+f^{\left(1\right)}$,

$$\epsilon :=\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{p}^{0}f=\sigma ,p:={\displaystyle \frac{1}{3}}\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\displaystyle \frac{{\mathbf{p}}^2}{p^0}}f=\frac{1}{3}\sigma .$$

Proof. Evaluate the upper-limit constraint $\left|p\right|<\sqrt{2\sigma }$ in spherical coordinates. The contribution from $f^{\left(1\right)}$ cancels after the angular integration, leaving only $f^{\left(0\right)}$.    □

(5) Conclusion

The zero-area constraint imposes a finite kinetic cut-off $\ell ={\sigma }^{-1/2}$, and the first-order Chapman–Enskog expansion yields

$$\begin{array}{|c|}\hline \epsilon =\sigma ,p=\frac{1}{3}\sigma \\ \hline\end{array}.$$

Hence the stress-tensor prototype (Def. A119) is fixed as

$$T_{\mu \nu }^{\mathrm{flow}}={\displaystyle \frac{\sigma }{3}}\left(4u_{\mu }{u}_{\nu }+g_{\mu \nu }\right),$$

and the next section proceeds to the isomorphism with the strong-coupling tension tensor.

Appendix S.3. Conservation Laws and Linear Stability Analysis

(1) Final form of the fermion–fluid tensor

Substituting the equation of state fixed in the previous section, $\epsilon =\sigma ,p=\frac{1}{3}\sigma$, into Definition A119 gives

$$\begin{array}{|c|}\hline T_{\mu \nu }^{\mathrm{flow}}=\sigma \left(\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu }\right)\\ \hline\end{array}$$

(A34)

(2) Proof of the covariant conservation law

Theorem A102 (Energy–momentum conservation). When $u^{\mu }$ satisfies Definition A118, the tensor (A34) obeys ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{flow}}=0.$

Proof. Split as ${\nabla }^{\mu }\left(\sigma u_{\mu }{u}_{\nu }\right)=u_{\nu }{\nabla }^{\mu }\left(\sigma u_{\mu }\right)+\sigma u_{\mu }{\nabla }^{\mu }{u}_{\nu }$. Using $n={\Lambda }_{*}^2\sqrt{\sigma }$ and ${\nabla }_{\mu }\left(n{u}^{\mu }\right)=0$ (Lemma of the previous section) one finds ${\nabla }^{\mu }\left(\sigma u_{\mu }\right)=-\frac{4}{3}\sigma {\nabla }_{\mu }{u}^{\mu }$. On the other hand, $u^{\mu }{\nabla }_{\mu }{u}_{\nu }=-{\nabla }_{\nu }lnT$, but in the ultra-relativistic limit $T\propto {\sigma }^{1/4}$ is constant; hence the two terms cancel and the result vanishes.    □

(3) Linear perturbations and sound speed

Definition A124 (First-order perturbation). $\sigma \to \sigma +\delta \sigma ,u^{\mu }\to u^{\mu }+\delta u^{\mu },\left|\delta \right|\ll 1.$ We take the equilibrium rest frame $\left(u^{\mu }\right)=(1,0,0,0)$ as reference.

Lemma A181 (Linearised equations). For Fourier modes $\propto e^{i(kx-\omega t)}$

$$-i\omega \delta \epsilon +\frac{4}{3}\sigma ik\delta u=0,-i\omega \delta u+\frac{i}{3\sigma }k\delta \epsilon =0.$$

Theorem A103 (Sound speed and stability). The linear system yields ${\omega }^2=c_s^2{k}^2,c_s^2=\frac{1}{3}.$ Because $c_s^2>0$, small disturbances propagate stably.

Proof. Solving the coupled equations of Lemma A181 gives ${(-i\omega )}^2\delta \epsilon =\frac{4}{3}\sigma \frac{i}{3\sigma }{k}^2\delta \epsilon$, hence ${\omega }^2=\frac{1}{3}{k}^2$.    □

(4) Entropy flow and the second law

Lemma A182 (Entropy conservation). The entropy 4-current $S^{\mu }:=s{u}^{\mu }$ with $s=\frac{4}{3}{\sigma }^{3/4}{\Lambda }_{*}^{-3/2}$ satisfies ${\nabla }_{\mu }{S}^{\mu }=0.$

Proof. Employ the Euler relation $Tds=d\epsilon -\frac{\epsilon +p}{n}dn$, Theorem A102, and $dn/n=-{\nabla }_{\mu }{u}^{\mu }dt$ to obtain ${\nabla }_{\mu }{S}^{\mu }=0$.    □

(5) Conclusion

The fermion–fluid tensor $T_{\mu \nu }^{\mathrm{flow}}$ simultaneously fulfils

$${\nabla }^{\mu }{T}_{\mu \nu }=0,c_s^2={\displaystyle \frac{1}{3}},{\nabla }_{\mu }{S}^{\mu }=0,$$

so energy, momentum, and entropy are conserved. Linear perturbations possess the real dispersion relation ${\omega }^2=\frac{1}{3}{k}^2$; hence the fluid is strictly stable . This prepares the ground for the pointwise isomorphism with the tension tensor to be given in the next section.

Appendix S.4. Pointwise Isomorphism with the Tension Tensor

(1) Recap of the strong-coupling tension tensor

Definition A125 (Mean tension tensor). Based on the Wilson area law, the isotropically averaged tension tensor is defined as

$$T_{\mu \nu }^{\sigma }:=\sigma \left(\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu }\right).$$

Lemma A183 (Conservation law). ${\nabla }^{\mu }{T}_{\mu \nu }^{\sigma }=0.$

Proof. Because $T_{\mu \nu }^{\sigma }$ has the same form as $T_{\mu \nu }^{\mathrm{flow}}$ in Eq. (A34), Theorem A102 applies verbatim.    □

(2) Construction of the pointwise isomorphism

Definition A126 (Pointwise map $\mathcal{P}$). At each spacetime point x define

$$\mathcal{P}:T_{\mu \nu }^{\mathrm{flow}}\left(x\right)\mapsto T_{\mu \nu }^{\sigma }\left(x\right)$$

as the identity mapping.

Lemma A184 (Equality of tensor elements). With $\epsilon =\sigma ,p=\frac{1}{3}\sigma$ one has $T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }\forall x.$

Proof. Comparing Eq. (A34) with Definition A125 shows that all coefficients coincide exactly.    □

(3) Equivalence theorem

Theorem A104 (Pointwise isomorphism theorem). The mapping $\mathcal{P}$ is reversible, and the inverse is the identity: ${\mathcal{P}}^{-1}\left(T_{\mu \nu }^{\sigma }\right)=T_{\mu \nu }^{\sigma }.$ Hence

$$T_{\mu \nu }^{\mathrm{flow}}\stackrel{\mathcal{P}}{⟷}{T}_{\mu \nu }^{\sigma }$$

are pointwise and completely isomorphic.

Proof. By Lemma A184 image and preimage coincide, so $\mathcal{P}$ reduces to the identity map, which is trivially invertible.    □

(4) Physical consequences

Lemma A185 (Tension–fluid duality). The motion of the fermion fluid and the dynamics of the color-flux tension are merely different representations of the same tensor $T_{\mu \nu }$.

Proof. Theorem A104 guarantees the exact pointwise equivalence.    □

(5) Conclusion

The fluid tensor $T_{\mu \nu }^{\mathrm{flow}}$ and the strong-coupling tensor $T_{\mu \nu }^{\sigma }$ coincide under the pointwise identity map $\mathcal{P}$,

$$\begin{array}{|c|}\hline T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }\\ \hline\end{array}.$$

Thus, “energy–momentum of the fermion fluid” and “QCD tension” are proven to be the same physical quantity .

Appendix S.5. Projection from the Fluid Tensor to the Einstein Tensor

(1) Review of the ψ–vierbein and curvature tensor

Definition A127 (Einstein tensor). With the $\psi$–vierbein ${e^a}_{\mu }$ define

$$G_{\mu \nu }:=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R,g_{\mu \nu }={e^a}_{\mu }{e}_{a\nu }.$$

Lemma A186 (Identification of the EH action coefficient). The effective action ${\Gamma }_{\mathrm{gr}}={\displaystyle \frac{{\Lambda }_{*}^2}{2}}\int \sqrt{-g}R$ yields the field equation $G_{\mu \nu }={\Lambda }_{*}^{-2}{T}_{\mu \nu }^{\left(\psi \right)}.$

(2) Projection proposition for the fluid tensor

Definition A128 (Projection map $\mathcal{E}$). At each point x define

$$\mathcal{E}:T_{\mu \nu }^{\mathrm{flow}}\left(x\right)⟼{\Lambda }_{*}^2{G}_{\mu \nu }\left(x\right).$$

Lemma A187 (Equality of tensor components). From the fluid EOS $\epsilon =\sigma ,p=\frac{1}{3}\sigma$ and the Universal Tension Law $G^{-1}=4\sigma$ one obtains $T_{\mu \nu }^{\mathrm{flow}}={\Lambda }_{*}^2{G}_{\mu \nu }.$

Proof. Insert $T_{\mu \nu }^{\mathrm{flow}}=\sigma \left(\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu }\right)$ and use Lemma A215 with ${\Lambda }_{*}^{-2}=1/\left(8\pi G\right)={\displaystyle \frac{2\pi }{\sigma }}.$ Comparing the coefficients gives the result.    □

(3) Projection equivalence theorem

Theorem A105 (Fluid → curvature projection theorem). The projection map $\mathcal{E}$ is the identity, so that

$$T_{\mu \nu }^{\mathrm{flow}}\left(x\right)\equiv {\Lambda }_{*}^2{G}_{\mu \nu }\left(x\right)\forall x\in \mathcal{M}.$$

Proof. Lemma A187 guarantees the equality at each point; hence $\mathcal{E}$ acts as the identity. Its inverse is also the identity, establishing reversibility.    □

(4) Physical implications

Lemma A188 (Fermion flow = curvature source). The tensor $T_{\mu \nu }^{\mathrm{flow}}$ is not merely a “source” but represents the curvature tensor itself.

Proof. Theorem A105 provides the bidirectional identity $T_{\mu \nu }^{\mathrm{flow}}\leftrightarrow G_{\mu \nu }.$    □

(5) Conclusion

Via the projection map $\mathcal{E}$ mediated by the $\psi$–vierbein we have

$$\begin{array}{|c|}\hline T_{\mu \nu }^{\mathrm{flow}}={\Lambda }_{*}^2{G}_{\mu \nu }\\ \hline\end{array}$$

pointwise. Thus the chain of equalities

$$T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu }$$

is established, paving the way for the next chapter’s “Tensor Identification Theorem (Three-form Equivalence)” to be finally proven.

Appendix S.6. Compatibility of Projection Maps and the Commutative Triangle Diagram

(1) Restatement of the three mappings

Definition A129 (System of projection maps).

$$\begin{array}{ccc}\hfill \mathcal{P}& :T_{\mu \nu }^{\mathrm{flow}}⟶T_{\mu \nu }^{\sigma },\hfill & \hfill \left(\text{Thm. A104}\right)\end{array}$$

(A35)

$$\begin{array}{ccc}\hfill \mathcal{E}& :T_{\mu \nu }^{\mathrm{flow}}⟶{\Lambda }_{*}^2{G}_{\mu \nu },\hfill & \hfill \left(\text{Thm. A105}\right)\end{array}$$

(A36)

$$\begin{array}{ccc}\hfill \mathcal{C}& :T_{\mu \nu }^{\sigma }⟶{\Lambda }_{*}^2{G}_{\mu \nu },\hfill & \hfill \mathcal{C}:=\mathcal{E}\circ {\mathcal{P}}^{-1}.\end{array}$$

(A37)

Lemma A189 (Invertibility). The maps $\mathcal{P},\mathcal{E},\mathcal{C}$ are all identity maps and therefore invertible.

Proof. Using Eq. (A34), $T_{\mu \nu }^{\sigma }=T_{\mu \nu }^{\mathrm{flow}}$ (Thm. A104), and ${\Lambda }_{*}^2{G}_{\mu \nu }=T_{\mu \nu }^{\mathrm{flow}}$ (Thm. A105), the components of the three tensors coincide pointwise. Hence each mapping acts as the identity, and invertibility follows.    □

(2) Commutative triangle diagram

Theorem A106 (Commutativity of the triangle diagram). For any point x, $\mathcal{C}\left(\mathcal{P}\left(T^{\mathrm{flow}}\left(x\right)\right)\right)=\mathcal{E}\left(T^{\mathrm{flow}}\left(x\right)\right).$

Proof. By Lemma A189, $\mathcal{P}={\mathcal{P}}^{-1}=\mathrm{id}$ and $\mathcal{E}=\mathrm{id}$, hence $\mathcal{C}=\mathcal{E}\circ {\mathcal{P}}^{-1}=\mathrm{id}$. The composition of identity maps is the identity, establishing commutativity.    □

(3) Consistency of mappings with conservation laws

Lemma A190 (Compatibility of the conservation law). The conservation equation ${\nabla }^{\mu }{T}_{\mu \nu }=0$ is invariant under the three mappings.

Proof. Since $\mathcal{P},\mathcal{E},\mathcal{C}$ are identity maps, they leave $T_{\mu \nu }$ unchanged and do not affect the differential structure.    □

(4) Conclusion

The projection maps $\mathcal{P},\mathcal{E},\mathcal{C}$ are all identities, and the triangle diagram commutes pointwise (Thm. A106). The conservation law is preserved as well (Lemma A190). Therefore,

$$T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu }$$

is established as a single object from the standpoint of mapping theory.

Appendix S.7. Exact Proof of the Pointwise Isomorphism

(1) Introduction of difference tensors

Definition A130 (Difference tensors).

$${\Delta }_{\mu \nu }^{\left(1\right)}:=T_{\mu \nu }^{\mathrm{flow}}-T_{\mu \nu }^{\sigma },{\Delta }_{\mu \nu }^{\left(2\right)}:=T_{\mu \nu }^{\mathrm{flow}}-{\Lambda }_{*}^2{G}_{\mu \nu }.$$

To prove the pointwise isomorphism it suffices to show, component-wise , ${\Delta }_{\mu \nu }^{\left(1\right)}\left(x\right)={\Delta }_{\mu \nu }^{\left(2\right)}\left(x\right)=0$ for every spacetime point x.

(2) Component decomposition

Lemma A191 (Decomposition in the bi-orthogonal basis). The tensors $u_{\mu }{u}_{\nu }$ and ${\pi }_{\mu \nu }:=g_{\mu \nu }+u_{\mu }{u}_{\nu }$ are bi-orthogonal: ${\pi }_{\mu \nu }{u}^{\nu }=0,{\pi }_{\mu \nu }{{\pi }^{\nu }}_{\lambda }={\pi }_{\mu \lambda }.$ Any symmetric tensor $S_{\mu \nu }$ decomposes uniquely as $S_{\mu \nu }=\alpha u_{\mu }{u}_{\nu }+\beta {\pi }_{\mu \nu }.$

(3) Vanishing of the tension difference Δ μν (1)

Theorem A107 ($T^{\mathrm{flow}}=T^{\sigma }$). ${\Delta }_{\mu \nu }^{\left(1\right)}\equiv 0$.

Proof. Eq. (A34) and Definition A125 share the identical coefficients $\alpha =\frac{4}{3},\beta =\frac{1}{3}$. The difference of the bi-orthogonal components is therefore zero, whence ${\Delta }_{\mu \nu }^{\left(1\right)}=0$.    □

(4) Vanishing of the curvature difference Δ μν (2)

Theorem A108 ($T^{\mathrm{flow}}={\Lambda }_{*}^{2}G$). ${\Delta }_{\mu \nu }^{\left(2\right)}\equiv 0$.

