Summary of Results and Structure of the QICT Closure Program

1. Summary of Results and Structure of the Paper

This work assembles and systematizes the mathematical and physical foundations of the Quantum Information Copy Time (QICT) framework into a single, logically coherent corpus. Under explicitly stated hypotheses on the microscopic lattice model, on the functional renormalization group (FRG) flow, and on the scalar–neutrino sector, we prove a sequence of theorems that together provide a conditionally closed and falsifiable theory. The article is organized into six tightly connected parts.

2. Microscopic QICT Model and Constructive Algebraic Closure

In this section we specify an explicit microscopic Hamiltonian in the Quantum Information Copy Time (QICT) class on a three-dimensional cubic lattice and prove that it satisfies the structural hypotheses (H1)–(H4) required for the algebraic formulation of the theory. We then construct the quasi-local C^*-algebra, the ground state GNS representation, and the associated local von Neumann algebras. Finally, using standard criteria from algebraic quantum field theory (AQFT)—in particular the split property, modular nuclearity, and Connes’ classification—we establish the Theorem of Constructive Algebraic Closure (TCAC): local algebras in the QICT continuum limit are hyperfinite factors of Type III₁.

Throughout this section we work in units where $\hslash =c=1$ whenever convenient and keep track of lattice spacing $a>0$ only when necessary. All limits $L\to \infty$ are taken along cubic boxes ${\Lambda }_L$ with periodic boundary conditions.

2.1. Lattice Geometry and Local Hilbert Spaces

Let $\Lambda ={(\mathbb{Z}/L\mathbb{Z})}^3$ be a three-dimensional cubic lattice of linear size $L\in 2\mathbb{N}$ with periodic boundary conditions and lattice spacing $a>0$. The set of vertices (sites) is denoted by $V(\Lambda )$, the set of edges by $E(\Lambda )$, and the set of elementary plaquettes (faces) by $P(\Lambda )$.

We place a two-dimensional Hilbert space (qubit) on each edge $e\in E(\Lambda )$:

$${\mathcal{H}}_e\simeq {\mathbb{C}}^2,$$

(1)

with Pauli matrices ${\sigma }_e^x,{\sigma }_e^y,{\sigma }_e^z$ acting on ${\mathcal{H}}_e$. For a finite region $\Gamma \subset E(\Lambda )$ we define

$${\mathcal{H}}_{\Gamma }=\underset{e\in \Gamma }{⨂}{\mathcal{H}}_e,{\mathcal{A}}_{\Gamma }=\mathcal{B}\left({\mathcal{H}}_{\Gamma }\right).$$

(2)

If ${\Gamma }_1\subset {\Gamma }_2$, we regard ${\mathcal{A}}_{{\Gamma }_1}$ as a subalgebra of ${\mathcal{A}}_{{\Gamma }_2}$ by tensoring with the identity on ${\Gamma }_2\setminus {\Gamma }_1$.

The local algebra associated with a finite set $\Gamma \subset E(\Lambda )$ is ${\mathcal{A}}_{\Gamma }$. The quasi-local C^*-algebra is defined as the norm-completion of the inductive limit

$$\mathfrak{A}={\overline{\bigcup _{\Gamma {\subset }_{\mathrm{fin}}E(\Lambda )}{\mathcal{A}}_{\Gamma }}}^{\parallel ·\parallel }.$$

(3)

2.2. QICT Hamiltonian: 3D Toric Code with Copy Twists

We now define the microscopic Hamiltonian that realizes the QICT dynamics. It is a frustration-free stabilizer Hamiltonian built from the three-dimensional toric code, augmented by local “copy twist” operators implementing QICT copy automorphisms.

2.2.1. Standard 3D Toric Code Terms

For each vertex $v\in V(\Lambda )$, let $\mathrm{Star}\left(v\right)$ denote the set of six edges incident at v. For each plaquette $p\in P(\Lambda )$, let $\partial p\subset E(\Lambda )$ denote the set of its four boundary edges.

We define the vertex (“electric”) and plaquette (“magnetic”) stabilizers

$$\begin{array}{cc}\hfill A_v& =\prod _{e\in \mathrm{Star}\left(v\right)}{\sigma }_e^x,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill B_p& =\prod _{e\in \partial p}{\sigma }_e^z.\hfill \end{array}$$

(5)

These operators are Hermitian, commute with each other, and square to the identity:

$$A_v^2=B_p^2=⊮,[A_v,B_p]=0\forall v,p.$$

(6)

The standard 3D toric code Hamiltonian is

$$H_{\mathrm{TC}}=-J\sum _{v\in V(\Lambda )}{A}_v-J\sum _{p\in P(\Lambda )}{B}_p,J>0.$$

(7)

Its ground-state subspace

$${\mathcal{C}}_L=\left\{|\psi \rangle \in {\mathcal{H}}_{E(\Lambda )}|A_v|\psi \rangle =B_p|\psi \rangle =|\psi \rangle \forall v,p\right\}$$

(8)

is exactly solvable and exhibits topological order [6? ].

2.2.2. Copy Paths and Twist Operators

To model QICT copy automorphisms, we introduce a finite collection of copy regions ${\left\{X_{\alpha }\right\}}_{\alpha }$, where each $X_{\alpha }\subset V(\Lambda )$ is a finite set of vertices such that $|X_{\alpha }|<\infty$ and the sets are mutually disjoint. We consider ordered pairs $(X_{\alpha },Y_{\alpha })$ with $|X_{\alpha }|=|Y_{\alpha }|$ and define copy pathsc connecting $X_{\alpha }$ to $Y_{\alpha }$ along the lattice. A copy path $c:X_{\alpha }\to Y_{\alpha }$ is a set of oriented edges $c\subset E(\Lambda )$ such that:

• each $x\in X_{\alpha }$ is connected to a unique $y\in Y_{\alpha }$ by a path of edges in c;
• the total length satisfies $\ell \left(c\right)\le L_{max}$ for some $L_{max}$ independent of L (uniform locality);
• any two distinct paths share at most finitely many edges.

For each copy path c, we define a twisted string operator

$$T_c=\prod _{e\in c}{\sigma }_e^{y}exp\left(i{\theta }_c\sum _{x\in X_{\alpha }}{\sigma }_{e\left(x\right)}^z\right),$$

(9)

where $e\left(x\right)$ is a distinguished edge adjacent to x, and ${\theta }_c=\pi /\left|X_{\alpha }\right|$. The exponential implements a protected ${\mathbb{Z}}_2$ flux twist associated with the copy operation.

The full QICT Hamiltonian is defined as

$$H=H_{\mathrm{TC}}-K\sum _{c\in \mathcal{C}}{T}_c,0<K\le K_{max},$$

(10)

where $\mathcal{C}$ is a fixed finite set of copy paths whose number and lengths are bounded independently of L, and $K_{max}$ is chosen sufficiently small compared to J (see below).

The model (10) is still a commuting-projector Hamiltonian up to local unitary conjugation: the $T_c$ commute with all $A_v$ and $B_p$ except on a finite number of edges near the endpoints of c, and their inclusion preserves frustration-freeness in the ground sector for sufficiently small K.

2.3. Structural Hypotheses (H1)–(H4)

We now formulate the structural hypotheses that the microscopic QICT Hamiltonian must satisfy in order to support the algebraic continuum limit and the TCAC theorem.

(H1)

Finite-range locality: There exists $R>0$ such that the interaction potential $\Phi \left(X\right)$ in the formal expansion $H={\sum }_{X\subset E(\Lambda )}\Phi \left(X\right)$ vanishes whenever $\mathrm{diam}\left(X\right)>R$.

(H2)

Uniform spectral gap: There exists $\Delta >0$ independent of L such that the spectral gap ${\Delta }_L$ of H above its ground-state energy $E_0\left(L\right)$ satisfies ${\Delta }_L\ge \Delta$ for all L.

(H3)

Lieb–Robinson bound: There exist constants $C>0$, $\mu >0$, $v_{\mathrm{LR}}>0$ such that for all local observables $A,B\in \mathfrak{A}$ supported in disjoint regions $X,Y\subset E(\Lambda )$ and all $t\in \mathbb{R}$,

$$∥[{\alpha }_t\left(A\right),B]∥\le C\parallel A\parallel \parallel B\parallel exp\left(-\mu (d(X,Y)-v_{\mathrm{LR}}|t\left|\right)\right),$$

(11)

where ${\alpha }_t(·)=e^{iHt}(·)e^{-iHt}$ denotes the Heisenberg dynamics and $d(X,Y)$ is the lattice distance between X and Y.

(H4)

Area law with topological correction: For any simply-connected region $\Gamma \subset E(\Lambda )$, the ground state entanglement entropy $S(\Gamma )$ satisfies

$$S(\Gamma )=\alpha |\partial \Gamma |-\gamma +o\left(\right|\partial \Gamma \left|\right),$$

(12)

where $|\partial \Gamma |$ is the number of boundary plaquettes, $\alpha >0$ is a non-universal constant, and $\gamma >0$ is a topological entanglement entropy independent of $\Gamma$.

We now verify that the Hamiltonian (10) satisfies these hypotheses for suitable choices of J and K.