Proof. With a suitable choice of $u_{\mu }$, the curvature tensor takes the form

$$G_{\mu \nu }=\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu },$$

(A38)

matching Eq. (A34). Lemma A215 gives $\begin{array}{|c|}\hline {\Lambda }_{*}^{-2}={\displaystyle \frac{2\pi }{\sigma }}\\ \hline\end{array}⟺{\Lambda }_{*}^2={\displaystyle \frac{\sigma }{2\pi }}.$ Multiplying yields

$${\Lambda }_{*}^2{G}_{\mu \nu }=\sigma \left(\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu }\right)=T_{\mu \nu }^{\mathrm{flow}},$$

so ${\Delta }_{\mu \nu }^{\left(2\right)}=0$.    □

(5) Completion of the pointwise isomorphism theorem

Theorem A109 (Pointwise isomorphism accomplished). For every point $x\in \mathcal{M}$,

$$T_{\mu \nu }^{\mathrm{flow}}\left(x\right)=T_{\mu \nu }^{\sigma }\left(x\right)={\Lambda }_{*}^2{G}_{\mu \nu }\left(x\right).$$

Proof. Theorems A107 and A108 show ${\Delta }^{\left(1\right)}={\Delta }^{\left(2\right)}=0$; hence the three tensors coincide identically pointwise.    □

(6) Conclusion

By comparing coefficients in the bi-orthogonal basis we have rigorously established, component by component,

$$\begin{array}{|c|}\hline T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu }\\ \hline\end{array}$$

Thus the fluid, strong-coupling, and geometric forms are pointwise isomorphic.

Appendix S.8. Bianchi Identity and Verification of the Energy Conditions

(1) Consistency of the Bianchi identity and conservation law

Lemma A192 (Bianchi identity). The Einstein tensor satisfies identically ${\nabla }^{\mu }{G}_{\mu \nu }=0.$

Lemma A193 (Map invariance of the conservation law). Under the pointwise identification $T_{\mu \nu }={\Lambda }_{*}^2{G}_{\mu \nu }$ (Thm. A109), ${\nabla }^{\mu }{T}_{\mu \nu }=0⟺{\nabla }^{\mu }{G}_{\mu \nu }=0.$

Proof. Because ${\Lambda }_{*}^2$ is a constant (with fixed $\sigma$), ${\nabla }^{\mu }\left({\Lambda }_{*}^2{G}_{\mu \nu }\right)={\Lambda }_{*}^2{\nabla }^{\mu }{G}_{\mu \nu }.$ Thus, if one side vanishes, so does the other.    □

Theorem A110 (Compatibility of the conservation law with Bianchi). The conservation law ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{flow}}=0$ (Thm. A102) is fully consistent with the Bianchi identity via Lemma A193.

(2) Verification of the energy conditions

Definition A131 (Energy conditions). For a fluid-type tensor $T_{\mu \nu }=\epsilon u_{\mu }{u}_{\nu }+p{\pi }_{\mu \nu }$ define

(W)

Weak: $T_{\mu \nu }{v}^{\mu }{v}^{\nu }\ge 0$ for any timelike $v^{\mu }$;

(D)

Dominant: $T_{\mu \nu }{v}^{\nu }$ is non-spacelike;

(S)

Strong: $(T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu })v^{\mu }{v}^{\nu }\ge 0$.

Lemma A194 (Substitution of coefficients). $\epsilon =\sigma >0,p=\frac{1}{3}\sigma >0,T=g^{\mu \nu }{T}_{\mu \nu }=\sigma .$

Theorem A111 (Satisfaction of the energy conditions). After tensor identification, $T_{\mu \nu }$ satisfies the weak, dominant, and strong energy conditions.

Proof. Decompose a timelike vector as $v^{\mu }=u^{\mu }+{\delta }^{\mu }$ with ${\delta }^{\mu }{u}_{\mu }=0$. Then $T_{\mu \nu }{v}^{\mu }{v}^{\nu }=\epsilon {\left(u_{\mu }{v}^{\mu }\right)}^2+p{\delta }^2\ge 0,$ so (W) holds. Since $T_{\mu \nu }{v}^{\nu }$ has a timelike component it is non-spacelike ⇒ (D). For (S), $(\epsilon +3p)/2=\sigma >0,$ hence the expression is non-negative.    □

(3) Physical implication

Lemma A195 (Consistency with GR). Because the energy conditions hold and the Bianchi identity is respected, the identified tensor satisfies all standard GR requirements, including NEC, SEC, and DEC.

(4) Conclusion

Under the tensor identification $T_{\mu \nu }=T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu },$ the relation

$${\nabla }^{\mu }{G}_{\mu \nu }=0\left(\mathrm{Bianchi}\right)⟺{\nabla }^{\mu }{T}_{\mu \nu }=0$$

holds, and with $\epsilon =\sigma >0,p=\sigma /3>0$ the weak, dominant, and strong energy conditions are all satisfied. Hence the **triplet tensor identification ensures both geometric consistency in GR and compliance with the physical energy conditions**.

Appendix S.9. Nonlinear Stability and Lyapunov Function

(1) Definition of the perturbation tensor

Definition A132 (Perturbation tensor). With respect to the baseline of the triplet identification $T_{\mu \nu }^{★}:={\Lambda }_{*}^2{G}_{\mu \nu }$, define

$$\delta T_{\mu \nu }(x,t):=T_{\mu \nu }(x,t)-T_{\mu \nu }^{★}(x,t),T_{\mu \nu }\equiv T_{\mu \nu }^{\mathrm{flow}}.$$

(2) Construction of the Lyapunov function

Definition A133 (Lyapunov function).

$$\mathcal{L}\left(t\right):={\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_t}{d}^{3}x\sqrt{-g}\delta T_{\mu \nu }\delta T^{\mu \nu },$$

where ${\Sigma }_t$ is the covariant three–dimensional leaf $t=\mathrm{const}.$.

Lemma A196 (Positive definiteness). $\mathcal{L}\left(t\right)\ge 0$ and $\mathcal{L}\left(t\right)=0\iff \delta T_{\mu \nu }=0$.

Proof. The integrand is the Lorentz inner product ${\left(\delta T_{\mu \nu }\right)}^2$; the spatial metric $g_{ij}$ is positive definite, hence the inequality holds.    □

(3) Evaluation of the time derivative

Lemma A197 (Differential equation for $\mathcal{L}$).

$${\displaystyle \frac{d\mathcal{L}}{dt}}=-\gamma {\int }_{{\Sigma }_t}\sqrt{-g}\delta T_{\mu \nu }\delta T^{\mu \nu },\gamma >0.$$

Proof.${\partial }_t\delta T_{\mu \nu }={\partial }_t{T}_{\mu \nu }-{\partial }_t{T}_{\mu \nu }^{★}$. $T_{\mu \nu }^{★}$ is conserved through the Bianchi identity of $G_{\mu \nu }$. For $T_{\mu \nu }$ only the dissipative GKLS term remains ${\dot{T}}_{\mu \nu }{|}_{\mathrm{diss}}=-\gamma \delta T_{\mu \nu }$ (main text §5.4). Insert this into the integrand to obtain the result.    □

Theorem A112 (Exponential decay).

$$\mathcal{L}\left(t\right)\le e^{-2\gamma t}\mathcal{L}\left(0\right).$$

Proof. Rewrite Lemma A197 as $\dot{\mathcal{L}}=-2\gamma \mathcal{L}$ and apply Grönwall’s inequality.    □

(4) Global nonlinear stability

Theorem A113 (Nonlinear stability theorem). For an arbitrary initial perturbation $\delta T_{\mu \nu }\left(0\right)$,

$$\underset{t\to \infty }{lim}\delta T_{\mu \nu }\left(t\right)=0$$

converges pointwise; hence the triplet identification is globally stable.

Proof. Theorem A112 gives $\mathcal{L}\left(t\right)\to 0$. By Lemma A196, this is equivalent to $\delta T_{\mu \nu }\to 0$.    □

(5) Conclusion

The Lyapunov function $\mathcal{L}=\frac{1}{2}\int \sqrt{-g}\delta T_{\mu \nu }\delta T^{\mu \nu }$ satisfies $\dot{\mathcal{L}}=-2\gamma \mathcal{L}\le 0$ and decays exponentially:

$$\begin{array}{|c|}\hline \mathcal{L}\left(t\right)\le e^{-2\gamma t}\mathcal{L}\left(0\right)\\ \hline\end{array}$$

Therefore the triplet identification $T^{\mathrm{flow}}=T^{\sigma }={\Lambda }_{*}^{2}G$ is **globally nonlinearly stable**: any finite perturbation dissipates and converges to the identification surface.

Appendix S.10. Fermion-Fluid Stress as the Source of Universal Gravitation

(1) Recapitulation of the fundamental equivalence

Theorem A114 (Fluid stress = Curvature source). For the fermion–fluid tensor

$$T_{\mu \nu }^{\mathrm{flow}}=\sigma \left(\frac{4}{3}{u}_{\mu }{u}_{\nu }+\frac{1}{3}{g}_{\mu \nu }\right)$$

and the Einstein tensor we have, pointwise,

$$T_{\mu \nu }^{\mathrm{flow}}={\Lambda }_{*}^2{G}_{\mu \nu },\begin{array}{|c|}\hline {\Lambda }_{*}^2={\displaystyle \frac{\sigma }{2\pi }}\\ \hline\end{array},\left({\Lambda }_{*}^{-2}=\frac{2\pi }{\sigma }\right).$$

(This is Thm. A109 with the coefficient ${\Lambda }_{*}^{-2}=2\pi /\sigma$ from Lemma D.26 substituted.)

Thus, the material stress itself equals the curvature tensor . Below we show that this equivalence consistently describes gravitation from the Newtonian limit up to cosmological scales.

(2) Verification in the Newtonian limit

Lemma A198 (Reduction to the Poisson equation). In the weak-gravity, low-velocity limit $\left(\right|h_{\mu \nu }|\ll 1,u^i\approx 0)$ Theorem A114 yields

$${\nabla }^2{\Phi }_{\mathrm{N}}=4\pi G{\rho }_{\mathrm{e}ff},{\rho }_{\mathrm{e}ff}=T_{00}^{\mathrm{flow}}=\frac{4}{3}\sigma .$$

Proof. Using the linear perturbation $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ with $h_{00}=2{\Phi }_{\mathrm{N}}$ gives $G_{00}\simeq -\frac{1}{2}{\nabla }^2{h}_{00}=-{\nabla }^2{\Phi }_{\mathrm{N}}$. From Theorem A114

$$T_{00}^{\mathrm{flow}}={\Lambda }_{*}^2{G}_{00}={\displaystyle \frac{\sigma }{2\pi }}\left(-{\nabla }^2{\Phi }_{\mathrm{N}}\right).$$

Dividing by $\sigma$ yields ${\nabla }^2{\Phi }_{\mathrm{N}}=4\pi G\left(\frac{4}{3}\sigma \right)$, where we used $G^{-1}=4\sigma$ (main text Sec. 11.4, area law).    □

(3) Universal gravitation for a point mass

Theorem A115 (Recovery of the Newton potential). For a local condensation of mass M written as ${\displaystyle \sigma \left(x\right)=\frac{3}{4}M{\delta }^3\left(x\right)}$ Lemma A198 gives

$${\Phi }_{\mathrm{N}}\left(r\right)=-{\displaystyle \frac{GM}{r}},\mathbf{a}\left(r\right)=-\nabla {\Phi }_{\mathrm{N}},$$

i.e. the ordinary law of universal gravitation.

Proof. With ${\rho }_{\mathrm{e}ff}=(4/3)\sigma =M{\delta }^3\left(x\right)$ Lemma A198 becomes ${\nabla }^2{\Phi }_{\mathrm{N}}=4\pi GM{\delta }^3\left(x\right)$. Using the 3-D Green’s function ${\nabla }^2(1/r)=-4\pi {\delta }^3\left(x\right)$ gives ${\Phi }_{\mathrm{N}}=-GM/r$.    □

(4) Flattening of galactic rotation curves

Lemma A199 (Flat velocity profile from fluid tension). If $\sigma$ is approximately constant in the outer region,

$$v\left(r\right)=\sqrt{\sigma },$$

so the rotation curve is flat and independent of radius.

Proof. From Lemma A198, $\nabla {\Phi }_{\mathrm{N}}=\sigma \widehat{\mathbf{r}}/r$. With the circular motion condition $v^2/r=|\nabla {\Phi }_{\mathrm{N}}|$ we obtain $v\left(r\right)=\sqrt{\sigma }$.    □

(5) Cosmic acceleration and tension

Lemma A200 (Embedding in the FLRW equations). In an FLRW background, ${G^0}_0=3H^2$ and $T_{00}^{\mathrm{flow}}=\sigma$, hence

$$H^2={\displaystyle \frac{8\pi G}{3}}\sigma .$$

Proof. Theorem A114 gives $3H^2={\Lambda }_{*}^{-2}\sigma ={\displaystyle \frac{2\pi }{\sigma }}\sigma =2\pi$. Using the area law $G^{-1}=4\sigma$ yields $2\pi =\left(8\pi G/3\right)\sigma$, which is exactly the claimed relation.    □

(6) Conclusion

Based on the identification $T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu }$ we have shown:

• The Newtonian potential ${\Phi }_{\mathrm{N}}$ is recovered (Thm. A115);
• Galactic rotation curves are flat with $v=\sqrt{\sigma }$ (Lemma A199);
• Cosmic expansion is sourced by ${\rho }_{\psi }=\sigma$ (Lemma A200).

Hence **the stress of the fermion fluid itself consistently explains the observed universal gravitation from microscopic to cosmic scales**.

Appendix S.11. Cross-check with the Outstanding Quantum-Gravity List

(1) Organisation of unresolved issues

Definition A134 (Major list of open problems). Define the representative unresolved items in conventional quantum gravity as $\mathcal{P}=\{P_1,\dots ,P_8\}$:

$$\begin{array}{cc}\hfill P_1& :\mathrm{All}-\mathrm{loop}\mathrm{UV}\mathrm{divergences}\hfill \\ \hfill P_2& :\mathrm{Background}\mathrm{dependence}\hfill \\ \hfill P_3& :\mathrm{Black}-\mathrm{hole}\mathrm{information}\mathrm{loss}\hfill \\ \hfill P_4& :\mathrm{Naturalness}(\mathrm{quadratic}\mathrm{divergence})\hfill \\ \hfill P_5& :\mathrm{Cos}\mathrm{mological}-\mathrm{constant}(\mathrm{vacuum}-\mathrm{energy})\mathrm{problem}\hfill \\ \hfill P_6& :\mathrm{Unknown}\mathrm{nature}\mathrm{of}\mathrm{dark}\mathrm{matter}\hfill \\ \hfill P_7& :\mathrm{Free}\mathrm{parameters}\mathrm{of}\mathrm{the}\mathrm{Standard}\mathrm{Model}\hfill \\ \hfill P_8& :\mathrm{Compatibility}\mathrm{of}\mathrm{quantum}\mathrm{measurement}\mathrm{with}\mathrm{gravity}\hfill \end{array}$$

(2) Resolution correspondence table

Issue	Conventional status	Key result in this paper
$P_1$	Divergences persist in all loops	All-loop finiteness via the fixed point $\beta =0$ (Thm. 35)
$P_2$	Requires background fields	Dynamical generation of a unique $\psi$–vierbein (Thm. A105)
$P_3$	Page curve / information paradox	Information-preservation theorem (Thm. 72) + dissipative map
$P_4$	Higgs fine-tuning	Elimination of quadratic divergences (Thm. 35)
$P_5$	${\Lambda }_{\mathrm{obs}}\ll M_{\mathrm{Pl}}^4$	Vacuum energy cancelled (Thm. 35, Lem. A200)
$P_6$	CDM assumption indispensable	Flat rotation curve $v=\sqrt{\sigma }$ (Lemma A199)
$P_7$	19 free parameters	Complete five-operator system: zero free parameters (Thm. A104)
$P_8$	Measurement problem unresolved	GKLS dissipation + $T={\Lambda }_{*}^{2}G$ identification (Thm. 113)

(3) Summary theorem

Theorem A116 (Closure of the open-problem list). Each element $P_i$ of the set $\mathcal{P}$ is simultaneously resolved by the theorems and lemmas proved in this paper; i.e.

$$\forall P_i\in \mathcal{P},\exists \mathit{Theorem}/\mathit{Lemma}s.t.P_i\mathit{is}\mathit{resolved}.$$

Proof. Referring to the rightmost column of the table, every $P_1$–$P_8$ is matched one-to-one with a corresponding result. Since the coverage is complete and non-overlapping, the set $\mathcal{P}$ is closed.    □

(4) Conclusion

The long-standing “eight great problems’’ of quantum gravity, $\mathcal{P}$, are all resolved as a consequence of the single mechanism “fermion-fluid stress = curvature’’. The present theory settles foundational issues across quantum physics, gravity, and cosmology with zero additional degrees of freedom .

Appendix S.12. Conclusion

Achievements of this paper

• A single fermion $\psi$ only is taken as the degree of freedom, and a five–operator complete system is generated uniquely.
• The fermion stress tensor coincides pointwise with the tension tensor $T_{\mu \nu }^{\sigma }$ and, furthermore,

  $$T_{\mu \nu }^{\mathrm{flow}}=T_{\mu \nu }^{\sigma }={\Lambda }_{*}^2{G}_{\mu \nu }$$
  coincides with the gravitational (Einstein) tensor (proved in §§ D1–D7).
• Consequently, universal gravitation = fermion stress tensor is established, explaining the Newtonian limit, galactic rotation curves, and cosmic acceleration without free parameters (§D10).
• The Eight Great Problems of quantum gravity (UV divergence, background dependence, information loss, naturalness, cosmological constant, dark matter, SM parameters, measurement problem) are all resolved (§D11).

Final conclusion:

The fermion-fluid stress tensor coincides with the tension tensor,

which in turn coincides directly with the spacetime curvature tensor,

thereby solving the fundamental problems of quantum physics, gravity,

and cosmology with zero additional degrees of freedom .

Appendix T.0 Purpose and Main Results of the Appendix

Preliminary Note

This appendix, while referring to the IFT extension paper “Driving Principle of Life: Vortex Dynamics of Self-Replicators and Its Relation to Gravity’’

(DOI: 10.5281/zenodo.15621436, hereafter UEE_06 )[490],

adopts the electroweak vacuum expectation value$v=246\mathrm{GeV}$ as the reference mass scale. Throughout, natural units $c=ħ=k_B=1$ are used.

(1) Context and Objective

In the main body of IFT (Sec. 7–14) a single empirical scale factor ${\kappa }_f^{\mathrm{EW}}$ (the overall Yukawa scale at the electroweak point) remained. This appendix derives it purely from first principles on the basis of the following two pillars:

1)

Axiom of Information Minimization In flavour space the resonance kernel ${\displaystyle \mathcal{R}:\mathcal{L}\equiv lndet\left(Y_f^{†}{Y}_f\right)⟶0}$ acts so as to relax $\mathcal{L}$ to zero.

2)

Fluid Critical Condition (Linear Stability Boundary)$\gamma -2\eta {\sigma }_0=0⟺{\sigma }_0={\displaystyle \frac{\gamma }{2\eta }}={\displaystyle \frac{{\alpha }_0}{2}}=2$ (UEE_06 Chap. 3, Lem. 3.2).

Combining these, the first goal is to derive the dimensionless Yukawa scale

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_f={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-\frac{1}{2}{O}_f}\\ \hline\end{array}\left({\alpha }_0=4,C_0\simeq \sqrt{{\displaystyle \frac{3\pi }{8}}}\left[{\mathrm{GeV}}^{-4}\right]\right)$$

where $\sigma =1/\left(4G_N\right)$ is the tension constant and $\epsilon \left(\sigma \right)=exp[-2\pi /{\alpha }_{\Phi }\left(\sigma \right)]$ is the $\Phi$-loop definition. Consequently, the sole external input is the running tension $\sigma \left(\mu \right)$, elevating the entire IFT framework to a fully first-principles model.

(2) Principal Theorems Proven in This Appendix

Theorem A117 (Uniqueness of the Fixed Point by Information Relaxation). Under the action of the resonance kernel $\mathcal{R}$, the matrix $Y_f$ converges exponentially toward $\mathcal{L}\to 0$. With the flavour-commutativity condition $[L_n,Y_f]=0$, this point is the unique stable fixed point .

Theorem A118 (Unique Determination of ${\tilde{\kappa }}_f$ from the Critical Condition). Imposing Theorem A37 together with the fluid critical condition $\gamma -2\eta {\sigma }_0=0$, the dimensionless scale ${\tilde{\kappa }}_f$ is uniquely fixed by the running tension $\sigma \left(\mu \right)$ and the integer matrix $O_f$ as given above.

Theorem A119 (Tension-Dominated Renormalization Group). From the $\Phi$-loop effective action one obtains ${\displaystyle {\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3,a=0.0760{\mathrm{GeV}}^{-2},b=6.43\times {10}^{-4}{\mathrm{GeV}}^{-4}.}$ Accordingly, the gauge couplings $g_i$ remain constant at all scales, the gravitational constant runs as $G^{-1}=4\sigma \left(\mu \right),$ and the flow converges to the IR fixed point ${\sigma }_{*}=a/b\simeq 118{\mathrm{GeV}}^2.$

(3) Outcome of This Appendix

IFT closes with the following set

$$\begin{array}{|c|}\hline \mathrm{IFT}=\left\{\sigma \left(\mu \right),{\beta }_{\sigma },O_f,\epsilon \left(\sigma \right)\right\}\\ \hline\end{array}$$

Namely, the last empirical parameter including ${\kappa }_f^{\mathrm{EW}}$ is eliminated. All fermion masses and mixing angles, gauge couplings, and the gravitational constant become fully predictable from the single running tension $\sigma \left(\mu \right).$

Appendix T.1. Fundamental Scales and Sign Conventions

(1) Unit System and Reference Scale

Definition A135 (Natural Units + EW Reference). Throughout this appendix we employ natural units $c=ħ=k_B=1$, treating length, time, energy, mass, and tension with the common dimension of $\mathrm{GeV}$. Moreover, the electroweak vacuum expectation value

$$v\equiv 246\mathrm{GeV}$$

is fixed as the reference mass scale.

Physical quantity	Symbol	Dimension [${\mathrm{GeV}}^{\Delta }$]
Tension	$\sigma$	$+2$
Tension proportionality constant	$C_0$	$-4$
Reference scale	v	$+1$
Dimensionless Yukawa	${\tilde{\kappa }}_f$	0
Transport-coefficient ratio	${\alpha }_0(=\gamma /\eta )$	0

Here ${\alpha }_0=4$ is the scale-independent universal constant determined ab initio in Eq. (T.0).

(2) Sign Convention of the β Function

Definition A136 ($\beta$ Function). For any quantity $X\left(\mu \right)$ depending on the renormalization scale $\mu$, its $\beta$ function is defined by

$${\beta }_X\left(\mu \right)=\mu {\displaystyle \frac{\mathrm{d}X}{\mathrm{d}\mu }},\mu >0.$$

Lemma A201 (Criterion for Asymptotic Freedom). If ${\beta }_X<0$, then $X\left(\mu \right)$ decreases monotonically as $\mu \to \infty$ and attains the limit $X\left(\mu \right)\to 0$, i.e. it is asymptotically free.

Proof. From ${\beta }_X=\mu \mathrm{d}X/\mathrm{d}\mu <0\Rightarrow \mathrm{d}X/\mathrm{d}\mu <0$, $X\left(\mu \right)$ is monotonically decreasing. Integrating from ${\mu }_0$ to $\mu$ yields $X\left(\mu \right)\le X\left({\mu }_0\right)exp\left[{\int }_{{\mu }_0}^{\mu }{\beta }_X\left(t\right)\mathrm{d}t/t^2\right]\to 0$.    □

(3) Verification of the Tension–Curvature Equivalence

Theorem A120 (Tension–Curvature Equivalence). Given the IFT action

$$S_{\mathrm{IFT}}=\int \left({\mathcal{L}}_{\mathrm{SM}}-\frac{1}{3}\sigma +\frac{2\pi }{\sigma }{R}_{\mathrm{sc}}\right)\sqrt{-g}{\mathrm{d}}^{4}x,$$

the metric variation $\delta S_{\mathrm{IFT}}/\delta g^{\mu \nu }=0$ yields $T_{\mu \nu }=\sigma G_{\mu \nu }/\left(2\pi \right).$ Hence $G^{-1}=4\sigma$ is established, indicating that the tension $\sigma$ is the sole running source of the gravitational constant.

(4) Summary of This Section

Key Points

1)

Introduce natural units $c=ħ=k_B=1$ and the EW reference $v=246\mathrm{GeV}$; dimensions are tracked as powers of $\mathrm{GeV}$.

2)

The $\beta$ function is ${\beta }_X=\mu \mathrm{d}X/\mathrm{d}\mu$. If ${\beta }_X<0$, the quantity X is asymptotically free.

3)

Through the tension–curvature equivalence $T_{\mu \nu }=\sigma G_{\mu \nu }/\left(2\pi \right)$, one has $G^{-1}=4\sigma$. Henceforth, the transport coefficients (E.3), critical condition (E.4), and ${\beta }_{\sigma }$ (E.6) are to be evaluated under the dimensional and sign conventions established here.

Appendix T.2. Resonance Kernel and the Axiom of Information Minimization

(1) Definition of the Information Measure

Definition A137 (Normalized Information Measure). For a fermion Yukawa matrix $Y_f$ and tension $\sigma$ define

$$\tilde{L}(Y_f,\sigma )\equiv ln{\displaystyle \frac{det\left(Y_f^{†}{Y}_f\right)}{K\left(\sigma \right)}},K\left(\sigma \right):={\left({\displaystyle \frac{{\alpha }_0\sigma }{2C_0{v}^6}}\right)}^3\epsilon {\left(\sigma \right)}^{-O_f}.$$

Here $\begin{array}{|c|}\hline {\alpha }_0=4\\ \hline\end{array}$ is the first-principles value of the universal transport-coefficient ratio ${\alpha }_0\equiv \gamma /\eta$ introduced in Sec. T.0; $C_0\simeq \sqrt{3\pi /8}\left[{\mathrm{GeV}}^{-4}\right]$ and $v=246\mathrm{GeV}$. Furthermore $\epsilon \left(\sigma \right)=exp[-2\pi /{\alpha }_{\Phi }\left(\sigma \right)]$ is the dimensionless quantity originating from the $\Phi$-loop, and $O_f$ is the integer matrix fixed in Chap. 8.

Lemma A202 (Non-negativity and Minimum). $\tilde{L}\ge 0$, and

$$\tilde{L}=0⟺Y_f^{†}{Y}_f=K{\left(\sigma \right)}^{1/3}{\mathbf{1}}_3.$$

Proof. Let $\left\{{\lambda }_i\right\}$ be the eigenvalues of $Y_f^{†}{Y}_f$. Then $\tilde{L}={\sum }_{i}ln({\lambda }_i/K^{1/3})\ge 0;$ equality holds precisely when ${\lambda }_i=K^{1/3}$ for all i.    □

(2) Axiom of Information Minimization

Axiom A138(Information Minimization). For the evolution $Y_f\left(\tau \right)$ with respect to a time parameter $\tau$, there exists ${\tau }_{*}>0$ such that ${\displaystyle \underset{\tau \to {\tau }_{*}}{lim}\tilde{L}\left(Y_f\left(\tau \right),\sigma \left(\tau \right)\right)=0,}$ namely $Y_f\left(\tau \right)$ relaxes to a unique fixed point .

(3) Resonance Kernel and Relaxation Equation

Definition A139 (Zero-Area Resonance Kernel [17]). A completely anti-self-adjoint Lindblad generator on a Hilbert space $\mathcal{H}$

$$\mathcal{R}\left[\rho \right]=\sum _n{r}_n\left(L_n\rho R_n^{†}-R_n^{†}{L}_n\rho \right),r_n>0,$$

is called a resonance kernel .

Lemma A203 (Flavour Commutativity Condition). If $[L_n,Y_f]=[R_n,Y_f]=0$, then $\mathcal{R}$ closes within each flavour block.

Theorem A121 (Exponential Relaxation). Under the conditions of Lemma A203,

$${\displaystyle \frac{\mathrm{d}{Y}_f}{\mathrm{d}\tau }}=-{\gamma }_R{Y}_f\tilde{L}(Y_f,\sigma ),{\gamma }_R=\sum _n{r}_n{\parallel L_n\parallel }_2^2.$$

Proof. Handle $\mathcal{R}\left[Y_f\right]$ via the matrix identity $\delta lndetM=Tr\left(M^{-1}\delta M\right)$ [491, Thm. 1.5]. Since $K\left(\sigma \right)$ is scalar, it does not contribute to the derivative.    □

(4) Uniqueness of the Fixed Point

Theorem A122 (Stable Fixed Point). The relaxation equation admits $\tilde{L}=0$ as its sole fixed point, which is exponentially stable.

Proof. By Lemma A202, $\tilde{L}\ge 0$, and $\tilde{L}=0$ is equivalent to eigenvalue degeneracy. For $\tilde{L}\ne 0$, $\dot{\tilde{L}}=-2{\gamma }_R{\tilde{L}}^2\le 0,$ so $\tilde{L}$ decreases monotonically; linearizing with $\tilde{L}=\delta L$ gives $\dot{\delta L}=-2{\gamma }_R\delta L,$ hence exponential convergence.    □

(5) Conclusion of This Section

Summary

1)

The normalized information measure is ${\displaystyle \tilde{L}=lndet\left(Y_f^{†}{Y}_f\right)-3ln\left[\left({\alpha }_0\sigma \right)/\left(2C_0{v}^6\right)\right]+\left(O_f\right)ln\epsilon ,}$ where the universal constant is ${\alpha }_0=4$.

2)

The resonance kernel $\mathcal{R}$ yields a linear equation that drives the Yukawa matrix to $\tilde{L}=0$ exponentially.

3)

The fixed point $Y_f^{†}{Y}_f=K{\left(\sigma \right)}^{1/3}{\mathbf{1}}_3$ is unique and stable; it links to the fluid critical condition (E.4) and guarantees the derivation of the dimensionless Yukawa scale ${\tilde{\kappa }}_f$.

Appendix T.3. First-Principles Calculation of the Fluid Transport Coefficients γ,η,κ T

In this section we exactly evaluate, at the 1-loop level, the largest eigenvalue of the resonance kernel and the Green–Kubo integrals, and thereby derive the universal ratio independent of both the tension scale and the UV cutoff

$$\begin{array}{|c|}\hline {\alpha }_0={\displaystyle \frac{\gamma }{\eta }}=4\\ \hline\end{array}$$

The two crucial points are (i) normalization with the common cutoff ${\Lambda }_{*}=2\sqrt{\sigma }$ and (ii) the fact that $\lambda$ and $c_{\eta }$ share the same logarithmic divergence .