Lemma 1

(Locality). The QICT Hamiltonian H in Eq. (10) satisfies (H1) with interaction range $R=2a$.

Proof. 

The vertex and plaquette terms $A_v$ and $B_p$ are supported on at most six and four edges, respectively, contained in a ball of radius a. Each twist operator $T_c$ is supported on a path of length $\ell \left(c\right)\le L_{max}$, with $L_{max}$ fixed by construction and independent of L. We can decompose H as

$$H=\sum _X\Phi \left(X\right),$$

(13)

where $\Phi \left(X\right)$ is a sum of stabilizer terms with support exactly X. By construction $\Phi \left(X\right)=0$ whenever $\mathrm{diam}\left(X\right)>R$, with $R=max\{a,L_{max}a\}$. Choosing $L_{max}=2$ (shortest paths), we can take $R=2a$. This yields (H1). □

Lemma 2

(Uniform spectral gap). There exists $K_0>0$ and $\Delta >0$ independent of L such that, for all $0\le K\le K_0$, the QICT Hamiltonian H has a spectral gap ${\Delta }_L\ge \Delta$ above its ground-state energy $E_0\left(L\right)$ for all L.

Proof. 

For $K=0$, the Hamiltonian reduces to the 3D toric code $H_{\mathrm{TC}}$, which is an exactly solvable commuting-projector model. The spectrum can be determined explicitly: any violation of a vertex or plaquette stabilizer costs an energy of order $2J$, and the elementary excitations (electric charges and magnetic fluxes) are gapped [6? ]. Hence $H_{\mathrm{TC}}$ has a uniform spectral gap ${\Delta }_{\mathrm{TC}}=2J$ above its ground-state sector, independent of L.

For nonzero K, we treat the twist term $V=-K{\sum }_c{T}_c$ as a perturbation. Each $T_c$ is supported on a finite set of edges with bounded norm $\parallel T_c\parallel =1$, and the number of twist terms is finite and independent of L. We thus have $\parallel V\parallel \le K{N}_{\mathrm{twist}}$, with $N_{\mathrm{twist}}$ independent of L.

By standard stability results for frustration-free Hamiltonians with topological quantum order [?? ], there exists $ϵ>0$, depending only on ${\Delta }_{\mathrm{TC}}$ and locality, such that any perturbation V with $\parallel V\parallel \le ϵJ$ preserves a nonzero spectral gap $\Delta \ge {\Delta }_{\mathrm{TC}}/2$ independent of L. Taking $K_0$ sufficiently small so that $K_0{N}_{\mathrm{twist}}\le ϵJ$ yields the desired result. □

Lemma 3

(Lieb–Robinson bound). The QICT dynamics ${\alpha }_t(·)=e^{iHt}(·)e^{-iHt}$ satisfies the Lieb–Robinson bound (11) with constants $C,\mu ,v_{\mathrm{LR}}$ independent of L.

Proof. 

The Hamiltonian H is a finite-range sum of bounded local operators on a lattice with finite coordination number. By the general Lieb–Robinson theory for quantum spin systems [??? ], any such Hamiltonian admits a bound of the form (11) with explicit constants depending only on the interaction strength and range. The additional twist terms $T_c$ do not alter the finite-range nature of the interaction, and their norm is uniformly bounded; they therefore only change the numerical values of $C,\mu ,v_{\mathrm{LR}}$, not the existence of the bound. □

Lemma 4

(Area law with topological correction). Let $|{\Omega }_L\rangle$ denote a ground state of H in ${\mathcal{H}}_{E(\Lambda )}$. For any simply-connected region $\Gamma \subset E(\Lambda )$ with smooth boundary, the reduced density matrix ${\rho }_{\Gamma }={\mathrm{Tr}}_{E(\Lambda )\setminus \Gamma }\left(\right|{\Omega }_L\rangle \langle {\Omega }_L\left|\right)$ satisfies the area law (111) with constants $\alpha >0$ and $\gamma >0$ independent of L.

Proof. 

For $K=0$ (pure toric code), the entanglement entropy of ground states is known to satisfy an exact area law with a universal constant correction, the topological entanglement entropy [8?? ]. In particular, for large regions $\Gamma$ one finds

$$S(\Gamma )={\alpha }_0|\partial \Gamma |-{\gamma }_0+o\left(1\right),$$

(14)

where ${\gamma }_0$ depends only on the type of topological order and is independent of $\Gamma$ and L.

For small but finite K, the Hamiltonian remains gapped and frustration-free on a code subspace that is unitarily equivalent (via a quasi-local unitary) to the toric code ground space [? ]. Entanglement entropies are stable under such quasi-local unitaries up to additive boundary corrections. Thus, the area law persists with constants $\alpha$ and $\gamma$ that differ from ${\alpha }_0$ and ${\gamma }_0$ by $O(K/J)$, and with boundary corrections bounded uniformly in L. This yields (111). □

Combining Lemmas 1–4, we conclude that the microscopic QICT Hamiltonian (10) satisfies (H1)–(H4) for $0\le K\le K_0$ with suitably chosen constants $(R,\Delta ,C,\mu ,v_{\mathrm{LR}},\alpha ,\gamma )$ independent of L.

2.4. Quasi-Local Algebra, Dynamics, and Thermodynamic Limit

We now pass to the thermodynamic limit and construct the quasi-local C^*-dynamical system $(\mathfrak{A},{\alpha }_t)$ associated with the QICT model.

Let ${\left\{{\Lambda }_n\right\}}_{n\in \mathbb{N}}$ be an increasing sequence of finite cubic volumes with periodic boundary conditions, exhausting ${\mathbb{Z}}^3$. For each ${\Lambda }_n$ we consider the Hamiltonian $H_{{\Lambda }_n}$ obtained by restricting (10) to edges in $E\left({\Lambda }_n\right)$, keeping the same interaction pattern at the boundary (standard thermodynamic limit prescription). The corresponding finite-volume dynamics is

$${\alpha }_t^{\left(n\right)}\left(A\right)=e^{i{H}_{{\Lambda }_n}t}A{e}^{-i{H}_{{\Lambda }_n}t},A\in {\mathcal{A}}_{E\left({\Lambda }_n\right)}.$$

(15)

By (H1) and standard results on the thermodynamic limit of local dynamics [? ], the sequence ${\alpha }_t^{\left(n\right)}\left(A\right)$ converges in operator norm for each local observable $A\in \mathfrak{A}$ and $t\in \mathbb{R}$, and the limit defines a strongly continuous group of ∗-automorphisms ${\alpha }_t:\mathfrak{A}\to \mathfrak{A}$.

2.5. Ground State and GNS Representation

Let $|{\Omega }_{{\Lambda }_n}\rangle$ denote a normalized ground state of $H_{{\Lambda }_n}$. Because of the uniform spectral gap (H2) and quasi-locality, the sequence of vector states

$${\omega }_n\left(A\right)=\langle {\Omega }_{{\Lambda }_n}\left|A\right|{\Omega }_{{\Lambda }_n}\rangle ,A\in \mathfrak{A},$$

(16)

is Cauchy in the weak-^* topology on ${\mathfrak{A}}^{*}$ and converges to a limiting state ${\omega }_0$ on $\mathfrak{A}$ [? ]. The state ${\omega }_0$ is invariant under the dynamics:

$${\omega }_0\left({\alpha }_t\left(A\right)\right)={\omega }_0\left(A\right),\forall A\in \mathfrak{A},t\in \mathbb{R}.$$

(17)

We denote by $({\pi }_0,{\mathcal{H}}_0,|{\Omega }_0\rangle )$ the GNS triple associated with $(\mathfrak{A},{\omega }_0)$. Thus ${\pi }_0:\mathfrak{A}\to \mathcal{B}\left({\mathcal{H}}_0\right)$ is a faithful representation, $|{\Omega }_0\rangle$ is a cyclic vector, and

$${\omega }_0\left(A\right)=\langle {\Omega }_0|{\pi }_0\left(A\right)|{\Omega }_0\rangle .$$

(18)

The dynamics ${\alpha }_t$ is implemented in the GNS representation by a strongly continuous unitary group $U\left(t\right)$:

$${\pi }_0\left({\alpha }_t\left(A\right)\right)=U\left(t\right){\pi }_0\left(A\right)U{\left(t\right)}^{-1},U\left(t\right)|{\Omega }_0\rangle =|{\Omega }_0\rangle .$$

(19)

The generator K defined by $U\left(t\right)=e^{iKt}$ is the GNS Hamiltonian.

2.6. Local Von Neumann Algebras and Net Structure

For any bounded region $\mathcal{O}\subset {\mathbb{R}}^3$ we define the corresponding local C^*-algebra

$$\mathfrak{A}\left(\mathcal{O}\right)={\overline{\bigcup _{\Gamma \subset E(\Lambda ):a\Gamma \subset \mathcal{O}}{\mathcal{A}}_{\Gamma }}}^{\parallel ·\parallel },$$

(20)

where $a\Gamma$ denotes the continuum region obtained by thickening the edges in $\Gamma$ to cylinders of radius $O\left(a\right)$. The associated local von Neumann algebra in the GNS representation is

$$\mathcal{M}\left(\mathcal{O}\right)={\pi }_0{\left(\mathfrak{A}\left(\mathcal{O}\right)\right)}^{″},$$

(21)

where the double prime denotes the bicommutant. The map $\mathcal{O}\mapsto \mathcal{M}\left(\mathcal{O}\right)$ defines a net of local algebras on ${\mathbb{R}}^3$ satisfying isotony, locality, and covariance under translations induced by the lattice symmetries.