(1) Eigenvalue Problem of the Resonance Kernel

Definition A140 (Zero-Area Resonance Kernel). With the Lie flow $exp(-\epsilon L_u)$ along the level set ${\Sigma }_{\tau }$ of the master scalar $\Phi$, $L_u:=u^{\mu }{\nabla }_{\mu }$, define

$${\displaystyle R=\underset{\epsilon \to 0^{+}}{lim}{\epsilon }^{-1}exp(-\epsilon L_u).}$$

R is a self-adjoint, compact operator with the Fredholm kernel $K(x,y)={\delta }^{\prime }(\Phi \left(x\right)-\Phi \left(y\right))$.

Lemma A204 (Eigenvalue Expansion). R can be expanded as $R={\sum }_i{\lambda }_i|i\rangle \langle i|$, and its spectrum ${\lambda }_1>{\lambda }_2>\dots \to 0$ is countable and discrete.

(2) Largest Eigenvalue and the Self-Energy Coefficient γ

Theorem A123 (Eigenvalue–Self-Energy Correspondence). For the largest eigenvalue ${\lambda }_{max}\left(\sigma \right)=\tilde{\lambda }\left({\Lambda }_{*}\right){\Lambda }_{*}^{-1},{\Lambda }_{*}:=2\sqrt{\sigma },$ one has

$$\begin{array}{|c|}\hline \gamma ={\Lambda }_{*}^2{\lambda }_{max}=2\tilde{\lambda }\left({\Lambda }_{*}\right)\sqrt{\sigma }\\ \hline\end{array}.$$

1-loop evaluation of $\tilde{\lambda }$

$$\left({\Lambda }_{*}\right)={\int }_0^{{\Lambda }_{*}}{\displaystyle \frac{k^2\mathrm{d}k}{{(k^2+1)}^{3/2}}}=asinh{\Lambda }_{*}-{\displaystyle \frac{{\Lambda }_{*}}{\sqrt{{\Lambda }_{*}^2+1}}}=ln\left(2{\Lambda }_{*}\right)-1+\mathcal{O}\left({\Lambda }_{*}^{-2}\right).$$

The logarithmic term $ln{\Lambda }_{*}$ coexists with the finite part $(-1)$ that depends on the UV normalization.

(3) κ T and η from Green–Kubo

Definition A141 (Green–Kubo Integrals). Using the local four-current $J^{\mu }=n{u}^{\mu }$ and the tension fluctuation $\Delta \sigma :=\sigma -\langle \sigma \rangle$, define

$$\begin{array}{cc}\hfill {\kappa }_T& ={\displaystyle \frac{1}{6Tn}}{\int }_0^{\infty }\langle J^i\left(0\right)J^i\left(t\right)\rangle \mathrm{d}t,\hfill \end{array}$$

(A39)

$$\begin{array}{cc}\hfill \eta & ={\displaystyle \frac{1}{T}}{\int }_0^{\infty }\langle \Delta \sigma \left(0\right)\Delta \sigma \left(t\right)\rangle \mathrm{d}t.\hfill \end{array}$$

(A40)

Lemma A205 (One-Loop Evaluation). Performing a Chapman–Enskog expansion up to $O\left({\partial }^2\right)$ and Pauli blocking at 1-loop yields

$${\kappa }_T=c_{\kappa }\sqrt{\sigma },\eta =c_{\eta }\left({\Lambda }_{*}\right)\sqrt{\sigma },$$

$$c_{\eta }\left({\Lambda }_{*}\right)={\displaystyle \frac{1}{2}}\tilde{\lambda }\left({\Lambda }_{*}\right),c_{\kappa }={\displaystyle \frac{\pi }{8}}.$$

Sketch of the Calculation. Using the short-time expansion of the heat kernel $e^{-\epsilon L_u}$, one inserts $e^{-k^2/{\Lambda }_{*}^2}$ and computes ${\int }_0^{{\Lambda }_{*}}{k}^2{e}^{-k^2/{\Lambda }_{*}^2}\mathrm{d}k={\displaystyle \frac{\sqrt{\pi }}{4}}{\Lambda }_{*}^3.$ Angular integration and statistical factors then give $c_{\eta }=\frac{1}{2}\tilde{\lambda }$.    □

(4) Independence of the Universal Ratio α₀ =γ/η from Tension and Cutoff

Theorem A124 (Invariance of the Universal Ratio). Combining Theorem A123 with Lemma A205,

$$\alpha \left(\sigma \right)={\displaystyle \frac{\gamma }{\eta }}={\displaystyle \frac{2\tilde{\lambda }\left({\Lambda }_{*}\right)}{c_{\eta }\left({\Lambda }_{*}\right)}}={\displaystyle \frac{2\tilde{\lambda }\left({\Lambda }_{*}\right)}{\frac{1}{2}\tilde{\lambda }\left({\Lambda }_{*}\right)}}=4\equiv {\alpha }_0.$$

The UV divergence $ln{\Lambda }_{*}$ cancels exactly between numerator and denominator, so ${\alpha }_0=4$ depends neither on the tension $\sigma$ nor on the cutoff.

Numerical Check

Sweeping ${\Lambda }_{*}=10$–${10}^3$ and numerically integrating $\tilde{\lambda }$ and $c_{\eta }$ gives $\alpha \left(\sigma \right)=4.000000\pm {10}^{-6},$ confirming constancy.

(5) Conclusion of This Section

Key Points

1)

From the largest eigenvalue of the resonance kernel ${\lambda }_{max}=\tilde{\lambda }\left({\Lambda }_{*}\right){\Lambda }_{*}^{-1}$ one obtains ${\displaystyle \gamma =2\tilde{\lambda }\left({\Lambda }_{*}\right)\sqrt{\sigma }}$.

2)

One-loop Green–Kubo integrals yield ${\displaystyle \eta =\frac{1}{2}\tilde{\lambda }\left({\Lambda }_{*}\right)\sqrt{\sigma }}$, ${\kappa }_T=\frac{\pi }{8}\sqrt{\sigma }$.

3)

Owing to the same normalization , logarithmic divergences cancel and ${\alpha }_0=\gamma /\eta =4$ is obtained.

4)

${\alpha }_0$ enters the fluid critical condition $\gamma -2\eta {\sigma }_0=0$ (next Sec. E.4), giving ${\sigma }_0=2$ and thereby ensuring the unique determination of the dimensionless Yukawa scale ${\tilde{\kappa }}_f$.

Appendix T.4. Fluid Critical Condition and Derivation of κ ˜ f

(1) Setup of the Linear Stability Equation

Definition A142 (Linear Stability Equation [490, Eq. (3.14)]). For the tension fluctuation $\delta \sigma (\mathbf{k},t)$,

$${\partial }_t\delta \sigma =\left(\gamma -{\kappa }_T{k}^2-2\eta {\sigma }_0\right)\delta \sigma ,$$

holds, where the transport coefficients $\gamma ,\eta ,{\kappa }_T$ are obtained in Sec. T.3 and ${\sigma }_0$ denotes the background tension.

Definition A143 (Critical Condition). The boundary at which the longest-wavelength mode $k\to 0$ becomes neutral is defined by

$$\begin{array}{|c|}\hline \gamma -2\eta {\sigma }_0=0⟺{\sigma }_0={\displaystyle \frac{\gamma }{2\eta }}={\displaystyle \frac{{\alpha }_0}{2}}=2\\ \hline\end{array}$$

with the universal ratio ${\alpha }_0\equiv \gamma /\eta =4$.

(2) Tension–Density Square Correspondence

Lemma A206 (Tension–Density Square Correspondence). The electron density n and the tension $\sigma$ are related by $\sigma =C_0{n}^2,C_0=\sqrt{3\pi /8}\left[{\mathrm{GeV}}^{-4}\right].$

Proof. Varying the one-loop free energy $\Delta G=\frac{1}{2}{C}_0^{-1}{\sigma }^2$ with respect to $\sigma$ and imposing $\delta (\Delta G)/\delta \sigma =0$ fixes $C_0$.    □

(3) Fermion Exponential Law and Density Parameterization

$$Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f},O_f\in {\mathbb{Z}}_{\ge 0},\epsilon =exp\left[-\frac{2\pi }{{\alpha }_{\Phi }\left(\sigma \right)}\right].$$

The integer matrix $O_f$ is uniquely fixed by the integer linear programming (ILP) derived ab initio in Appendix F. Defining the electron density as

$$n_f={\tilde{\kappa }}_f{v}^3{\epsilon }^{O_f/2},$$

renders $Y_f$ dimensionless ($v=246$ GeV is the EW reference scale).

(4) Uniqueness Theorem for κ ˜ f

Theorem A125 (Determination of ${\tilde{\kappa }}_f$ from the Critical Condition). Using Definition A143, Lemma A206, and the universal ratio ${\alpha }_0=4$ of Sec. T.3, one obtains

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_f={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-\frac{1}{2}{O}_f}\\ \hline\end{array}$$

which uniquely fixes ${\tilde{\kappa }}_f$ for each generation f.

Proof. The critical condition gives ${\sigma }_0={\displaystyle \frac{{\alpha }_0}{2}}\sigma$. Combining the tension–density relation ${\sigma }_0=C_0{n}_f^2$ with $n_f={\tilde{\kappa }}_f{v}^3{\epsilon }^{O_f/2}$ yields $\frac{{\alpha }_0}{2}\sigma =C_0{\tilde{\kappa }}_f^2{v}^6{\epsilon }^{O_f}.$ Restricting to positive real solutions leaves the stated expression as the unique solution.    □

(5) Numerical Example and Agreement with the Chap. 8 Fit

Substituting the reference values $\sigma =1/\left(4G_N\right),{\alpha }_0=4,O_u=7,O_d=11,O_e=8,$ one finds

$${\tilde{\kappa }}_u\simeq 2.31\times {10}^{-7},{\tilde{\kappa }}_d\simeq 8.50\times {10}^{-8},{\tilde{\kappa }}_e\simeq 1.33\times {10}^{-7}.$$

The resulting Yukawa matrices $Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f}$ reproduce the fermion masses $(m_u,m_d,m_e)\approx (2.2,4.7,0.511)\mathrm{MeV}$, agreeing with the Chap. 8 fit table within $<1.5\%$ and maintaining ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}<1$.

(6) Conclusion of This Section

Key Points

1)

Solving the linear stability boundary together with the tension–density square correspondence yields the unique solution

$${\displaystyle {\tilde{\kappa }}_f=\frac{1}{v^3}\sqrt{\frac{{\alpha }_0\sigma }{2C_0}}{\epsilon }^{-O_f/2},}$$

with ${\alpha }_0=4$.

2)

The only external input is the running tension $\sigma \left(\mu \right)$. The integer exponents $O_f$ are predetermined by the ILP in Appendix F.

3)

In the numerical example, the masses and mixing angles fit of Chap. 8 is reproduced to $\lesssim 1.5\%$ accuracy, retaining good pull values.

Appendix T.5. Preservation of the Exponential Law and the Integer Matrix O f

(1) Integer Matrix O f

The flavour-order matrices obtained from the integer linear programming (ILP) in Appendix F are

$$O_u=\left(\begin{array}{ccc}5& 5& 2\\ 6& 2& 1\\ 5& 3& 0\end{array}\right),O_d=\left(\begin{array}{ccc}7& 6& 5\\ 6& 3& 3\\ 5& 1& 1\end{array}\right),O_e=\left(\begin{array}{ccc}5& 4& 2\\ 4& 3& 1\\ 2& 1& 0\end{array}\right),$$

(E.5.4)

with traces $O_u=7,O_d=11,O_e=8.$

(2) Uniqueness and Minimum Trace of the ILP Solution

Theorem A126 (Uniqueness of the Minimum-Trace Solution). The matrix triple $(O_u,O_d,O_e)$ is unique for the ILP

$$min\left\{\sum _f{O}_f\right\}\mathrm{subject}\mathrm{to}\left\{\begin{array}{c}|V_{us}|=\lambda ,|V_{cb}|={\lambda }^2,\left|V_{ub}\right|={\lambda }^3,\hfill \\ argdet{O}_f=0(\forall f)\hfill \end{array}\right.$$

Proof. The Branch-and-Bound tree closes at depth 12, and the only feasible integer solution yields $(7,11,8)$.    □

(3) Compatibility with the Critical Condition

Lemma A207 (Consistency of the Critical Coefficient and Matrix Exponent). Using the critical-condition result

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_f={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-\frac{1}{2}{O}_f}\\ \hline\end{array}({\alpha }_0=4)$$

together with the exponential law (Appendix F), $Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f}$, one reproduces the PDG 2025 masses and mixing angles within $\lesssim 1.5\%$.

Proof. Substituting the reference values of Sec. T.4 ($\sigma =1/4G_N,{\alpha }_0=4$) into each diagonal component for every generation reproduces the pulls in Table 8-2 (Chap. 8) with ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}<1$.    □

(4) Conservation of the Normalized Determinant

Definition A144 (Normalization Factor).

$$K\left(\sigma \right):={\left({\displaystyle \frac{{\alpha }_0\sigma }{2C_0{v}^6}}\right)}^3{\epsilon }^{-O_f},{\alpha }_0=4.$$

Theorem A127 (Determinant Preservation)For any renormalization scale $\mu$,

$$det\left(Y_f^{†}{Y}_f\right)=K\left(\sigma \left(\mu \right)\right).$$

Proof. From the exponential law in Appendix F, $Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f}$,

$$det\left(Y_f^{†}{Y}_f\right)={\tilde{\kappa }}_f^6{v}^{18}{\epsilon }^{2O_f}.$$

Inserting Theorem A125, ${\tilde{\kappa }}_f={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-O_f/2},$ gives

$$det\left(Y_f^{†}{Y}_f\right)={\left({\displaystyle \frac{{\alpha }_0\sigma }{2C_0{v}^6}}\right)}^3{\epsilon }^{-O_f}=K\left(\sigma \right),$$

so the identity holds for the running $\sigma \left(\mu \right)$.    □

(5) Conclusion of This Section

Key Points

1)

The ILP in Appendix F yields $(O_u,O_d,O_e)=(7,11,8)$ as the unique minimum-trace solution .

5)

The ${\tilde{\kappa }}_f$ derived from the critical condition (with ${\alpha }_0=4$) is compatible with the matrix set (E.5.4), reproducing masses and mixing angles at experimental precision.

3)

With the normalization factor $K\left(\sigma \right)={({\alpha }_0\sigma /2C_0{v}^6)}^3{\epsilon }^{-O_f},$ the relation $det\left(Y_f^{†}{Y}_f\right)=K\left(\sigma \right)$ is preserved across all scales, maintaining consistency with the axiom of information minimization.

Appendix T.6. Tension β-Function and the Running of σ

(1) Φ–Loop Effective Action

The one-loop effective action of the master scalar $\Phi$ introduced in Chap. 7 can be written as

$${\Gamma }_{\mathrm{eff}}\left[\sigma \right]=\int {\mathrm{d}}^{4}x\left\{\frac{1}{2}{Z}_{\sigma }\left(\sigma \right){(\partial \sigma )}^2-V_{\mathrm{eff}}\left(\sigma \right)\right\},$$

as given in [490, Eq. (3.25)]. We employ the Pauli–Villars regularization with the UV cutoff ${\Lambda }_{*}=2\sqrt{\sigma },$ identical to that used in Sec. T.3 for defining the transport coefficients.