2.7. Split Property

We now show that the QICT local algebras satisfy the split property, which is a key input for the constructive passage from lattice models to continuum AQFT.

Theorem 1

(Split property for QICT). Let ${\mathcal{O}}_1,{\mathcal{O}}_2\subset {\mathbb{R}}^3$ be bounded open regions with $\overline{{\mathcal{O}}_1}\subset {\mathcal{O}}_2$ and $\mathrm{dist}(\partial {\mathcal{O}}_1,\partial {\mathcal{O}}_2)\ge \delta >0$. Then there exists a Type I factor $\mathcal{N}\subset \mathcal{B}\left({\mathcal{H}}_0\right)$ such that

$$\mathcal{M}\left({\mathcal{O}}_1\right)\subset \mathcal{N}\subset \mathcal{M}\left({\mathcal{O}}_2\right).$$

(22)

In particular, the inclusion $\mathcal{M}\left({\mathcal{O}}_1\right)\subset \mathcal{M}\left({\mathcal{O}}_2\right)$ is split.

Proof. 

We sketch the argument and refer to [??? ] for the general theory of the split property and to [9? ] for its adaptation to lattice systems with Lieb–Robinson bounds and area laws.

The split property can be derived from a nuclearity condition (or modular nuclearity) for maps of the form

$$T_{\beta }:A\mapsto e^{-\beta K}{\pi }_0\left(A\right)|{\Omega }_0\rangle ,A\in \mathcal{M}\left(\mathcal{O}\right),$$

(23)

for suitable $\beta >0$ and regions $\mathcal{O}$ comparable to ${\mathcal{O}}_1$ and ${\mathcal{O}}_2$. The map $T_{\beta }$ is nuclear if it can be approximated by finite-rank operators with summable singular values. Buchholz and Wichmann showed that nuclearity for an inclusion ${\mathcal{O}}_1\subset \subset {\mathcal{O}}_2$ implies the existence of a Type I factor $\mathcal{N}$ interpolating between $\mathcal{M}\left({\mathcal{O}}_1\right)$ and $\mathcal{M}\left({\mathcal{O}}_2\right)$ [? ].

In the QICT model, the combination of (H2) a uniform spectral gap, (H3) a Lieb–Robinson bound, and (H4) an area law implies that the density of low-energy excitations in a region $\mathcal{O}$ grows at most exponentially in the boundary area $|\partial \mathcal{O}|$. More precisely, the number of orthogonal states with energy below E in a region of volume V obeys

$$N(E,V)\le C_{1}exp(C_2|\partial \mathcal{O}|E/\Delta ),$$

(24)

for constants $C_1,C_2>0$. This estimate follows from standard quasi-particle counting arguments in gapped systems with topological order [? ].

Using this bound and the finite propagation speed from the Lieb–Robinson estimate (11), one shows that the map $T_{\beta }$ is nuclear for $\beta$ larger than a constant depending on $\Delta$ and $v_{\mathrm{LR}}$, see [9?? ]. In particular, for two regions ${\mathcal{O}}_1\subset \subset {\mathcal{O}}_2$ with finite separation, the nuclearity index of $T_{\beta }$ is finite and depends only on the boundary data and the separation. Applying the Buchholz–Wichmann criterion for the split property to the local QICT net then yields the existence of an interpolating Type I factor $\mathcal{N}$, as claimed. □

2.8. Theorem of Constructive Algebraic Closure (TCAC)

We now prove the main result of this section: under the additional assumption of ultraviolet scale invariance (as provided by the NGFP in the FRG analysis), the local algebras of the QICT continuum theory are hyperfinite factors of Type III₁.

Theorem 2

(TCAC: Constructive Algebraic Closure). Let H be the microscopic QICT Hamiltonian defined in Eq. (10) with $0\le K\le K_0$. Assume that the continuum limit of the QICT theory is scale invariant in the ultraviolet, as implied by the existence of a non-Gaussian fixed point for the FRG flow. Then, for each bounded open region $\mathcal{O}\subset {\mathbb{R}}^3$, the local von Neumann algebra $\mathcal{M}\left(\mathcal{O}\right)$ in the ground state GNS representation is ahyperfinite factor of Type III₁. In particular, the QICT continuum model realizes a standard AQFT net with Type III₁ local algebras.

Proof. 

We divide the proof into three steps: factoriality, hyperfiniteness, and classification of the type.

(i)

Factoriality.

The ground state ${\omega }_0$ describes a pure phase of the QICT model in the thermodynamic limit. By construction, different topological sectors corresponding to global toric code fluxes are projected out by selecting a translation-invariant ground state. This yields a primary representation in the sense of [? ], and each local algebra $\mathcal{M}\left(\mathcal{O}\right)$ has a trivial center:

$$\mathcal{Z}\left(\mathcal{M}\left(\mathcal{O}\right)\right)=\mathcal{M}\left(\mathcal{O}\right)\cap \mathcal{M}{\left(\mathcal{O}\right)}^{\prime }=\mathbb{C}=⊮$$

(25)

so $\mathcal{M}\left(\mathcal{O}\right)$ is a factor.

(ii)

Hyperfiniteness.

By definition, hyperfiniteness of a von Neumann algebra means that it can be approximated in the strong operator topology by an increasing sequence of finite-dimensional ∗-subalgebras. In the present setting, the local algebra $\mathfrak{A}\left(\mathcal{O}\right)$ is an inductive limit of finite tensor products of $2\times 2$ matrices, and its image under the GNS representation ${\pi }_0$ is dense in $\mathcal{M}\left(\mathcal{O}\right)$. Therefore, for any finite set of vectors $\left\{\right|{\psi }_k\rangle \}$ and any finite set of operators $\left\{A_j\right\}\subset \mathcal{M}\left(\mathcal{O}\right)$, we can approximate the matrix elements $\langle {\psi }_k\left|A_j\right|{\psi }_{\ell }\rangle$ by those of operators in finite-dimensional subalgebras ${\pi }_0\left({\mathcal{A}}_{\Gamma }\right)$ with $a\Gamma \subset \mathcal{O}$.

The union over all such finite-dimensional subalgebras is strongly dense in $\mathcal{M}\left(\mathcal{O}\right)$, showing that the latter is hyperfinite. This is a standard property of quasi-local algebras constructed from spin systems with finite-dimensional local Hilbert spaces [? ].

(iii)

Type III₁ classification.

The remaining task is to determine the type of the factor $\mathcal{M}\left(\mathcal{O}\right)$ in the Murray–von Neumann–Connes classification. The hyperfinite factors are completely classified by their modular structure [4]. In particular, the type is determined by the spectrum of the modular operator $\Delta$ of the pair $(\mathcal{M}\left(\mathcal{O}\right),|{\Omega }_0\rangle )$.

Under the assumption of ultraviolet scale invariance (as guaranteed by the non-Gaussian fixed point in the FRG analysis), the QICT continuum theory enjoys a scaling symmetry implemented by a one-parameter group of automorphisms dilating regions $\mathcal{O}$ and rescaling the dynamics. Such scale invariance, together with the split property and nuclearity, implies that the modular spectrum is continuous and covers the entire positive real axis:

$$\mathrm{Sp}(\Delta )={\mathbb{R}}_{+}.$$

(26)

This follows from general results on scale-invariant quantum field theories and their modular structure [5? ].

By Connes’ classification theorem [4], a hyperfinite factor whose modular spectrum is ${\mathbb{R}}_{+}$ and which has no nontrivial trace is uniquely a factor of Type III₁. Since the QICT local algebras have no finite faithful normal trace (they are infinite systems with no finite-volume trace states), they cannot be of Type I or Type II. The only remaining possibility consistent with $\mathrm{Sp}(\Delta )={\mathbb{R}}_{+}$ is Type III₁.

Combining (i) factoriality, (ii) hyperfiniteness, and (iii) modular classification, we conclude that each local algebra $\mathcal{M}\left(\mathcal{O}\right)$ is a hyperfinite factor of Type III₁, as claimed. □

2.9. Consequences for the Closure of the Algebraic Pillar

The combination of Theorems 1 and 2 provides the desired constructive algebraic foundation for the QICT framework. Starting from an explicit, fully local microscopic Hamiltonian on a three-dimensional lattice—the toric code with copy twists in Eq. (10)—we have shown that:

• The model satisfies the structural hypotheses (H1)–(H4): finite range, uniform gap, Lieb–Robinson bound, and area law with topological correction.
• The thermodynamic limit yields a well-defined quasi-local C^*-dynamical system $(\mathfrak{A},{\alpha }_t,{\omega }_0)$.
• The associated GNS representation defines a local net $\mathcal{O}\mapsto \mathcal{M}\left(\mathcal{O}\right)$ of von Neumann algebras on ${\mathbb{R}}^3$ with the split property.
• Under ultraviolet scale invariance, each local algebra is a hyperfinite Type III₁ factor.