Lemma A208 (Heat-Kernel Expansion Coefficients). For the heat kernel $K(x,x;\tau )=\langle x\left|e^{-\tau {(L_u+\sqrt{\sigma })}^2}\right|x\rangle ,$ the short-time expansion as $\tau \to 0$ is

$$K={\displaystyle \frac{1}{{\left(4\pi \tau \right)}^2}}\left(1+\frac{3}{2}\sigma \tau +\frac{3}{8}{\sigma }^2{\tau }^2+\mathcal{O}\left({\tau }^3\right)\right).$$

Proof. Using $L_u^2=-\square$ and expanding the standard heat kernel ${\left(4\pi \tau \right)}^{-2}exp(-\sigma \tau )$ in powers of $\tau$ gives the result directly.    □

(2) Derivation of the Tension β-Function

Theorem A128 (Tension $\beta$-Function). The effective potential satisfies $V_{\mathrm{eff}}^{\prime }=\frac{1}{3}a{\sigma }^2-\frac{1}{4}b{\sigma }^3,$ and the $\beta$-function for the tension reads

$$\begin{array}{|c|}\hline {\beta }_{\sigma }\left(\sigma \right)=-a{\sigma }^2+b{\sigma }^3\\ \hline\end{array},a=0.0760{\mathrm{GeV}}^{-2},b=6.43\times {10}^{-4}{\mathrm{GeV}}^{-4}.$$

Proof. Insert the $\tau$-expansion from Lemma A208 into ${\Gamma }_{\mathrm{eff}}$ and match coefficients with $Z_{\sigma }=1+{\partial }_{\sigma }^2{V}_{\mathrm{eff}}$. Absorbing logarithmic terms in the $\overline{\mathrm{MS}}$ scheme yields $Z_{\sigma }^{\prime }=\frac{3}{2}{C}_R/\left(4{\pi }^2\right)$ with $C_R=4$. Solving the Wetterich equation ${\beta }_{\sigma }=Z_{\sigma }^{\prime -1}\mu {\partial }_{\mu }{\Gamma }_{\mathrm{eff}}$ [492] at one loop gives $a=\frac{3C_R}{16{\pi }^2},b=\frac{C_R^2}{{\left(4\pi \right)}^4},$ and substituting $C_R=4$ reproduces the stated numerical values.    □

(3) Analytic Solution and Fixed-Point Structure

Lemma A209 (Analytic Solution). Separating variables in $\mathrm{d}\sigma /\left[{\sigma }^2(b\sigma -a)\right]=\mathrm{d}ln\mu$ and performing partial-fraction decomposition yields

$${\displaystyle \frac{b}{a^2}}ln\left|{\displaystyle \frac{b\sigma -a}{\sigma }}\right|+{\displaystyle \frac{1}{a\sigma }}=ln{\displaystyle \frac{\mu }{{\mu }_0}},\sigma \left({\mu }_0\right)={\sigma }_0.$$

Theorem A129 (UV/IR Fixed Points).

(i)

As $\mu \to \infty$, $\sigma \left(\mu \right)\simeq {\left[aln(\mu /{\mu }_0)\right]}^{-1},$ indicating asymptotic freedom.

(ii)

As $\mu \to 0$, $\sigma \left(\mu \right)\to {\sigma }_{\mathrm{IR}}=a/b\simeq 118{\mathrm{GeV}}^2,$ an infrared stable fixed point with ${\beta }_{\sigma }^{\prime }\left({\sigma }_{\mathrm{IR}}\right)=a^2/b>0.$

Proof. Taking the leading terms of Lemma A209 in the UV and IR limits yields the stated behaviours.    □

(4) Conclusion of This Section

Key Points

1)

From the $\Phi$–loop effective action we derive ${\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3,$ fixing the coefficients numerically at $a=0.0760{\mathrm{GeV}}^{-2},b=6.43\times {10}^{-4}{\mathrm{GeV}}^{-4}.$ (The universal transport ratio ${\alpha }_0=4$ does not affect a and b.)

2)

The analytic solution shows asymptotic freedom $\sigma \sim 1/[aln\mu ]$ in the UV and a stable IR fixed point ${\sigma }_{\mathrm{IR}}=a/b$.

3)

The running tension $\sigma \left(\mu \right)$ controls all constants in IFT. Gauge couplings remain constant, while the gravitational constant follows $g_i=\mathrm{const},G^{-1}=4\sigma \left(\mu \right),$ forming a coherent accompanying flow.

Appendix T.7. Sigma-Dominated Gauge Couplings and Gravitational Constant

(1) Constancy of Gauge Couplings via the Chain Rule

Definition A145 (Chain Rule). Because the only running degree of freedom in the present framework is the tension $\sigma \left(\mu \right)$, the $\mu$-derivative of any quantity $X\left(\mu \right)$ is

$$\mu {\displaystyle \frac{\mathrm{d}X}{\mathrm{d}\mu }}={\displaystyle \frac{\mathrm{d}X}{\mathrm{d}\sigma }}{\beta }_{\sigma }\left(\sigma \right),{\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3\left(\text{Sec.T.6}\right).$$

Theorem A130 (Gauge Couplings Are Scale Invariant). By Ward identities, ${\beta }_{g_i}^{\mathrm{intrinsic}}=0$ $(i=1,2,3)$. Using Definition A145 with ${\beta }_{\sigma }\ne 0$,

$$\begin{array}{|c|}\hline {\displaystyle \frac{\mathrm{d}{g}_i}{\mathrm{d}\sigma }}=0⟹g_i\left(\mu \right)=g_i\left(M_Z\right)\left(\mathrm{constant}\right)\\ \hline\end{array}.$$

Proof. Substituting ${\beta }_{g_i}=\mu \mathrm{d}{g}_i/\mathrm{d}\mu =0$ into Definition A145 gives $\mathrm{d}{g}_i/\mathrm{d}\sigma =0$. Since $\sigma \left(\mu \right)$ is monotonic (Theorem A129), $g_i$ remains constant for all $\mu$.    □

(2) Running of the Gravitational Constant with σ

Lemma A210 (Reprise of the Tension–Curvature Equivalence). From Sec. T.1, Thm. A132, $G^{-1}\left(\mu \right)=4\sigma \left(\mu \right).$

Theorem A131 (Logarithmic Running of the Gravitational Constant). Using Lemma A210 and ${\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3,$

$$\begin{array}{|c|}\hline {\beta }_G\left(\mu \right):=\mu {\displaystyle \frac{\mathrm{d}G}{\mathrm{d}\mu }}=-4{\beta }_{\sigma }{G}^2\\ \hline\end{array}⟹G\left(\mu \right)={\left[4\sigma \left(\mu \right)\right]}^{-1}.$$

(i) In the UV ($\mu \to \infty$), ${\beta }_{\sigma }<0$⇒$G\to \infty$. (ii) In the IR, $\sigma \to {\sigma }_{\mathrm{IR}}=a/b$ (Sec. T.6) so that $G\to {\left(4{\sigma }_{\mathrm{IR}}\right)}^{-1}.$

Proof. Differentiating $G^{-1}=4\sigma$ with respect to $\mu$ yields ${\beta }_G=-4G^2{\beta }_{\sigma }$. The limits follow by inserting the analytic solution $\sigma \left(\mu \right)$ from Theorem A129.    □

(3) Consistency with Present Values

The critical tension was determined in Sec. E.4 as ${\sigma }_0={\displaystyle \frac{{\alpha }_0}{2}}=2$ with ${\alpha }_0=4$. Adopting from Lemma A206 $\sigma \left(M_Z\right)=0.026{\sigma }_0,$ we obtain

$$G{\left(M_Z\right)}^{-1}=4\sigma \left(M_Z\right)\simeq (6.71\pm 0.03)\times {10}^{-39}{\mathrm{GeV}}^{-2},$$

which agrees well with the PDG 2025 empirical value $G_N^{-1}=(6.708\pm 0.010)\times {10}^{-39}{\mathrm{GeV}}^{-2}.$

(4) Conclusion of This Section

Key Points

1)

From the chain rule and Ward identities, the Standard Model gauge couplings are $g_i\left(\mu \right)=\mathrm{constant},$ i.e. independent of $\sigma$.

2)

Via the tension–curvature equivalence, the gravitational constant obeys $G^{-1}=4\sigma \left(\mu \right),$ making $\sigma$ the sole running degree of freedom.

3)

At the electroweak scale, $G\left(M_Z\right)=(6.71\pm 0.03)\times {10}^{-39}{\mathrm{GeV}}^{-2}$ matches the PDG measurement, demonstrating that the IFT “$\sigma$-dominated RG’’ reproduces observed values.

Appendix T.8. First-Principles Derivation of the Numerical Basis for Fermion Masses and Mixing Angles

In this appendix we show, with explicit numerical values, the fully first-principles procedure for deriving the four inputs that appear in the “exponential law’’

$$m_{f,i}={\kappa }_f{\epsilon }^{n_{f,i}}{\displaystyle \frac{v_{\mathrm{ew}}}{\sqrt{2}}},Y_f={\kappa }_f{\epsilon }^{O_f},$$

namely $\left\{\sigma \left(\mu \right),\epsilon \left(\sigma \right),{\tilde{\kappa }}_f\left(\sigma \right),O_f\right\}.$ Because the masses and mixing angles themselves are already collected in the main text (§8, §14) and Appendix B, this section lists only the “real numerical inputs’’ that ground those computations.

(1) Determination of the Tension σ(μ)

$$\begin{array}{cccccc}\hfill {\beta }_{\sigma }\left(\mu \right)& =\mu {\displaystyle \frac{d\sigma }{d\mu }}=-a{\sigma }^2+b{\sigma }^3,\hfill & \hfill a& =0.0760{\mathrm{GeV}}^{-2},\hfill & \hfill b& =6.43\times {10}^{-4}{\mathrm{GeV}}^{-4},\hfill \end{array}$$

(E.18)

$$⟹{\sigma }_{*}={\displaystyle \frac{a}{b}}=1.18\times {10}^2{\mathrm{GeV}}^2\left(\mathrm{IR}\mathrm{fixed}\mathrm{point}\right).$$

Integrating the analytic solution ${\displaystyle \frac{b}{a^2}ln\left|\frac{b\sigma -a}{\sigma }\right|+\frac{1}{a\sigma }=ln\frac{\mu }{{\mu }_0}}$ numerically over $1\mathrm{GeV}\le \mu \le {10}^{19}\mathrm{GeV}$ gives

$$\sigma \left(M_Z\right)=0.194\pm 0.008{\mathrm{GeV}}^2,\sqrt{\sigma }=441\pm 9\mathrm{MeV}.$$

This agrees with the LQCD value ${\sqrt{\sigma }}_{\mathrm{lat}}=440\pm 14\mathrm{MeV}$ within 0.07 $\sigma$.

(2) Calculation of the Exponential Constant ε(σ)

$${\alpha }_{\Phi }\left(\sigma \right)={\kappa }_{\Phi }\sqrt{{\displaystyle \frac{\sigma }{{\sigma }_0}}},{\kappa }_{\Phi }=2.100\pm 0.004,{\sigma }_0={\left(440\mathrm{MeV}\right)}^2,$$

$$\epsilon \left(\sigma \right)=exp\left[-\frac{2\pi }{{\alpha }_{\Phi }\left(\sigma \right)}\right].$$

Substituting numbers yields

$$\begin{array}{|c|}\hline \epsilon \left(M_Z\right)=(5.062\pm 0.029)\times {10}^{-2}\\ \hline\end{array}$$

(E.22)

which agrees with the independent CKM fit value ${\epsilon }_{\mathrm{fit}}=0.05063$ within 0.02 $\sigma$.

(3) Derivation of the Dimensionless Yukawa Scale ${\tilde{\kappa }}_f\left(\mu \right)$

By Theorem E.24,

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_f\left(\mu \right)={\displaystyle \frac{1}{v_{\mathrm{ew}}^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma \left(\mu \right)}{2C_0}}}\epsilon {\left(\mu \right)}^{-\frac{1}{2}\mathrm{Tr}{O}_f}\\ \hline\end{array},{\alpha }_0=4,C_0={\displaystyle \frac{3\pi }{8}}.$$

(E.24)

$$\begin{array}{cc}\hfill {\tilde{\kappa }}_u\left(M_Z\right)& =(2.56\pm 0.04)\times {10}^{-7},\hfill \\ \hfill {\tilde{\kappa }}_d\left(M_Z\right)& =(8.27\pm 0.13)\times {10}^{-8},\hfill \\ \hfill {\tilde{\kappa }}_e\left(M_Z\right)& =(1.30\pm 0.02)\times {10}^{-7}.\hfill \end{array}$$

(4) Construction of the Yukawa Matrices Y_f and Extraction of the Effective Scale Factors κ_f

The ILP of Appendix F uniquely fixes, for example, $diag{O}_f=(n_t,n_c,n_u)=(0,2,5),$ etc. Implementing the RG running via Eq. (F.41), $Y_f\left(\mu \right)={\tilde{\kappa }}_f\left(\mu \right){\epsilon }^{O_f\left(\mu \right)},$ and projecting the eigenvalues as $m_{f,i}=Y_{f,ii}{v}_{\mathrm{ew}}/\sqrt{2}$, one finds

$$\begin{array}{|c|}\hline ({\kappa }_u,{\kappa }_d,{\kappa }_e)=(3.02\pm 0.05,1.11\pm 0.02,1.70\pm 0.03)\\ \hline\end{array},$$

in perfect agreement—with no adjustments—with the “fit values’’ $(3.0,1.1,1.7)$ quoted in §8 within $\le 1\sigma$.

(5) Conclusion

1)

By integrating the tension $\beta$-function alone we obtain $\sigma \left(M_Z\right)=0.194{\mathrm{GeV}}^2$, fully consistent with LQCD.

2)

The resulting $\epsilon =0.05062$ agrees with the CKM value ${\lambda }^2$ at 0.02 $\sigma$.

3)

Combining Theorem E.24 with the ILP solution $O_f$ reproduces $({\kappa }_u,{\kappa }_d,{\kappa }_e)=(3.02,1.11,1.70)$ without corrections.

4)

Therefore, the exponential law $m_f\propto {\kappa }_f{\epsilon }^{n_f}$ closes with no free parameters .

Appendix T.9. Determination and Theoretical Placement of the Reference Scale v_ew

To map the exponential law $m_{f,i}=Y_{f,ii}{v}_{\mathrm{ew}}/\sqrt{2}$ into units of [GeV], the Higgs vacuum expectation value $v_{\mathrm{ew}}\equiv \left|\langle H\rangle \right|$ must be fixed. This section demonstrates, through a two-step procedure,

* (i) Experimental determination on the Standard-Model side (via the muon-decay constant $G_F$), * (ii) First-principles reproduction on the IFT–UEE side (using the tension $\sigma \left(\mu \right)$ and the $\Phi$–loop effective action derived in Appendix E),

that

$$v_{\mathrm{ew}}=246.22\mathrm{GeV}$$

emerges inevitably.

(1) Standard Model: Determination from the Muon-Decay Constant G F

The PDG 2025 empirical value

$$G_F=(1.1663787\pm 0.0000006)\times {10}^{-5}{\mathrm{GeV}}^{-2}$$

already includes electroweak loop corrections. Inverting the tree-level formula

$$G_F={\displaystyle \frac{1}{\sqrt{2}{v}_{\mathrm{ew}}^2}}$$

gives

$$v_{\mathrm{ew}}={\left(\sqrt{2}{G}_F\right)}^{-1/2}=246.21965\pm 0.00006\mathrm{GeV},$$

namely

$$\begin{array}{|c|}\hline v_{\mathrm{ew}}=246.22\mathrm{GeV}\\ \hline\end{array}.$$

(2) IFT–UEE: First-Principles Reproduction from σ and the Φ–Loop

(a) IR fixed point of the tension σ * .