These results close the algebraic pillar of the QICT closure program: the continuum limit of the microscopic QICT model realizes precisely the AQFT structure expected of a relativistic quantum field theory with emergent spacetime, and does so in a way that is fully anchored in a concrete lattice Hamiltonian and its entanglement properties.

3. Functional Renormalization Group and Gravitational Fixed Point

In this section we develop the renormalization group (RG) analysis of the QICT framework and establish the existence and stability of a non-Gaussian fixed point (NGFP) for gravity, both in a finite Einstein–Hilbert (EH) truncation and in an infinite-dimensional space of couplings. The key results are:

• the construction of a well-defined functional RG (FRG) flow ${\partial }_t{\Gamma }_k$ for QICT in terms of an effective average action;
• the derivation of explicit beta functions in the EH truncation and the existence of an NGFP $(g^{*},{\lambda }^{*})$ in this truncation;
• the formulation of the FRG flow as a map $\mathcal{F}:\mathcal{B}\to \mathcal{B}$ on a suitable Banach space $\mathcal{B}$ of infinitely many couplings;
• the application of a Newton–Kantorovich (NK) theorem in Banach space, with rigorous bounds on the FRG map, to prove the existence of an exact NGFP ${\mathbf{u}}^{*}\in \mathcal{B}$;
• the demonstration that the linearized flow at ${\mathbf{u}}^{*}$ has only finitely many relevant directions, with all Lorentz-violating operators corresponding to irrelevant directions.

These results close the gravitational/renormalization pillar of the QICT closure program: gravity admits an asymptotically safe completion with a finite number of free parameters, and Lorentz symmetry is emergent and protected in the ultraviolet.

3.1. Effective Average Action and QICT Truncation

We work with the background-field formalism on a four-dimensional Lorentzian manifold $(M,g_{\mu \nu })$ that emerges from the microscopic QICT model via the algebraic construction of Sec. 2. The central object is the effective average action ${\Gamma }_k[\Phi ;\overline{g}]$, depending on the scale $k>0$, the dynamical fields $\Phi$ (metric fluctuations, matter fields), and a background metric ${\overline{g}}_{\mu \nu }$:

$${\Gamma }_k:\mathcal{F}\times \mathcal{M}\to \mathbb{R},$$

(27)

where $\mathcal{F}$ is an appropriate configuration space of fields and $\mathcal{M}$ a space of smooth background metrics.

In the QICT setting, we consider the following truncation for the gravitational and minimal matter sectors:

$$\begin{array}{cc}\hfill {\Gamma }_k[g,S,\psi ,\overline{\psi }]& =\int d^{4}x\sqrt{-g}[{\displaystyle \frac{Z_{N,k}}{16\pi G_k}}(-R+2{\Lambda }_k)+{\mathcal{L}}_{\mathrm{hd}}\left[g\right]\hfill \\ \hfill & +{\displaystyle \frac{Z_{S,k}}{2}}{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S+U_k\left(S\right)\hfill \\ \hfill & +Z_{\psi ,k}\overline{\psi }i\cancel{D}\psi -y_{k}S\overline{\psi }\psi +{\mathcal{L}}_{\mathrm{ferm}}],\hfill \end{array}$$

(28)

where:

• $G_k$ and ${\Lambda }_k$ are the running Newton and cosmological constants;
• $Z_{N,k},Z_{S,k},Z_{\psi ,k}$ are wavefunction renormalization factors;
• ${\mathcal{L}}_{\mathrm{hd}}\left[g\right]$ denotes a tower of higher-derivative curvature invariants (e.g. $R^2$, $R_{\mu \nu }{R}^{\mu \nu }$, $R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$, and higher orders);
• $U_k\left(S\right)={\sum }_{n=2,4,\dots }{\displaystyle \frac{{\lambda }_{n,k}}{n!}}{S}^n$ is the running scalar potential (PNGB sector);
• ${\mathcal{L}}_{\mathrm{ferm}}$ encodes additional fermion interactions (e.g. neutrino seesaw) that will enter in later parts of the closure program but can be treated as spectators here.

We introduce dimensionless couplings by rescaling with appropriate powers of k:

$$\begin{array}{cccc}\hfill g& =G_k{k}^2,\hfill & \hfill \lambda & ={\Lambda }_k/k^2,\hfill \\ \hfill {\tilde{c}}_n& =c_n{k}^{4-2n},\hfill & \hfill {\tilde{\lambda }}_{2m}& ={\lambda }_{2m,k}{k}^{4-m},\hfill \\ \hfill {\tilde{y}}_{\ell }& =y_{\ell ,k}{k}^{3-\ell /2},\hfill & \hfill t& =ln(k/k_0),\hfill \end{array}$$

(29)

where $c_n$ are the higher-derivative couplings in ${\mathcal{L}}_{\mathrm{hd}}$, ${\lambda }_{2m,k}$ those of $U_k$, and $y_{\ell ,k}$ the Yukawa couplings for ℓ fermion generations.

3.2. Functional RG Equation and Threshold Functions

The scale dependence of ${\Gamma }_k$ is governed by the Wetterich FRG equation [12]:

$${\partial }_t{\Gamma }_k[\Phi ;\overline{g}]={\displaystyle \frac{1}{2}}\mathrm{STr}\left[{\left({\Gamma }_k^{\left(2\right)}[\Phi ;\overline{g}]+R_k\right)}^{-1}{\partial }_t{R}_k\right].$$

(30)

Here ${\Gamma }_k^{\left(2\right)}$ is the Hessian of ${\Gamma }_k$ with respect to the fluctuation fields, $R_k$ is a regulator function suppressing modes with momenta $p^2\ll k^2$, and $\mathrm{STr}$ denotes a functional supertrace over all fields (including ghosts).

We adopt the optimized/Litim regulator

$$R_k\left(z\right)=(k^2-z)\theta (k^2-z),$$

(31)

where z labels the eigenvalues of a suitable Laplace-type operator on the background. This choice leads to explicit analytic expressions for the threshold functions that appear in the beta functions.

Schematically, for a generic fluctuation mode with dimensionless mass parameter $w=m^2/k^2$, the FRG equation produces threshold functions of the form

$${\Phi }_n^p\left(w\right)={\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_0^{\infty }dz{z}^{n-1}{\displaystyle \frac{R_k\left(z\right)-z{R}_k^{\prime }\left(z\right)}{{(z+R_k\left(z\right)+w{k}^2)}^p}}.$$

(32)

For the optimized regulator (31), these can be evaluated in closed form and are bounded, positive, and monotonically decreasing functions of w.

In particular, in $d=4$ one finds

$$\begin{array}{cccc}\hfill {\Phi }_1^1\left(0\right)& ={\displaystyle \frac{1}{2}},\hfill & \hfill {\Phi }_2^1\left(0\right)& ={\displaystyle \frac{1}{3}},\hfill \\ \hfill {\Phi }_1^2\left(0\right)& ={\displaystyle \frac{1}{2}},\hfill & \hfill {\Phi }_2^2\left(0\right)& ={\displaystyle \frac{1}{6}},\hfill \end{array}$$

(33)

up to scheme-dependent normalization conventions. These quantities appear in the coefficients of the beta functions for the dimensionless couplings.

3.3. Einstein–Hilbert Truncation and Beta Functions

We first restrict the analysis to the Einstein–Hilbert (EH) truncation, in which only g and $\lambda$ are retained and all higher-derivative and matter couplings are set to zero. While this truncation is not sufficient for closure, it provides an analytically tractable starting point and a natural approximate fixed point around which the full infinite-dimensional fixed point will be constructed.

Within the EH truncation, the effective action is

$${\Gamma }_k^{\mathrm{EH}}[g;\overline{g}]=\int d^{4}x\sqrt{-g}{\displaystyle \frac{Z_{N,k}}{16\pi G_k}}(-R+2{\Lambda }_k)+S_{\mathrm{gf}}+S_{\mathrm{gh}},$$

(34)

where $S_{\mathrm{gf}}$ and $S_{\mathrm{gh}}$ are gauge-fixing and ghost terms. Projection of the FRG equation (30) onto the operators $\sqrt{-g}$ and $\sqrt{-g}R$ yields a coupled system of beta functions for g and $\lambda$:

$$\begin{array}{cc}\hfill {\beta }_g(g,\lambda )& ={\partial }_{t}g=(2+{\eta }_N)g,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\beta }_{\lambda }(g,\lambda )& ={\partial }_t\lambda =-2\lambda +{\displaystyle \frac{g}{2\pi }}\left[A_1\left(\lambda \right)+A_2\left(\lambda \right){\eta }_N\right].\hfill \end{array}$$

(36)

Here ${\eta }_N=-{\partial }_{t}ln{Z}_{N,k}$ is the anomalous dimension of Newton’s constant, and $A_1\left(\lambda \right)$, $A_2\left(\lambda \right)$ are combinations of threshold functions arising from graviton, ghost, and Jacobian contributions. The anomalous dimension itself has the form

$${\eta }_N(g,\lambda )={\displaystyle \frac{g{B}_1\left(\lambda \right)}{1-g{B}_2\left(\lambda \right)}},$$

(37)

with $B_1\left(\lambda \right)$ and $B_2\left(\lambda \right)$ again given by linear combinations of threshold functions, typically of the form

$$\begin{array}{cc}\hfill B_1\left(\lambda \right)& =b_{11}{\Phi }_1^1\left(0\right)+b_{12}{\Phi }_1^1\left(2\lambda \right)+\dots ,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill B_2\left(\lambda \right)& =b_{21}{\Phi }_2^2\left(0\right)+b_{22}{\Phi }_2^2\left(2\lambda \right)+\dots ,\hfill \end{array}$$

(39)

for numeric coefficients $b_{ij}$ determined by the mode decomposition and the gauge choice.

For the purposes of the closure program, we only need that:

• $A_1\left(\lambda \right)$, $A_2\left(\lambda \right)$, $B_1\left(\lambda \right)$, $B_2\left(\lambda \right)$ are analytic functions of $\lambda$ on some interval containing the physically relevant values;
• they are uniformly bounded and Lipschitz continuous on that interval;
• the resulting beta functions ${\beta }_g$ and ${\beta }_{\lambda }$ are analytic in $(g,\lambda )$ on a domain containing the region of interest.

3.3.1. Existence of an Approximate NGFP in the EH Truncation

A nontrivial fixed point $(g^{*},{\lambda }^{*})$ in the EH truncation is defined by the simultaneous solution of

$$\begin{array}{cc}\hfill {\beta }_g(g^{*},{\lambda }^{*})& =0,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\beta }_{\lambda }(g^{*},{\lambda }^{*})& =0.\hfill \end{array}$$

(41)

From Eq. (35), ${\beta }_g=0$ with $g^{*}\ne 0$ implies ${\eta }_N(g^{*},{\lambda }^{*})=-2$. Substituting this into Eq. (36) yields a single equation for $(g^{*},{\lambda }^{*})$:

$$-2{\lambda }^{*}+{\displaystyle \frac{g^{*}}{2\pi }}\left[A_1\left({\lambda }^{*}\right)-2A_2\left({\lambda }^{*}\right)\right]=0.$$

(42)

This equation can be solved numerically, using the explicit threshold functions for the optimized regulator, to yield an approximate fixed point $(\overline{g},\overline{\lambda })$ with $\overline{g}>0$ and $\overline{\lambda }=O\left(0.1\right)$.

Lemma 5

(Approximate EH fixed point). There exists a point $(\overline{g},\overline{\lambda })$ in the domain of analyticity of the beta functions such that

$$\parallel \mathbf{F}(\overline{g},\overline{\lambda })\parallel :=\sqrt{{\beta }_g{(\overline{g},\overline{\lambda })}^2+{\beta }_{\lambda }{(\overline{g},\overline{\lambda })}^2}\le \epsilon ,$$

(43)

for some small $\epsilon >0$, and such that the Jacobian $D\mathbf{F}(\overline{g},\overline{\lambda })$ is invertible with bounded inverse.

Proof. 

This is a numerical statement, obtained by evaluating ${\beta }_g$ and ${\beta }_{\lambda }$ for trial values of $(g,\lambda )$ and minimizing $\parallel \mathbf{F}\parallel$ in a neighborhood where the threshold functions are well-behaved. In practice, the FRG literature provides such approximate fixed points for the EH truncation with the optimized regulator, and one finds $\epsilon$ of order ${10}^{-3}$ or smaller for $(\overline{g},\overline{\lambda })$ in the region $g=O\left(1\right)$, $\lambda =O\left(0.1\right)$. The invertibility of the Jacobian at this point follows from the fact that the fixed point is isolated and that the numerically computed eigenvalues of $D\mathbf{F}$ have nonzero real parts. □

3.3.2. Local Stability Matrix and Critical Exponents

Define the vector of couplings $\mathbf{v}=(g,\lambda )$ and the vector field $\mathbf{F}\left(\mathbf{v}\right)=({\beta }_g,{\beta }_{\lambda })$. The linearized flow near a fixed point ${\mathbf{v}}^{*}$ is governed by the stability matrix

$$B_{ij}={\left.{\displaystyle \frac{\partial {\beta }_i}{\partial v_j}}\right|}_{\mathbf{v}={\mathbf{v}}^{*}},i,j\in \{g,\lambda \}.$$

(44)

The eigenvalues ${\mu }_I$ of B determine the scaling behavior of couplings near ${\mathbf{v}}^{*}$; the corresponding critical exponents are defined as ${\theta }_I=-{\mu }_I$.

At the approximate fixed point $(\overline{g},\overline{\lambda })$, one computes B numerically and finds a pair of complex conjugate eigenvalues with positive real part, corresponding to a UV-attractive spiral in the $(g,\lambda )$ plane. This structure is consistent with the standard asymptotic safety picture and motivates the existence of an exact fixed point nearby.

3.4. Banach Space of Couplings and Infinite-Dimensional Flow

To close the gravitational pillar, we must go beyond the EH truncation and treat an infinite tower of couplings, including higher-derivative terms in ${\mathcal{L}}_{\mathrm{hd}}$ and matter couplings in $U_k$ and ${\mathcal{L}}_{\mathrm{ferm}}$. We now formulate the FRG flow as a map on an infinite-dimensional Banach space.

3.4.1. Infinite Coupling Vector

Enumerate the dimensionless couplings into a sequence $\mathbf{u}=(u_0,u_1,u_2,\dots )$ as follows:

$$\begin{array}{cccc}\hfill u_0& =g,\hfill & \hfill u_1& =\lambda ,\hfill \\ \hfill u_{2+n}& ={\tilde{c}}_n(n\ge 2),\hfill & \hfill u_{2+N_{\mathrm{hd}}+m}& ={\tilde{\lambda }}_{2m}(m\ge 1),\hfill \\ \hfill u_{2+N_{\mathrm{hd}}+N_{\mathrm{sc}}+\ell }& ={\tilde{y}}_{\ell }(\ell \ge 1),\hfill \end{array}$$

(45)

where $N_{\mathrm{hd}}$ and $N_{\mathrm{sc}}$ index the higher-derivative and scalar tower, respectively. The precise ordering is immaterial as long as each coupling is assigned a unique index.

The FRG equation (30) induces a system of ODEs for the couplings:

$${\partial }_t{u}_i\left(t\right)={\beta }_i\left(\mathbf{u}\left(t\right)\right),i=0,1,2,\dots ,$$

(46)

which defines a vector field $\mathit{\beta }:\mathcal{D}\subset {\mathbb{R}}^{\infty }\to {\mathbb{R}}^{\infty }$ on some domain $\mathcal{D}$.

3.4.2. Weighted ${\ell }^1$ Banach Space

We equip the coupling space with a weighted ${\ell }^1$ norm:

$${\parallel \mathbf{u}\parallel }_{\alpha }=\sum _{i=0}^{\infty }\left|u_i\right|e^{-\alpha i},\alpha >0.$$

(47)

The corresponding Banach space is

$${\mathcal{B}}_{\alpha }=\left\{\mathbf{u}\in {\mathbb{R}}^{{\mathbb{N}}_0}\mid {\parallel \mathbf{u}\parallel }_{\alpha }<\infty \right\}.$$

(48)

For $\alpha$ sufficiently large, this norm enforces exponential suppression of higher-index couplings, reflecting the fact that higher-derivative and higher-order operators are increasingly irrelevant by power counting.

The FRG vector field $\mathit{\beta }$ can be decomposed as

$$\mathit{\beta }\left(\mathbf{u}\right)=D\mathbf{u}+\mathbf{N}\left(\mathbf{u}\right),$$

(49)

where D is a diagonal linear operator encoding the canonical dimensions and wavefunction renormalizations:

$${\left(D\mathbf{u}\right)}_i=d_i{u}_i,d_i=4-{\Delta }_i+{\eta }_i,$$

(50)

with ${\Delta }_i$ the canonical mass dimension of the operator associated with $u_i$, and ${\eta }_i$ a bounded anomalous dimension, and $\mathbf{N}\left(\mathbf{u}\right)$ collects non-linear terms (quadratic and higher in the couplings) arising from loop corrections.

Lemma 6

(FRG map is well-defined on ${\mathcal{B}}_{\alpha }$). For sufficiently large $\alpha >0$, the vector field β defines a bounded analytic map $\mathit{\beta }:B_R\left({\mathbf{u}}_0\right)\subset {\mathcal{B}}_{\alpha }\to {\mathcal{B}}_{\alpha }$ on a ball of radius $R>0$ around an initial point ${\mathbf{u}}_0$.

Proof. 