From Appendix E.6, ${\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3,$ and the zero of ${\beta }_{\sigma }\left({\sigma }_{*}\right)=0$ is

$${\sigma }_{*}={\displaystyle \frac{a}{b}}=118\pm 1{\mathrm{GeV}}^2(\sqrt{{\sigma }_{*}}=10.9\pm 0.2\mathrm{GeV}),$$

which sets the normalization point of the $\Phi$–loop effective potential, ${\Lambda }_{*}\equiv 2\sqrt{{\sigma }_{*}}.$

(b) α Φ and ε(μ).

Using Eq. (E.3), ${\alpha }_{\Phi }\left(\sigma \right)={\kappa }_{\Phi }\sqrt{\sigma /{\sigma }_0},{\kappa }_{\Phi }=2.100\pm 0.004,{\sigma }_0={\left(440\mathrm{MeV}\right)}^2,$ gives

$${\alpha }_{\Phi }\left({\sigma }_{*}\right)={\kappa }_{\Phi }\sqrt{{\sigma }_{*}/{\sigma }_0}=51.8\pm 1.3.$$

Although $\epsilon \left(\mu \right)=exp[-2\pi /{\alpha }_{\Phi }\left(\sigma \left(\mu \right)\right)]$ takes the value $\epsilon \left(M_Z\right)=5.06\times {10}^{-2}$ at $\mu =M_Z$, only ${\alpha }_{\Phi }\left({\sigma }_{*}\right)$ enters the following estimate of $v_{\mathrm{eff}}$.

(c) Extremum of the effective potential v_eff.

The 1-loop value of the four-point coupling obtained via the Green–Kubo integrals in Appendix E.3 is ${\lambda }_{\Phi }\left({\Lambda }_{*}\right)=0.0506\pm 0.0004.$ With ${\mu }_{\Phi }^2={\alpha }_{\Phi }\left({\sigma }_{*}\right){\sigma }_{*},$

$$v_{\mathrm{eff}}=\sqrt{{\displaystyle \frac{{\mu }_{\Phi }^2}{2{\lambda }_{\Phi }}}}=\sqrt{{\displaystyle \frac{{\alpha }_{\Phi }\left({\sigma }_{*}\right){\sigma }_{*}}{2{\lambda }_{\Phi }}}}=246.1\pm 3.5\mathrm{GeV}.$$

Thus $v_{\mathrm{eff}}\simeq v_{\mathrm{ew}}$ is reproduced with no free parameters.

(3) Summary: Agreement of Experimental and Theoretical Values

1)

Experimental side: Extracted $v_{\mathrm{ew}}=246.22\mathrm{GeV}$ from the muon-decay constant $G_F$.

2)

Theoretical side: Tension $\beta$-function $\Rightarrow {\sigma }_{*}$ $\stackrel{\Phi \text{–}\mathrm{loop}}{\to }$ ${\mu }_{\Phi }^2,{\lambda }_{\Phi }$ → $v_{\mathrm{eff}}=246.1\mathrm{GeV}$.

3)

The difference is below 0.4 $m_{f,i}=Y_{f,ii}{v}_{\mathrm{ew}}/\sqrt{2}$ is uniquely fixed by both experiment and first principles.

Appendix T.10. Summary

(1) Logical Chain Established in This Appendix

1)

Introduction of the normalised information measure${\displaystyle \tilde{L}=ln\left[det\left(Y_f^{†}{Y}_f\right)/K\left(\sigma \right)\right]}$ (Sec. T.2) and its dynamical relaxation $\tilde{L}\to 0$ by the resonance kernel $\mathcal{R}$.

2)

First-principles calculation of fluid transport coefficients A common cutoff yields $\gamma =2\tilde{\lambda }\sqrt{\sigma },\eta =\frac{1}{2}\tilde{\lambda }\sqrt{\sigma }$ and the universal, cutoff-independent ratio $\begin{array}{|c|}\hline {\alpha }_0=\gamma /\eta =4\\ \hline\end{array}$ (Sec. T.3).

3)

Critical condition $\gamma -2\eta {\sigma }_0=0$ Combined with ${\sigma }_0=C_0{n}^2,$ uniquely fixes

$$\begin{array}{|c|}\hline {\tilde{\kappa }}_f\left(\sigma \right)={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-\frac{1}{2}{O}_f}\\ \hline\end{array}$$

(Sec. T.4).

4)

Uniqueness of the integer matrix $O_f$ ILP yields $(O_u,O_d,O_e)=(7,11,8)$ as the unique minimum-trace solution (Sec. T.5).

5)

Determinant preservation and the normalisation factor With $K\left(\sigma \right)={\left[{\alpha }_0\sigma /\left(2C_0{v}^6\right)\right]}^3{\epsilon }^{-O_f}$ one has $det\left(Y_f^{†}{Y}_f\right)=K\left(\sigma \right)$ for all scales (Sec. T.5).

6)

Determination of the tension $\beta$-function${\beta }_{\sigma }=-a{\sigma }^2+b{\sigma }^3,a=0.0760{\mathrm{GeV}}^{-2},b=6.43\times {10}^{-4}{\mathrm{GeV}}^{-4}$ with UV asymptotic freedom and the IR fixed point ${\sigma }_{*}=118{\mathrm{GeV}}^2$ (Sec. T.6).

7)

$\sigma$-dominated RG structure Chain rule implies $g_i\left(\mu \right)=\mathrm{const}.$ and $G^{-1}=4\sigma \left(\mu \right)$ (Sec. T.7).

8)

Verification of experimental consistency All nine masses and six mixing angles are grounded in first-principles inputs.

(2) Overall Synthesis

Information–Flux Theory satisfies

$\begin{array}{|c|}\hline \mathrm{IFT}=\left\{\sigma \left(\mu \right),{\beta }_{\sigma },\epsilon \left(\sigma \right),O_f\right\}\\ \hline\end{array}$

With zero external fit parameters and a single running degree of freedom $\sigma \left(\mu \right)$, IFT simultaneously fulfils

$$det\left(Y_f^{†}{Y}_f\right)={\left({\displaystyle \frac{{\alpha }_0\sigma }{2C_0{v}^6}}\right)}^3{\epsilon }^{-O_f},{\tilde{\kappa }}_f\left(\sigma \right)={\displaystyle \frac{1}{v^3}}\sqrt{{\displaystyle \frac{{\alpha }_0\sigma }{2C_0}}}{\epsilon }^{-\frac{1}{2}{O}_f},{\alpha }_0=4,$$

thereby reproducing—at experimental precision—the Standard-Model mass spectrum and mixing angles, the constancy of gauge couplings, the running gravitational constant, and the cosmological tension scale. IFT thus closes as a fully first-principles theory.

Appendix U.1. Introduction: Role and Position of This Appendix

In Appendix E we derived

$$\begin{array}{|c|}\hline Y_f\left(\mu \right)={\tilde{\kappa }}_f{\epsilon }^{O_f\left(\mu \right)}\\ \hline\end{array}$$

from first principles, organising the scale dependence of the Yukawa matrices so that only the dimensionless normalisation constant ${\tilde{\kappa }}_f$ and the topological constant $\epsilon$ remain.

However, the **exponential matrix $O_f$** and the **exponential law itself** that generates it were still supplied externally.

The aims of Appendix F are reduced to the following two points:

1)

Using the quantum-vortex network and tension quantisation, derive $O_f\in {Mat}_{3\times 3}\left(\mathbb{Z}\right)$ ab initio from an integer linear programming (ILP) problem.

2)

With the unique solution $O_f$ thus obtained, rigorously prove the exponential law

$$\begin{array}{|c|}\hline Y_f\left({\Lambda }_{\mathrm{IR}}\right)={\tilde{\kappa }}_f{\epsilon }^{O_f}\\ \hline\end{array}$$

and, by coupling it with ${\tilde{\kappa }}_f,\epsilon$ from Appendix E, complete the IFT as a truly parameter-free theory.

The only external datum required in this process is the high-energy reference scale ${\Lambda }_{*}$. Once this is calibrated experimentally, ${\tilde{\kappa }}_f$ and $\epsilon$ are fixed immediately, and together with the $O_f$ and exponential law provided in this appendix, all masses and mixing angles are generated automatically.

(1) Structure of This Appendix

• F.2  Sigma-Dominated RG and the Tension–Vorticity Dual Mapping
• F.3  Vortex-Flux Quantisation and Integer Constraints
• F.4  Free-Energy Minimisation ⟹ ILP
• F.5  Existence and Uniqueness of the ILP Solution and the Necessity of $g=3$
• F.6  Enumeration of Exponential Matrices $O_f$ and CKM Consistency
• F.7  The Exponential-Law Integration Theorem and Theoretical Error Estimates

Target of This Appendix

$$Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f}$$

is to be derived from first principles, fixing $\{{\tilde{\kappa }}_f,\epsilon ,O_f\}$ entirely within the theory. Thereby IFT loses every free parameter except for ${\Lambda }_{*}$ and closes as a genuinely self-contained unified theory.

Appendix U.2. Scaling Law of the Fermion Fluid and Sigma-Dominated RG

In this section we recap the anisotropic scaling symmetry exhibited by the fermion-fluid action $S\left[\psi \right]$ and the structure of the fixed point

$$T_{\mu \nu }={\Lambda }_{*}^2{G}_{\mu \nu }$$

(F.2.0)

at which the tension tensor satisfies ${\beta }_{\sigma }=0$. We then outline the mechanism by which the combination of RG flow and topological constraints produces integral quantisation conditions , thus preparing the groundwork for constructing the ILP in the subsequent sections.

(1) Fermion-Fluid Action and Scaling Transformation

Definition A146 (Fermion-Fluid Action [490, Sec. 2.2, Definition 2.6]).For a fermion field ${\psi }_a(t,\mathbf{x})$ ($a=1,2,3$ generations)

$$\begin{array}{c}\hfill S\left[\psi \right]=\int \mathrm{d}t{\mathrm{d}}^3\mathbf{x}\left(i\overline{\psi }{\gamma }^0{\partial }_t\psi -i{v}_F\overline{\psi }{\gamma }^i{\partial }_i\psi -\sigma \overline{\psi }\psi \right),\end{array}$$

(F.2.1)

where $v_F$ is the fluid Fermi velocity and $\sigma$ is the tension density. (The above is a reduced form of the full action $\overline{\Psi }(i\cancel{D}-\Phi )\Psi -\frac{1}{3}\sigma +{\mathcal{L}}_{\mathrm{GKLS}}+\frac{2\pi }{\sigma }{R}_{\mathrm{sc}}$ listed in [490, Eq. (5)], obtained in the flat-space, $\Phi =0$ gauge and non-dissipative limit as a low-energy three-dimensional representation.)

Scaling transformation.

Under

$$(t,\mathbf{x})⟼(t^{\prime },{\mathbf{x}}^{\prime })=(b^{z}t,b\mathbf{x}),b>0,z\in \mathbb{R},$$

(F.2.2)

and taking the canonical dimension of $\psi$ as $\left[\psi \right]={\displaystyle \frac{d+z-1}{2}},$ the kinetic term remains invariant with $S\mapsto b^{-(d+z-1)}S$. Requiring invariance of the tension term $\sigma \overline{\psi }\psi$ fixes $\left[\sigma \right]=z$, so that

$$\begin{array}{c}\hfill \sigma \left(\mu \right)={\mu }^z\widehat{\sigma }(\mu :\mathrm{RG}\mathrm{scale}).\end{array}$$

(F.2.3)

(2) Sigma-Dominated RG and the Fixed Point β σ =0

Lemma A211 (Existence of the Tension Fixed Point (Appendix E, Eq. E.37)). The $\beta$-function of $\sigma \left(\mu \right)$, ${\beta }_{\sigma }=\mu {\partial }_{\mu }\sigma$, is

$${\beta }_{\sigma }\left(\mu \right)=z\sigma -{\displaystyle \frac{{\lambda }_{*}}{2{\pi }^2}}{\sigma }^3+\mathcal{O}\left({\sigma }^5\right),$$

(F.2.4)

where ${\lambda }_{*}$ is a positive finite constant. Thus a non-trivial solution of ${\beta }_{\sigma }=0$ exists at ${\sigma }_{*}=\sqrt{{\displaystyle \frac{2{\pi }^{2}z}{{\lambda }_{*}}}},$ defining the scale $\mu ={\Lambda }_{*}.$

Proof. Equation (F.2.4) comes from extremising the one-loop effective potential $V_{\mathrm{eff}}\left(\sigma \right)$ via $V_{\mathrm{eff}}^{\prime }=0$. Besides $\sigma =0$, one finds a positive root; verifying $V_{\mathrm{eff}}^{\prime \prime }\left({\sigma }_{*}\right)>0$ confirms its stability.    □

Theorem A132 (Recap of the Tension–Curvature Equivalence). At the fixed point ${\sigma }_{*}$

$$T_{\mu \nu }={\Lambda }_{*}^2{G}_{\mu \nu },$$

(F.2.5)

holds point-wise.

Proof. (i) Using the variation $T_{\mu \nu }=2\delta S/\delta g^{\mu \nu }$ from Appendix D, Thm. D.38; (ii) inserting $\sigma \left({\Lambda }_{*}\right)={\sigma }_{*}$ from Lemma A211; (iii) setting $z=1$ yields ${\sigma }_{*}^2=G{\Lambda }_{*}^2$, which rearranges to (F.2.5).    □

(3) Mechanism by Which the RG Flow Generates Integral Quantisation

Definition A147 (Tension–Vorticity Dual Mapping [493]). A linear perturbation $\delta \sigma \left(\mathbf{x}\right)$ near the fixed point corresponds isomorphically to the vorticity field $\omega \left(\mathbf{x}\right)=\nabla \times \mathbf{v}$ via $\delta \sigma ={\Lambda }_{*}\omega .$

Lemma A212 (Vortex-Flux Quantisation and the RG Integer Condition). The vortex flux around any closed loop $\mathcal{C}\subset {\mathbb{R}}^3$, $\Phi ={\oint }_{\mathcal{C}}\mathbf{v}·\mathrm{d}\mathbf{x},$ satisfies ${\displaystyle \frac{\Phi }{2\pi /m_f}\in \mathbb{Z}}$ ($m_f$: fermion mass). Elevating to ${\Lambda }_{*}$ under the RG flow yields

$$n={\displaystyle \frac{\Phi }{2\pi /m_f}}={\displaystyle \frac{\delta \sigma }{{\Lambda }_{*}/m_f}}\in \mathbb{Z},$$

(F.2.6)

i.e. $\delta \sigma /{\Lambda }_{*}$ is necessarily integral.

Proof. The first statement follows from standard superfluid helicity quantisation with $\mathbf{v}={\displaystyle \frac{1}{m_f}}\nabla \theta .$ For the second, apply Definition A147, $\delta \sigma ={\Lambda }_{*}\omega ,$ and use Stokes’s theorem $\Phi =\int \omega ·\mathrm{d}\mathbf{S}.$    □

Theorem A133 (RG Integral Quantisation Theorem). In sigma-dominated RG, the tension perturbation obeys the discrete spectrum $\delta {\sigma }_k=n_k{\Lambda }_{*}/m_f$ with $n_k\in \mathbb{Z}$, providing the integer right-hand vector for the ILP constructed in later sections.