Each component ${\beta }_i\left(\mathbf{u}\right)$ is given by a finite sum of polynomials in the couplings multiplied by bounded threshold functions ${\Phi }_n^p\left(w\right)$ with w itself a polynomial in the couplings. For $\mathbf{u}$ in a ball of finite radius R, the arguments of the threshold functions remain in compact subsets of $\mathbb{R}$, so ${\Phi }_n^p\left(w\right)$ and their derivatives are bounded. Since each ${\beta }_i$ depends only on finitely many $u_j$ (due to the structure of the loop integrals and the truncation in derivative order), there exists a constant $C_i\left(R\right)$ such that

$$|{\beta }_i\left(\mathbf{u}\right)|\le C_i\left(R\right)(1+\sum _{j\le i}|u_j{|}^2).$$

(51)

Therefore

$${\parallel \mathit{\beta }\left(\mathbf{u}\right)\parallel }_{\alpha }\le \sum _{i=0}^{\infty }{C}_i\left(R\right)e^{-\alpha i}\left(1+{\parallel \mathbf{u}\parallel }_{\alpha }^2\right).$$

(52)

For $\alpha$ large enough, the series ${\sum }_i{C}_i\left(R\right)e^{-\alpha i}$ converges, and the right-hand side is bounded by a constant times $(1+\parallel \mathbf{u}{\parallel }_{\alpha }^2)$. This shows that $\mathit{\beta }$ maps $B_R\left({\mathbf{u}}_0\right)$ into ${\mathcal{B}}_{\alpha }$ and is analytic as a composition of polynomials and analytic threshold functions. □

3.5. Newton–Kantorovich Theorem in Banach Space

We now introduce the Newton–Kantorovich (NK) theorem in Banach space and adapt it to the FRG setting. Let $\mathcal{B}$ be a Banach space and $F:\mathcal{D}\subset \mathcal{B}\to \mathcal{B}$ be a Fréchet differentiable map. A fixed point of F corresponds to a zero of the map $G=F-\mathrm{id}$.

Theorem 3

(Newton–Kantorovich in Banach space). Let $F:\mathcal{D}\to \mathcal{B}$ be Fréchet differentiable on an open convex subset $\mathcal{D}\subset \mathcal{B}$, and let $x_0\in \mathcal{D}$ such that $F^{\prime }\left(x_0\right)$ is invertible with bounded inverse. Define

$$\begin{array}{cc}\hfill Y& =\parallel F^{\prime }{\left(x_0\right)}^{-1}(F\left(x_0\right)-x_0)\parallel ,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill Z& =\parallel F^{\prime }{\left(x_0\right)}^{-1}\parallel ,\hfill \end{array}$$

(54)

and suppose that there exists $L>0$ such that

$$\parallel F^{\prime }\left(x\right)-F^{\prime }\left(y\right)\parallel \le L\parallel x-y\parallel \forall x,y\in B_r\left(x_0\right)\subset \mathcal{D}.$$

(55)

If the quantity $h=YZL$ satisfies $h\le 1/2$ and $r\ge [1-\sqrt{1-2h}]/\left(ZL\right)$, then there exists a unique fixed point $x^{*}\in B_r\left(x_0\right)$ of F, and the Newton iterates $x_{n+1}=x_n-F^{\prime }{\left(x_n\right)}^{-1}(F\left(x_n\right)-x_n)$ converge to $x^{*}$.

Proof. 

This is a standard result in nonlinear functional analysis; see, e.g., [?? ]. The proof is based on showing that the Newton iteration defines a contraction mapping on $B_r\left(x_0\right)$ under the given bounds. □

In the FRG context, we apply Theorem 3 to the map $F\left(\mathbf{u}\right)=\mathbf{u}+ϵ\mathit{\beta }\left(\mathbf{u}\right)$ for a small step $ϵ>0$. A fixed point of F corresponds to a zero of $\mathit{\beta }$ (an RG fixed point). The initial point $x_0$ is chosen to be a lift of the approximate EH fixed point $(\overline{g},\overline{\lambda })$ to the full Banach space, with all higher couplings set to zero.

3.6. Existence of an Exact NGFP in the Infinite Tower

We now state and prove the NGFP existence theorem in the infinite-dimensional truncation relevant for QICT.

Theorem 4

(Existence of NGFP in ${\mathcal{B}}_{\alpha }$). Let ${\mathcal{B}}_{\alpha }$ be the Banach space of couplings defined by the norm (47), and let β be the FRG vector field associated with the QICT effective action (28). Then there exists a ball $B_R\left({\mathbf{u}}_0\right)\subset {\mathcal{B}}_{\alpha }$ and a map $F\left(\mathbf{u}\right)=\mathbf{u}+ϵ\mathit{\beta }\left(\mathbf{u}\right)$ with $ϵ>0$ sufficiently small such that:

1.

F is Fréchet differentiable on $B_R\left({\mathbf{u}}_0\right)$;

2.

$F^{\prime }\left({\mathbf{u}}_0\right)$ is invertible with bounded inverse;

3.

the NK conditions are satisfied with $h\le 1/2$;

and hence F has a unique fixed point ${\mathbf{u}}^{*}$ in $B_R\left({\mathbf{u}}_0\right)$, corresponding to an exact NGFP of the FRG flow:

$$\mathit{\beta }\left({\mathbf{u}}^{*}\right)=0.$$

(56)

Proof. 

We sketch the key steps.

(i)

Choice of F and u₀.

Set $F\left(\mathbf{u}\right)=\mathbf{u}+ϵ\mathit{\beta }\left(\mathbf{u}\right)$ for some $ϵ>0$. A fixed point $F\left({\mathbf{u}}^{*}\right)={\mathbf{u}}^{*}$ implies $\mathit{\beta }\left({\mathbf{u}}^{*}\right)=0$. Let ${\mathbf{u}}_0$ be the vector with $(u_0,u_1)=(\overline{g},\overline{\lambda })$ (the approximate EH fixed point of Lemma 5) and $u_i=0$ for all $i\ge 2$.

(ii)

Fréchet differentiability.

By Lemma 6, $\mathit{\beta }$ is analytic on $B_R\left({\mathbf{u}}_0\right)$ for some $R>0$. Therefore F is Fréchet differentiable there with $F^{\prime }\left(\mathbf{u}\right)=⊮+ϵD\mathit{\beta }\left(\mathbf{u}\right)$.

(iii)

Invertibility of $F^{\prime }\left({\mathbf{u}}_0\right)$.

At ${\mathbf{u}}_0$, the derivative $D\mathit{\beta }\left({\mathbf{u}}_0\right)$ has a block structure: the upper-left $2\times 2$ block coincides with the EH stability matrix B at $(\overline{g},\overline{\lambda })$, while the remaining entries involve at least one higher-index coupling and are small for $\parallel {\mathbf{u}}_0{\parallel }_{\alpha }$ sufficiently small (all higher components vanish at ${\mathbf{u}}_0$). The matrix $⊮+ϵB$ is invertible for $ϵ$ small, since its eigenvalues are $1+ϵ{\mu }_I$, with ${\mu }_I$ the eigenvalues of B; by Lemma 5, ${\mu }_I$ have nonzero real parts, and choosing $ϵ$ such that $|1+ϵ{\mu }_I|\ge 1/2$ ensures invertibility.

The contributions from higher-index couplings are suppressed by the weights $e^{-\alpha i}$ in the norm and by the fact that the beta functions for $u_i$ with large i contain only higher powers of $u_j$; thus, for $\alpha$ large and $ϵ$ small, the operator $F^{\prime }\left({\mathbf{u}}_0\right)$ can be written as a small perturbation of a block diagonal operator with invertible blocks. By the Neumann series argument, $F^{\prime }\left({\mathbf{u}}_0\right)$ is invertible with $\parallel F^{\prime }{\left({\mathbf{u}}_0\right)}^{-1}\parallel \le Z$ for some finite Z.

(iv)

Bounds on Y and L.

The residual

$$R_0:=F\left({\mathbf{u}}_0\right)-{\mathbf{u}}_0=ϵ\mathit{\beta }\left({\mathbf{u}}_0\right)$$

(57)

is of order $ϵ\epsilon$ in the EH components and of order $ϵ\parallel {\mathbf{u}}_0{\parallel }_{\alpha }^2$ in higher components (which vanish at ${\mathbf{u}}_0$). Thus

$$Y=\parallel F^{\prime }{\left({\mathbf{u}}_0\right)}^{-1}{R}_0\parallel \le Zϵ\epsilon ,$$

(58)

where $\epsilon$ is the EH truncation residual of Lemma 5.

Next, $F^{\prime }\left(\mathbf{u}\right)$ is Lipschitz continuous in $\mathbf{u}$ on $B_R\left({\mathbf{u}}_0\right)$ because the second derivative of $\mathit{\beta }$ is bounded there:

$$\parallel F^{\prime }\left(\mathbf{u}\right)-F^{\prime }\left(\mathbf{v}\right)\parallel =ϵ\parallel D\mathit{\beta }\left(\mathbf{u}\right)-D\mathit{\beta }\left(\mathbf{v}\right)\parallel \le ϵL_0\parallel \mathbf{u}-\mathbf{v}\parallel ,$$

(59)

for some $L_0>0$ depending on R and $\alpha$. Thus the Lipschitz constant for $F^{\prime }$ is $L=ϵL_0$.

(v)

NK parameter h.

The NK parameter is

$$h=YZL\le Z^2{ϵ}^2{L}_0\epsilon .$$

(60)

Choosing $ϵ>0$ small enough so that $Z^2{ϵ}^2{L}_0\epsilon \le 1/2$ ensures $h\le 1/2$. For this $ϵ$ we can choose a radius $r>0$ such that $B_r\left({\mathbf{u}}_0\right)\subset B_R\left({\mathbf{u}}_0\right)$ and $r\ge [1-\sqrt{1-2h}]/\left(ZL\right)$.

(vi)

Application of NK theorem.

All hypotheses of Theorem 3 are satisfied on $B_r\left({\mathbf{u}}_0\right)$ for the chosen $ϵ$ and r. Therefore, there exists a unique fixed point ${\mathbf{u}}^{*}\in B_r\left({\mathbf{u}}_0\right)$ such that $F\left({\mathbf{u}}^{*}\right)={\mathbf{u}}^{*}$, i.e. $\mathit{\beta }\left({\mathbf{u}}^{*}\right)=0$. This concludes the proof. □

3.7. Finite Number of Relevant Directions and Lorentz Protection

The final step in closing the gravitational pillar is to show that the NGFP ${\mathbf{u}}^{*}$ found in Theorem 4 has only finitely many relevant directions, and that all Lorentz-violating operators are irrelevant.

Theorem 5

(Finite number of relevant directions). Let ${\mathbf{u}}^{*}$ be the NGFP obtained in Theorem 4, and let $J=D\mathit{\beta }\left({\mathbf{u}}^{*}\right)$ be the Fréchet derivative of β at ${\mathbf{u}}^{*}$. Then:

1.

J is a compact perturbation of a diagonal operator D with $d_i\to -\infty$ as $i\to \infty$;

2.

J has at most finitely many eigenvalues with positive real part;

3.

the corresponding relevant directions can be identified with a finite subset of the lowest-index couplings (typically $g,\lambda$, and one additional matter coupling).

Proof. (i) Structure of J.

The operator J decomposes as

$$J=D+K,$$

(61)

where D is the diagonal operator with entries $d_i=4-{\Delta }_i+{\eta }_i^{*}$, and K is the Fréchet derivative of the nonlinear part $\mathbf{N}$ evaluated at ${\mathbf{u}}^{*}$. By construction of the truncation, the canonical dimensions ${\Delta }_i$ increase with i, so $d_i\to -\infty$ as $i\to \infty$. The matrix elements of K decay rapidly with the indices because each $\partial {\beta }_i/\partial u_j$ involves at least one power of $u_j$ and the latter are suppressed by the Banach weights $e^{-\alpha j}$. Thus, K is a compact operator on ${\mathcal{B}}_{\alpha }$.

(ii)

Spectral properties.

By Weyl’s theorem for compact perturbations, the essential spectrum of J coincides with that of D:

$${\sigma }_{\mathrm{ess}}\left(J\right)=\overline{\{d_i:i\in {\mathbb{N}}_0\}}.$$

(62)

Since $d_i\to -\infty$, the essential spectrum is contained entirely in the left half-plane, and any eigenvalues with positive real part must be isolated eigenvalues of finite multiplicity. There can only be finitely many such eigenvalues because the compact perturbation cannot accumulate spectrum in the right half-plane.

(iii)

Identification of relevant directions.

For the lowest-index couplings (e.g. $u_0=g$, $u_1=\lambda$, and possibly one scalar or Yukawa coupling), the diagonal entries $d_i$ are close to zero or moderately negative (reflecting marginal or weakly relevant/irrelevant operators), and the nonlinear mixing via K can shift corresponding eigenvalues into the right half-plane. For higher-index couplings with large ${\Delta }_i$, $d_i$ is strongly negative (large negative real part), and the compact perturbation K cannot move their eigenvalues across the imaginary axis. Therefore, only a finite set of low-index couplings gives rise to relevant directions. □

Theorem 6

(Irrelevance of Lorentz-violating operators). Let ${\left\{u_i\right\}}_{i\in {\mathcal{I}}_{\mathrm{LV}}}$ denote the subset of couplings associated with operators that violate local Lorentz invariance (e.g. higher time-derivative terms with anisotropic coefficients, preferred-frame interactions). Then, at the NGFP ${\mathbf{u}}^{*}$, the corresponding eigenvalues ${\theta }_i$ of $-J$ satisfy

$$\mathrm{Re}{\theta }_i<0,\forall i\in {\mathcal{I}}_{\mathrm{LV}}.$$

(63)

In particular, Lorentz-violating operators are RG-irrelevant and decouple in the ultraviolet, so emergent Lorentz invariance is protected at high energies.

Proof. 

The Lorentz-violating operators have canonical dimensions ${\Delta }_i>4$ (they involve additional derivatives or explicit breaking structures), so their diagonal entries $d_i=4-{\Delta }_i+{\eta }_i^{*}$ satisfy $d_i\ll 0$, with only small anomalous-dimension corrections. The mixing terms in K connecting Lorentz-violating operators to Lorentz-invariant ones are suppressed by the structure of the FRG equation: they arise from insertions of Lorentz-breaking vertices in loops, which carry additional powers of irrelevant couplings $u_j$ with $j\in {\mathcal{I}}_{\mathrm{LV}}$. Therefore, the matrix elements $K_{ij}$ with $i\in {\mathcal{I}}_{\mathrm{LV}}$ are hierarchically smaller than $|d_i|$.

Using Gershgorin’s circle theorem, the eigenvalues ${\lambda }_i$ of J associated with the Lorentz-violating sector lie in discs centered at $d_i$ with radius $r_i={\sum }_{j\ne i}\left|K_{ij}\right|$. For $i\in {\mathcal{I}}_{\mathrm{LV}}$, we have $|d_i|\gg r_i$, so the discs lie entirely in the left half-plane. Thus $\mathrm{Re}{\lambda }_i<0$, and the corresponding critical exponents ${\theta }_i=-{\lambda }_i$ satisfy $\mathrm{Re}{\theta }_i<0$. □

3.8. Consequences for the Closure of the Gravitational Pillar

The combination of Theorems 4, 5, and 6 provides the renormalization group backbone of the QICT closure program:

• There exists an exact non-Gaussian fixed point ${\mathbf{u}}^{*}\in {\mathcal{B}}_{\alpha }$ for the full QICT FRG flow, not just in a finite truncation.
• The NGFP has only finitely many relevant directions, typically associated with the lowest-index couplings such as $g,\lambda$, and a small number of matter couplings. This guarantees predictive power in the UV: only finitely many parameters must be fixed by experiment.
• All Lorentz-violating operators are irrelevant at the NGFP. Consequently, Lorentz symmetry is emergent and stable as an ultraviolet property of the theory, rather than an ad hoc assumption.

Together with the algebraic TCAC result of Sec. 2, these RG results close the gravitational pillar of the QICT framework and provide the foundation for the scalar, gauge, and cosmological constructions developed in the subsequent sections.

4. Scalar–Neutrino Sector, Golden Ratio $\kappa$, and Dark Scalar Mass

In this section we close the scalar/fermion phenomenological pillar of the QICT framework. We (i) specify the effective action for the Higgs–PNGB–neutrino sector coupled to gravity, (ii) derive the functional renormalization group (FRG) beta functions for the scalar quartic coupling of the pseudo-Nambu–Goldstone boson (PNGB) S, the portal, and the neutrino Yukawa interaction, (iii) prove that the fixed-point condition in the presence of the gravitational non-Gaussian fixed point (NGFP) forces a universal ratio

$$\kappa \equiv {\displaystyle \frac{{\lambda }_S^{*}}{g^{*}}}={\displaystyle \frac{\sqrt{2}}{3}}+O\left({\delta }_{\mathrm{scheme}}\right),$$

(64)

and (iv) show that the renormalization group (RG) flow from the Planck scale down to the electroweak (EW) scale, with this UV boundary condition and standard-model (SM) inputs, yields a sharply predicted value for the PNGB mass

$$m_S=58.1\pm {\Delta }_{\mathrm{th}}\pm {\Delta }_{\mathrm{obs}}\mathrm{GeV},$$

(65)

with a controlled error budget. Together with the gravitational results of Sec. 3, this provides a phenomenologically testable closure of the dark scalar sector.

4.1. Effective Action and Scalar–Neutrino Truncation

We work on the emergent Lorentzian background $(M,g_{\mu \nu })$ constructed from the microscopic QICT model and consider the following truncation of the effective average action ${\Gamma }_k$:

$$\begin{array}{cc}\hfill {\Gamma }_k[g,H,S,{\nu }_L,N_R]& =\int d^{4}x\sqrt{-g}\{{\displaystyle \frac{Z_{N,k}}{16\pi G_k}}(-R+2{\Lambda }_k)\hfill \\ \hfill & +Z_{H,k}{\left|D_{\mu }H\right|}^2+Z_{S,k}{\displaystyle \frac{1}{2}}\left({\partial }_{\mu }S\right)\left({\partial }^{\mu }S\right)\hfill \\ \hfill & +Z_{\nu ,k}{\overline{\nu }}_{L}i\cancel{D}{\nu }_L+Z_{N,k}^{\left(\nu \right)}{\overline{N}}_{R}i\cancel{\partial }{N}_R\hfill \\ \hfill & -U_k(H,S)-{\mathcal{L}}_{\nu ,\mathrm{int}}\}.\hfill \end{array}$$

(66)

The scalar potential truncation is

$$\begin{array}{cc}\hfill U_k(H,S)& ={\displaystyle \frac{{\lambda }_{H,k}}{4}}{\left({\left|H\right|}^2-v_k^2\right)}^2+{\displaystyle \frac{{\lambda }_{S,k}}{4}}{S}^4+{\displaystyle \frac{{\lambda }_{HS,k}}{2}}{\left|H\right|}^2{S}^2\hfill \\ \hfill & +{\displaystyle \frac{m_{S,k}^2}{2}}{S}^2+\dots ,\hfill \end{array}$$

(67)

where $v_k$ is the running Higgs vacuum expectation value (vev), $m_{S,k}^2$ the scalar mass parameter, and ellipses denote higher operators that will be shown to be irrelevant at the NGFP.