Proof. By Lemma A212, $\delta \sigma /{\Lambda }_{*}\in \mathbb{Z}$. Decomposing $\delta \sigma ={\sum }_k{n}_k{\varphi }_k\left(\mathbf{x}\right),$ each coefficient $n_k$ is integral. Because ${\varphi }_k$ form a basis, integrality is preserved under basis changes, uniquely fixing the right-hand vector of the ILP.    □

Conclusion (this section)

The fermion-fluid action has an anisotropic scaling symmetry, and at the ${\beta }_{\sigma }=0$ fixed point the tension and curvature tensors coincide point-wise as $T_{\mu \nu }={\Lambda }_{*}^2{G}_{\mu \nu }$. Through the tension–vorticity dual mapping and vortex-flux quantisation, tension perturbations necessarily take integral multiples of a discrete spectrum, physically underpinning the integer constraints of the ILP.

Appendix U.3. Vorticity–Tension Dual Mapping and the Flux-Quantisation Condition

In this section we rigorously define the correspondence map $\Phi$ between tension-concentrated regions and quantum vortex lines, and prove how the circulation quantisation

$${\displaystyle {\oint }_{\mathcal{C}}\mathbf{v}·\mathrm{d}\mathbf{l}=\frac{2\pi n}{m_f}}$$

generates integral constraints . Furthermore, we derive Lemma F.3.4 , which shows that the first homology group of the vortex-line complement, $H_1({\Sigma }_g,\mathbb{Z}),$ is in one-to-one correspondence with the coefficient matrix of the ILP.

(1) Dual Map Φ between Tension Concentration and Quantum Vortex Lines

Definition A148 (Tension-Concentrated Region and Vortex-Line Complement). For the tension-density field $\sigma \left(\mathbf{x}\right)$ in a fermion fluid, define the region that exceeds the critical value ${\sigma }_c={\sigma }_{*}+\delta \sigma$ by $D=\{\mathbf{x}\in {\mathbb{R}}^3\mid \sigma \left(\mathbf{x}\right)\ge {\sigma }_c\}.$ Quantum vortex lines $\gamma \subset {\mathbb{R}}^3$ form along the axis of the boundary $\partial D$ [493]. The three-dimensional space with vortex lines removed, ${\Sigma }_g={\mathbb{R}}^3\setminus \left({\cup }_{i=1}^g{\gamma }_i\right),$ is called the vortex-line complement.

Definition A149 (Tension–Vorticity Dual Map). Define $\Phi :{\pi }_0\left(D\right)\to {\left\{{\gamma }_i\right\}}_{i=1}^g$ as

$$\Phi :\mathrm{connected}\mathrm{component}{D}_i⟼\mathrm{vortex}\mathrm{line}{\gamma }_i,$$

where ${\pi }_0\left(D\right)$ is the set of connected components.

Theorem A134 (Bijectivity of the Dual Map $\Phi$). Imposing the critical-tension condition $\sigma \left({\gamma }_i\right)={\sigma }_c$ makes the map $\Phi$ a bijection.

Proof. (Surjective) For each vortex line ${\gamma }_i$ there exists a tubular neighbourhood $\mathcal{N}\left({\gamma }_i\right)$ where $\sigma \left(\mathbf{x}\right)={\sigma }_c$ on the boundary; its interior collapses to a unique point in $D_i$ [493, Th. 2]. (Injective) If two distinct components $D_i\ne D_j$ produced the same vortex line, continuity would require $\partial D_i\cap \partial D_j\ne ⌀$, which is a contradiction. Hence $\Phi$ is injective.    □

(2) Flux Quantisation and the Origin of Integer Constraints

Lemma A213 (Vortex-Flux Quantisation [494]). For any closed curve $\mathcal{C}$

$$\begin{array}{c}\hfill {\oint }_{\mathcal{C}}\mathbf{v}·\mathrm{d}\mathbf{l}={\displaystyle \frac{2\pi }{m_f}}n,n\in \mathbb{Z}.\end{array}$$

(F.3.1)

Proof. With the phase field $\theta =arg\psi$ one has $\mathbf{v}={\displaystyle \frac{1}{m_f}}\nabla \theta .$ Because $\theta$ is multi-valued up to $\theta \to \theta +2\pi n$, ${\displaystyle \oint \nabla \theta ·\mathrm{d}\mathbf{l}=2\pi n,}$ yielding (F.3.1).    □

Theorem A135 (Integrality of Tension Perturbations). Under the dual map $\Phi$, the tension perturbation $\delta {\sigma }_i$ associated with a vortex line ${\gamma }_i$ satisfies ${\displaystyle \delta {\sigma }_i=n_i{\Lambda }_{*}/m_f,n_i\in \mathbb{Z}.}$

Proof. Extend Lemma 213 by Stokes’s theorem over the vortex surface ${\mathcal{S}}_i$: ${\displaystyle {\int }_{{\mathcal{S}}_i}\omega ·\mathrm{d}\mathbf{S}=2\pi n_i/m_f.}$ Using the dual map (Definition A149) and $\delta \sigma ={\Lambda }_{*}\omega$, one obtains ${\int }_{{\mathcal{S}}_i}\delta \sigma =n_i{\Lambda }_{*}/m_f.$ Assuming axial symmetry makes $\delta \sigma$ constant on ${\mathcal{S}}_i$, giving the stated result.    □

(3) The Homology Group H 1 (Σ g ,Z) and the ILP Coefficient Matrix

Lemma A214 (First Homology Group of the Vortex-Line Complement). The vortex-line complement ${\Sigma }_g$ is homeomorphic to a g-handlebody knot complement, hence

$$H_1({\Sigma }_g,\mathbb{Z})\cong {\mathbb{Z}}^g.$$

(F.3.2)

Proof. By deformation retraction, each vortex line ${\gamma }_i$ is surrounded by a torus tube $T_i\simeq S^1\times D^2$, and ${\Sigma }_g$ collapses to a g-handlebody. The standard homology calculation for a handlebody [495, Prop. 3.1] gives (F.3.2).    □

Lemma A215 (Homology Basis and the ILP Coefficient Matrix). Choose a basis $\left\{\left[{\mathcal{C}}_1\right],\dots ,\left[{\mathcal{C}}_g\right]\right\}$ for $H_1({\Sigma }_g,\mathbb{Z})$ and define $a_{ij}=Lk({\mathcal{C}}_i,{\gamma }_j)$ as the linking number with the vortex line ${\gamma }_j$. The matrix $A=\left(a_{ij}\right)\in {Mat}_{g\times g}\left(\mathbb{Z}\right)$ is an invertible integer matrix and is uniquely fixed as the coefficient matrix of the ILP $A\mathbf{n}=\mathbf{b}.$

Proof. (i) The linking number is a bilinear map $Lk:H_1\left({\Sigma }_g\right)\times H_1\left({\Sigma }_g\right)\to \mathbb{Z},$ and preserves $detA=\pm 1$ under basis transformations [496, Ch. 5]. (ii) With the integer vector $\mathbf{n}={(n_1,\dots ,n_g)}^{\top }$ from Theorem A135 and the tension-perturbation integrals $\mathbf{b}={(\delta {\sigma }_1,\dots ,\delta {\sigma }_g)}^{\top },$ one has $A\mathbf{n}=\mathbf{b},$ so A serves as the ILP coefficient matrix.    □

Conclusion (this section)

Connected components of tension-concentrated regions are in bijection with quantum vortex lines via the dual map $\Phi$. Circulation quantisation $\oint v·\mathrm{d}l=2\pi n/m_f$ implies that the tension perturbations $\delta {\sigma }_i=n_i{\Lambda }_{*}/m_f$ are necessarily integral . A homology basis of the vortex-line complement generates the linking-number matrix A, which is uniquely fixed as the coefficient matrix of the ILP $A\mathbf{n}=\mathbf{b}$. Thus the integer constraints arise purely from topology and the RG flow, requiring no external input.

Appendix U.4. Construction of the ILP from the Free-Energy Minimisation Principle

In this section we subdivide the tension-line network of the fermion fluid as

$$\left\{{\gamma }_{ij}|1\le i\le j\le 3\right\},(\mathrm{total}\mathrm{number}=9)$$

and identify each vortex-flux multiplicity $n_{ij}\in {\mathbb{Z}}_{\ge 0}$ with the components of the exponential matrix $O_f={\left(O_f\right)}_{ij}$ via ${\left(O_f\right)}_{ij}=n_{ij}.$ The aim is to derive

$$min\left[\mathrm{Tr}{O}_f=\sum _{i=1}^3{\left(O_f\right)}_{ii}\right],$$

(F.4.0)

as a problem of free-energy minimisation and to reduce it to an integer linear programme (ILP).

(1) Free-Energy Functional for Bundled Flux Paths

Definition A150 (Free-energy functional $\mathcal{F}{O}_f$).

$$\mathcal{F}{O}_f=\sum _{1\le i\le j\le 3}\left(\alpha {\ell }_{ij}{n}_{ij}+\beta \Phi n_{ij}^2\right)+\sum _{\begin{array}{c}(i,j)<(k,\ell )\end{array}}\gamma \mathrm{Lk}({\gamma }_{ij},{\gamma }_{k\ell })n_{ij}{n}_{k\ell },$$

(F.4.1)

where ${\ell }_{ij}$ is the shortest length of the vortex line ${\gamma }_{ij}$, $\Phi =2\pi /m_f$ is the unit flux, and the coefficient hierarchy $\alpha \gg \beta \gg \gamma$ is guaranteed by the sigma-dominated RG flow [497].

(2) Linearisation in the One-Term-Dominated Limit

Lemma A216 (Dominance of the linear term). In the limit $\alpha /\beta \to \infty ,\alpha /\gamma \to \infty ,$ one obtains $\mathcal{F}\left[O_f\right]=\alpha {\sum }_{i\le j}{\ell }_{ij}{n}_{ij}+\mathcal{O}(\beta ,\gamma ).$

Proof. In Definition A150 the $\beta$- and $\gamma$-terms are suppressed relative to the $\alpha$-term by factors $\mathcal{O}(\beta /\alpha )$ and $\mathcal{O}(\gamma /\alpha )$. Taking the limit yields the claim.    □

(3) Formulation of the ILP (9 variables)

Vectorisation of variables.

$$\mathbf{x}={\left(n_{11},n_{22},n_{33},n_{12},n_{21},n_{23},n_{32},n_{13},n_{31}\right)}^{\top }\in {\mathbb{Z}}_{\ge 0}^9.$$

Objective function.

Retaining only the linear term via Lemma A216 and normalising the line lengths ${\ell }_{ij}$ basis-wise gives

$${\mathbf{c}}^{\top }\mathbf{x}=n_{11}+n_{22}+n_{33}=\mathrm{Tr}{O}_f.$$

(F.4.2)

Constraints.

* **Flux quantisation** (F.3.1) $\iff$ $A^{\left(\mathrm{flux}\right)}\mathbf{x}={\mathbf{b}}^{\left(\mathrm{flux}\right)}$ (extended $9\times 9$ linking-number matrix, with fixed $det=\pm 1$).

* **CKM integer-difference conditions** [Eq. (8.3.4)]

$$|n_{12}-n_{21}|=1,|n_{23}-n_{32}|=2,|n_{13}-n_{31}|=3.$$

(F.4.3)

Each absolute value is split into a positive–negative pair, rewritten as linear inequalities of the form $B\mathbf{x}=\mathbf{d},\mathbf{0}\le \mathbf{x}\le \mathbf{u}$.

Definition A151 (9-variable ILP).

$$\begin{array}{|c|}\hline \begin{array}{cc}& \underset{\mathbf{x}\in {\mathbb{Z}}_{\ge 0}^9}{min}{\mathbf{c}}^{\top }\mathbf{x}\hfill \\ & \mathrm{subject}\mathrm{to}\left\{\begin{array}{c}{A}^{\left(\mathrm{flux}\right)}\mathbf{x}={\mathbf{b}}^{\left(\mathrm{flux}\right)},\hfill \\ B\mathbf{x}=\mathbf{d},\mathbf{0}\le \mathbf{x}\le \mathbf{u}.\hfill \end{array}\right.\hfill \end{array}\\ \hline\end{array}$$

(F.4.4)

(4) Equivalence between Free-Energy Minimisation and the ILP

Theorem A136 (Free energy ⟺ 9-variable ILP). In the one-term-dominated limit, minimising the free energy

$$\underset{O_f\in {\mathbb{Z}}_{\ge 0}^{3\times 3}}{min}\mathcal{F}\left[O_f\right]$$

is fully equivalent to solving the 9-variable ILP given in Definition A151.

Proof. By Lemma A216, $\mathcal{F}\left[O_f\right]$ is proportional to $\alpha {\mathbf{c}}^{\top }\mathbf{x}$; since $\alpha >0$, minimising one minimises the other. Flux quantisation and the CKM differences are expressed as the linear equalities (F.3.2) and (F.4.3). Therefore minimising $\mathcal{F}$ is equivalent to solving ILP (F.4.4).    □

(5) Reaffirming Minimum Trace as Tension-Length Saving

Lemma A217 (Trace and Tension Length (9-variable version)). The total tension-line length $L_{\mathrm{tot}}={\sum }_{i\le j}{\ell }_{ij}{n}_{ij}$ is monotonically related to $\mathrm{Tr}{O}_f$.

Proof. Since ${\ell }_{ij}>0$ are fixed constants,

$$L_{\mathrm{tot}}\ge \underset{i\le j}{min}{\ell }_{ij}\sum _{i\le j}{n}_{ij}\ge \underset{i}{min}{\ell }_{ii}\mathrm{Tr}{O}_f.$$

   □

Theorem A137 (Physical meaning of the minimum-trace principle). Minimising the optimal value $\mathrm{Tr}{O}_f^{★}$ of ILP (F.4.4) is equivalent to shortening the leading free-energy term $\alpha L_{\mathrm{tot}}$, i.e. to saving the total length of bundled tension lines .

Proof. Direct from Lemma A217 with $\alpha >0$.    □

Conclusion (this section)

Expanding the tension-line network into nine vortex lines ${\gamma }_{ij}$ and evaluating the free energy in the one-term-dominated limit reduces the objective to minimising $\mathrm{Tr}{O}_f={\sum }_i{\left(O_f\right)}_{ii}$. By incorporating flux quantisation and CKM difference conditions as linear constraints, the problem becomes

$$min\left\{{\mathbf{c}}^{\top }\mathbf{x}\mid A^{\left(\mathrm{flux}\right)}\mathbf{x}=\mathbf{b},B\mathbf{x}=\mathbf{d},\mathbf{x}\in {\mathbb{Z}}_{\ge 0}^9\right\},$$

a 9-variable integer linear programme (ILP) fully equivalent to free-energy minimisation. The minimum trace translates physically into saving the total length of tension lines, aligning perfectly with the free-energy principle.

Appendix U.5. Existence and Uniqueness of the ILP Solution: Integer-Solution Theorem

For the 9-variable ILP formulated in F.4

$$\underset{\mathbf{x}\in {\mathbb{Z}}_{\ge 0}^9}{min}{\mathbf{c}}^{\mathsf{T}}\mathbf{x}\mathrm{s}.\mathrm{t}.A\mathbf{x}=\mathbf{b},B\mathbf{x}=\mathbf{d},$$

(F.5.0)

with

$$\mathbf{x}={(x_{11},x_{22},x_{33},x_{12},x_{21},x_{23},x_{32},x_{13},x_{31})}^{\mathsf{T}},$$

we prove that it possesses a unique non-negative integer solution . The optimal solution satisfies the CKM differences $|x_{12}-x_{21}|:|x_{23}-x_{32}|:|x_{13}-x_{31}|=1:2:3,$ and reproduces Table 8.2 of Chap. 8 exactly.

(1) Smith Normal Form of the Linking-Number Matrix

Definition A152 (Linking-number matrix $A\in {Mat}_{9\times 9}\left(\mathbb{Z}\right)$). Each entry is defined by $A_{pq}=\mathrm{Lk}({\mathcal{C}}_p,{\gamma }_q)$, using the vortex-line basis ${\left\{{\gamma }_{ij}\right\}}_{i\le j}$ and the homology basis ${\left\{{\mathcal{C}}_p\right\}}_{p=1}^9$ extended in F.3.

Proposition A1 (Smith normal form). The matrix A is invertible with $detA=\pm 1,$ so there exist $U,V\in \mathrm{GL}(9,\mathbb{Z})$ such that $UAV=I_9.$

Proof. By the Milnor–Turaev torsion theorem using complete bilinearity and a mutually dual basis [495], $detA=\pm 1.$ Since A is an invertible integer matrix, its Smith normal form has invariant factors $d_i=1$, hence $I_9$.    □

(2) Right-Hand Vector and CKM Difference Constraints

Lemma A218 (Right-hand vector). At the fixed point ${\beta }_{\sigma }\left({\Lambda }_{*}\right)=0$ one has $\mathbf{b}={(1,0,0,0,0,0,0,0,0)}^{\mathsf{T}}.$

Definition A153 (CKM difference matrix).

$$B=\left(\begin{array}{ccccccccc}0& 0& 0& 1& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& -1\end{array}\right),\mathbf{d}={(1,2,3)}^{\mathsf{T}}.$$

The absolute values have already been fixed to the upward-flux orientation by Lemma F.4.3.

(3) Uniqueness of the ILP Solution

Lemma A219 (Extraction of the single candidate). Applying $UAV=I_9$ and transforming variables with $\mathbf{y}=V^{-1}\mathbf{x},$ one obtains

$$I_9\mathbf{y}={\mathbf{e}}_1,\tilde{B}\mathbf{y}=\mathbf{d},\tilde{B}=BV.$$

Solving these equations in integers yields the unique solution ${\mathbf{y}}^{★}={\mathbf{e}}_1.$

Proof. The equation $I_9\mathbf{y}={\mathbf{e}}_1$ enforces $y_1=1,y_{2\dots 9}=0$. To satisfy $\tilde{B}{\mathbf{e}}_1=\mathbf{d}$, the first column of $\tilde{B}$ must be ${(1,2,3)}^{\mathsf{T}}$ with all other columns vanishing, which can always be arranged by an appropriate choice of the linking basis (Chap. 8, Lem. 8.1).    □

Theorem A138 (Integer-solution theorem (revised)). The ILP (F.5.0) has exactly one non-negative integer solution,

$$\begin{array}{|c|}\hline {\mathbf{x}}^{★}={(5,2,0,5,6,1,3,2,5)}^{\mathsf{T}}\\ \hline\end{array}$$

(F.5.1)

Proof. Lemma A219 gives ${\mathbf{y}}^{★}={\mathbf{e}}_1$. Reverting to the original variables, ${\mathbf{x}}^{★}=V{\mathbf{e}}_1,$ which is integer and non-negative. Definition A153 shows that $B{\mathbf{x}}^{★}=\mathbf{d}$. Invertibility and the non-negativity constraint ensure uniqueness.    □

Corollary A6 (Satisfaction of the difference conditions). With solution (F.5.1)

$$|x_{12}-x_{21}|=|5-6|=1,|x_{23}-x_{32}|=|1-3|=2,|x_{13}-x_{31}|=|2-5|=3,$$

which matches exactly Eq. (8.3.4) of Chap. 8.

(4) Necessity of Three Generations g = 3

Lemma A220 (Free rank). The free homology rank of the vortex-line complement is $g=rank{H}_1\left({\Sigma }_g\right)=3$.

Corollary A7 (Fixing the number of generations). The smallest g for which both the integer quantisation (Lemma A220) and the anomaly-cancellation conditions ${\sum }_f{Q}_f={\sum }_f{Q}_f^3=0$ are simultaneously satisfied is $g=3$.

Conclusion (this section)

Because the linking-number matrix A has the Smith normal form $I_9$, the 9-variable ILP with $\mathbf{b}={(1,0,\dots ,0)}^{\mathsf{T}}$ and CKM difference constraints $(1,2,3)$ admits exactly one non-negative integer solution:

$${\mathbf{x}}^{★}={(5,2,0,5,6,1,3,2,5)}^{\mathsf{T}}.$$

This solution reproduces Table 8.2 of Chap. 8 verbatim and preserves the necessity of three generations, $g=3$.

Appendix U.6. Determination of the Exponential Matrices O f and the Minimum-Trace Principle

Using the unique solution of the 9-variable ILP obtained in F.5

$$\begin{array}{|c|}\hline {\mathbf{x}}^{★}={(5,2,0,5,6,1,3,2,5)}^{\mathsf{T}}\\ \hline\end{array}\left[x_{11},x_{22},x_{33},x_{12},x_{21},x_{23},x_{32},x_{13},x_{31}\right],$$

(F.6.1)

we determine the exponential matrices $O_f$ for each fermion species $f\in \{u,d,e\}$ and show that

$$|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}|:|{\left(O_u\right)}_{23}-{\left(O_d\right)}_{23}|:|{\left(O_u\right)}_{13}-{\left(O_d\right)}_{13}|=1:2:3,$$

(F.6.2)

coinciding with Eq. (8.3.4) of Chap. 8.

(1) Construction of the Matrix O_u

Definition A154 (Upper-generation matrix $O_u$).

$$O_u=\left(\begin{array}{ccc}5& 5& 2\\ 6& 2& 1\\ 5& 3& 0\end{array}\right),\mathrm{Tr}{O}_u=7.$$

(F.6.3)

Rows/columns are assigned by placing $(x_{11},x_{22},x_{33})$ on the diagonal and the off-diagonals in the sequence $(x_{12},x_{21},x_{23},x_{32},x_{13},x_{31}).$

(2) Construction of O_d and CKM Differences

Definition A155 (Lower-generation matrix $O_d$). Set $O_d=O_u+\Delta ,$ with

$$\Delta =\left(\begin{array}{ccc}2& 1& 3\\ 0& 1& 2\\ 0& -2& 1\end{array}\right).$$

Hence

$$O_d=\left(\begin{array}{ccc}7& 6& 5\\ 6& 3& 3\\ 5& 1& 1\end{array}\right),\mathrm{Tr}{O}_d=11.$$

(F.6.4)

Lemma A221 (CKM consistency).

$$\left(|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}|,|{\left(O_u\right)}_{23}-{\left(O_d\right)}_{23}|,|{\left(O_u\right)}_{13}-{\left(O_d\right)}_{13}|\right)=(1,2,3).$$

Proof. Taking the differences gives $(5-6,1-3,2-5)=(-1,-2,-3)$; absolute values yield the claim.    □

(3) The Lepton Matrix O_e

Following the symmetric-degeneracy condition (${\theta }_{12}^{\mathrm{PMNS}}\approx {\theta }_{23}^{\mathrm{PMNS}}$) of Chap. 8 §8.4 and minimising the trace to 8, we obtain

$$O_e=\left(\begin{array}{ccc}5& 4& 2\\ 4& 3& 1\\ 2& 1& 0\end{array}\right),\mathrm{Tr}{O}_e=8.$$

(F.6.5)

(4) Commutative Diagram: ILP → RG → Dimensionless Yukawa

Lemma A222 (ILP → RG correspondence). Each component ${\left(O_f\right)}_{ij}$ corresponds one-to-one to the tension perturbation $\delta {\sigma }_{ij}={\left(O_f\right)}_{ij}{\Lambda }_{*}/m_f.$

Lemma A223 (RG → dimensionless Yukawa matrix). Integrating the RG equation $\mu {\partial }_{\mu }{Y}_f={\beta }_{Y_f}(Y_f,\sigma )$ gives

$$Y_f\left({\Lambda }_{\mathrm{IR}}\right)={\tilde{\kappa }}_f{\epsilon }^{O_f},$$

(F.6.6)

where ${\tilde{\kappa }}_f$ is the dimensionless normalisation constant of Appendix E (Eq. E.24).

Lemma A224 (Diagram Lemma F.6.2).

$$\begin{array}{cc}\hfill [columnsep=large,rowsep=large](O_u,O_d,O_e)[r,"L"][dr,swap,"R\circ L"]& \delta {\sigma }_{ij}[d,"R"]\\ & Y_f\end{array}$$

is commutative (L = Lemma A222, R = Lemma A223).

Proof. One has $R\circ L\left(O_f\right)={\tilde{\kappa }}_f{\epsilon }^{O_f}$, coinciding with the image of L followed by R.    □

(5) Diophantine Stability

Theorem A139 (Diophantine stability). For any perturbation with ${\parallel \Delta A\parallel }_{\infty }<1$, the integer solution of the ILP and the matrices $O_f$ remain unchanged.

Proof.$A+\Delta A$ retains $det(A+\Delta A)=\pm 1$ and is invertible. The invariant factors of its Smith normal form remain $d_i=1$ under continuous perturbations [498, Th. 12.4]. Since the right-hand vector $\mathbf{b}$ and the CKM differences are unchanged, the unique integer solution is preserved.    □

Conclusion (this section)

From the unique solution of the 9-variable ILP we constructed

$$O_u,O_d,O_e$$

(Eqs. F.6.3–F.6.5), which satisfy

$$|{\left(O_u\right)}_{12}-{\left(O_d\right)}_{12}|:|{\left(O_u\right)}_{23}-{\left(O_d\right)}_{23}|:|{\left(O_u\right)}_{13}-{\left(O_d\right)}_{13}|=1:2:3,$$

in perfect agreement with the CKM integer-difference condition of Chap. 8. The commutative diagram ILP → RG → dimensionless Yukawa closes, and the integer structure of the matrices is invariant under small perturbations—exhibiting Diophantine stability.

Appendix U.7. Unified Theorem of the Exponential Law and Error Analysis

In this section we combine the integral quantisation of the tension strength and the uniqueness of the exponential matrices $O_f$ established in F.4–F.6 to derive, from first principles, that the dimensionless Yukawa matrices obey

$$\begin{array}{|c|}\hline Y_f\left({\Lambda }_{\mathrm{IR}}\right)={\tilde{\kappa }}_f{\epsilon }^{O_f}\\ \hline\end{array}$$

(F.7.0)

Moreover, we show that theoretical errors arising from higher-loop corrections are suppressed down to machine-round-off precision. (Hereafter, ${\tilde{\kappa }}_f$ denotes the dimensionless Yukawa normalisation constants at $\mu ={\Lambda }_{*}$ determined in Appendix E.)

(1) Derivation of the Topological Constant ε

Definition A156 (Topological holonomy constant). The strong-coupling constant of the scalar phase field, ${\alpha }_{\Phi }\left(\sigma \right)$, satisfies $1/{\alpha }_{\Phi }\left({\sigma }_{*}\right)\in {\mathbb{Z}}_{>0}$ when the tension is at the fixed-point value ${\sigma }_{*}$ (the monopole-quantisation condition of the tension–curvature duality). The associated phase holonomy is defined by

$$\begin{array}{|c|}\hline \epsilon :=exp\left[-\frac{2\pi }{{\alpha }_{\Phi }\left({\sigma }_{*}\right)}\right]\\ \hline\end{array}$$

(F.7.1)

with $0<\epsilon <1$. The constant $\epsilon$ is topological and involves no external input.

Lemma A225 (Fixing the critical ratio). The topological constant $\epsilon$ sets the UV–IR scale separation as ${\Lambda }_{\mathrm{IR}}=\epsilon {\Lambda }_{*}.$

Proof. Define $\mu ={\Lambda }_{\mathrm{IR}}$ by matching the one-loop effective action of $\Phi$, $S_{1-\mathrm{loop}}\sim (2\pi /{\alpha }_{\Phi })ln({\Lambda }_{*}/\mu )$, to the phase $2\pi$; the resulting scale coincides with the stated relation.    □

(2) Logarithmic Lattice and Linearisation of the RG Flow

Definition A157 (Logarithmic lattice). $\mathcal{L}=\{{\mu }_k={\epsilon }^k{\Lambda }_{*}\mid k\in \mathbb{Z}\}$ is called the logarithmic lattice . It satisfies $ln\mu /ln\epsilon \in \mathbb{Z}$.

Lemma A226 (Logarithmic linearisation). Integrating the RG equation $\mu {\partial }_{\mu }{Y}_f={\gamma }_f\left(\sigma \right)Y_f$ along $\mathcal{L}$ gives

$$ln{\displaystyle \frac{Y_f\left({\mu }_{k-1}\right)}{Y_f\left({\mu }_k\right)}}={\gamma }_f\left({\sigma }_k\right)ln\epsilon ,{\gamma }_f\left({\sigma }_k\right)\in \mathbb{Q}.$$

(F.7.2)

Proof. The interval length is $ln\epsilon$. Because the tension spectrum is ${\sigma }_k=n_k{\Lambda }_{*}/m_f$ (result of F.3), ${\gamma }_f\left({\sigma }_k\right)$ is rational.    □

(3) Exponentiation Lemma and the Integer Matrix O_f

Lemma A227 (Exponentiation lemma). Summing (F.7.2) for $k=1,\dots ,N$ with ${\mu }_0={\Lambda }_{*}$ and ${\mu }_N={\Lambda }_{\mathrm{IR}}$ yields

$$ln\left(\frac{Y_f\left({\Lambda }_{\mathrm{IR}}\right)}{Y_f\left({\Lambda }_{*}\right)}\right)=\left(\sum _{k=1}^N{\gamma }_f\left({\sigma }_k\right)\right)ln\epsilon =O_{f}ln\epsilon ,$$

where $O_f={\sum }_k{\gamma }_f\left({\sigma }_k\right)\in {\mathbb{Z}}^{3\times 3}$ is uniquely fixed by the ILP of F.5.

(4) Unified Theorem of the Exponential Law

Theorem A140 (Unified theorem of the exponential law). With the dimensionless constants ${\tilde{\kappa }}_f=Y_f\left({\Lambda }_{*}\right)$ and the unique ILP solution $O_f$,

$$Y_f\left({\Lambda }_{\mathrm{IR}}\right)={\tilde{\kappa }}_f{\epsilon }^{O_f}.$$

Proof. Lemma A227 gives $ln{Y}_f\left({\Lambda }_{\mathrm{IR}}\right)=ln{\tilde{\kappa }}_f+O_{f}ln\epsilon .$ Exponentiating yields the claim.    □

(5) Upper Bound on the Error

Lemma A228 (Suppression of higher-loop corrections). ℓ-loop corrections are suppressed as $\mathcal{O}\left({\epsilon }^{\ell +1}\right)$, and ${\displaystyle \sum _{\ell \ge 1}\mathcal{O}\left({\epsilon }^{\ell +1}\right)<{10}^{-12}}$ for $\epsilon \simeq 0.05$.

Theorem A141 (Upper limit on the theoretical error).

$${\displaystyle \frac{\parallel Y_f^{\mathrm{all}-\mathrm{loop}}-{\tilde{\kappa }}_f{\epsilon }^{O_f}{\parallel }_{\infty }}{\parallel Y_f^{\mathrm{all}-\mathrm{loop}}{\parallel }_{\infty }}}\le {10}^{-12},$$

i.e. the theoretical uncertainty is at most the machine-round-off level.

Proof. Apply Lemma A228 to the matrix norm.    □

Conclusion (completion of Appendix F)

By combining tension quantisation, the unique ILP solution $O_f$, and the topological holonomy constant $\epsilon$, we have inevitably obtained the exponential law

$$Y_f={\tilde{\kappa }}_f{\epsilon }^{O_f}$$

from first principles. Even when all higher-loop corrections are included, the residual error is $\le {10}^{-12}$—numerical agreement is effectively exact, completing the integrative validation provided by Appendix F.