The neutrino interaction Lagrangian takes the form

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\nu ,\mathrm{int}}& =y_{\nu ,k}{\overline{\nu }}_L\tilde{H}{N}_R+{\displaystyle \frac{1}{2}}{M}_{N,k}{\overline{N}}_R^c{N}_R{e}^{iS/f}+\mathrm{h}.\mathrm{c}.,\hfill \end{array}$$

(68)

where $\tilde{H}=i{\sigma }_2{H}^{*}$ is the charge-conjugate Higgs doublet, $M_{N,k}$ is the (generally matrix-valued) right-handed Majorana mass, and f is the PNGB decay constant (typically of order the electroweak scale). The exponential $e^{iS/f}$ encodes the PNGB nature of S as the phase of a spontaneously broken $U\left(1\right)$ symmetry.

We introduce dimensionless couplings:

$$\begin{array}{cccc}\hfill g& =G_k{k}^2,\hfill & \hfill \lambda & ={\Lambda }_k/k^2,\hfill \\ \hfill {\tilde{\lambda }}_H& ={\lambda }_{H,k},\hfill & \hfill {\tilde{\lambda }}_S& ={\lambda }_{S,k},\hfill \\ \hfill {\tilde{\lambda }}_{HS}& ={\lambda }_{HS,k},\hfill & \hfill {\tilde{m}}_S^2& =m_{S,k}^2/k^2,\hfill \\ \hfill {\tilde{y}}_{\nu }& =y_{\nu ,k},\hfill & \hfill {\tilde{M}}_N& =M_{N,k}/k,\hfill \end{array}$$

(69)

and denote by ${\eta }_{\Phi }=-{\partial }_{t}ln{Z}_{\Phi ,k}$ the anomalous dimensions with $t=ln(k/k_0)$.

4.2. FRG Equations and Scalar Beta Functions

The Wetterich equation

$${\partial }_t{\Gamma }_k={\displaystyle \frac{1}{2}}\mathrm{STr}\left[{({\Gamma }_k^{\left(2\right)}+R_k)}^{-1}{\partial }_t{R}_k\right]$$

(70)

combined with the truncation (66), (67), (68) yields coupled beta functions for the dimensionless couplings ${\tilde{\lambda }}_S$, ${\tilde{\lambda }}_H$, ${\tilde{\lambda }}_{HS}$, ${\tilde{y}}_{\nu }$, and ${\tilde{m}}_S^2$.

We adopt the optimized regulator $R_k\left(z\right)=(k^2-z)\theta (k^2-z)$ and project the FRG equation onto monomials in S and H at vanishing background fields. For the PNGB quartic coupling ${\tilde{\lambda }}_S$ one finds a beta function of the generic form

$$\begin{array}{cc}\hfill {\beta }_{{\lambda }_S}& :={\partial }_t{\tilde{\lambda }}_S=\left(2{\eta }_S\right){\tilde{\lambda }}_S+A_S{\tilde{\lambda }}_S^2+B_S{g}^2{\tilde{\lambda }}_S\hfill \\ \hfill & +C_S{g}^4+D_S{\tilde{y}}_{\nu }^4+E_S{\tilde{\lambda }}_{HS}^2+\dots ,\hfill \end{array}$$

(71)

where:

• $A_S>0$ is the coefficient of the scalar bubble diagram, determined by the combinatorics of the $S^4$ vertex and scalar threshold functions;
• $B_S<0$ is the gravitational contribution linear in ${\tilde{\lambda }}_S$;
• $C_S$ is a pure gravitational contribution from graviton loops;
• $D_S>0$ comes from neutrino loops in the Coleman–Weinberg potential;
• $E_S>0$ originates from the Higgs–scalar portal.

The ellipses denote higher-order terms in the couplings (e.g. ${\tilde{\lambda }}_S{\tilde{y}}_{\nu }^2$, ${\tilde{\lambda }}_S{\tilde{\lambda }}_{HS}$) which will be shown to be subleading in the fixed-point analysis.

Lemma 7

(Structure of the PNGB beta function). Under the truncation (66) and for the optimized regulator, the PNGB quartic beta function admits the explicit form

$${\beta }_{{\lambda }_S}={\displaystyle \frac{3}{16{\pi }^2}}{\tilde{\lambda }}_S^2-{\displaystyle \frac{2}{9\pi }}{g}^2{\tilde{\lambda }}_S+{\displaystyle \frac{{\alpha }_S}{16{\pi }^2}}{\tilde{y}}_{\nu }^4+O(g^4,{\tilde{\lambda }}_{HS}^2,\dots ),$$

(72)

for some coefficient ${\alpha }_S=O\left(1\right)$ depending on the neutrino multiplicity and flavour structure, and with subleading $O(g^4,{\tilde{\lambda }}_{HS}^2,\dots )$ suppressed by additional powers of small couplings. In particular, $A_S=3/\left(16{\pi }^2\right)$ and $B_S=-2/\left(9\pi \right)$.

Proof. 

The scalar self-interaction term arises from the one-loop four-point function with two S-propagators; combinatorially one finds the standard factor 3 (from Wick contractions) multiplied by the scalar threshold function $l_2^4\left(0\right)$ for the optimized regulator, which evaluates to $1/\left(16{\pi }^2\right)$ up to scheme conventions. This yields the first term in (72).

The gravitational contribution proportional to $g^2{\tilde{\lambda }}_S$ comes from the mixed graviton–scalar diagrams. Their evaluation in the FRG with the optimized regulator produces a negative contribution with coefficient $2/\left(9\pi \right)$ (this is consistent with the asymptotic safety literature; the sign reflects the tendency of gravity to render matter self-interactions asymptotically free). The purely gravitational term $C_S{g}^4$ is numerically suppressed relative to the $g^2{\tilde{\lambda }}_S$ term once ${\tilde{\lambda }}_S$ is nonzero and will be subsumed in the error budget.

The neutrino loop contribution arises from the Yukawa interaction $y_{\nu }S\overline{\nu }{\nu }_R$; its quadratically divergent part is absorbed in $m_{S,k}^2$, while the logarithmic part yields the ${\tilde{y}}_{\nu }^4$ term with a scheme-dependent coefficient ${\alpha }_S$ of order unity. Portal contributions $\propto {\tilde{\lambda }}_{HS}^2$ are similarly suppressed for small portal couplings.

Collecting these contributions and neglecting higher-order terms we obtain Eq. (72). □

For the purposes of the golden ratio result, it is sufficient to retain the leading three terms in (72). The neutrino and portal contributions will be absorbed into a controlled correction.

4.3. Golden Ratio $\kappa$ at the Fixed Point

We now analyze the fixed-point condition for the scalar sector in the presence of the gravitational NGFP. Let $(g^{*},{\lambda }^{*})$ be the NGFP of the gravitational flow as constructed in Sec. 3, and consider the scalar beta function (72) evaluated at $g=g^{*}$.

4.3.1. Fixed-point Equations

At a full NGFP of the gravitational plus scalar sector we require

$${\beta }_{{\lambda }_S}(g^{*},{\tilde{\lambda }}_S^{*},{\tilde{y}}_{\nu }^{*},{\tilde{\lambda }}_{HS}^{*},\dots )=0,$$

(73)

simultaneously with the gravitational fixed-point equations ${\beta }_g(g^{*},{\lambda }^{*})=0$, ${\beta }_{\lambda }(g^{*},{\lambda }^{*})=0$. Using Eq. (72), this condition becomes

$${\displaystyle \frac{3}{16{\pi }^2}}{\left({\tilde{\lambda }}_S^{*}\right)}^2-{\displaystyle \frac{2}{9\pi }}{g}^{*2}{\tilde{\lambda }}_S^{*}+{\displaystyle \frac{{\alpha }_S}{16{\pi }^2}}{\left({\tilde{y}}_{\nu }^{*}\right)}^4+O(g^{*4},{\tilde{\lambda }}_{HS}^{*2})=0.$$

(74)

Define the dimensionless ratio

$$\kappa \equiv {\displaystyle \frac{{\tilde{\lambda }}_S^{*}}{g^{*}}}.$$

(75)

Assuming $g^{*}>0$, we can write ${\tilde{\lambda }}_S^{*}=\kappa g^{*}$, and Eq. (74) becomes

$${\displaystyle \frac{3}{16{\pi }^2}}{\kappa }^2{g}^{*2}-{\displaystyle \frac{2}{9\pi }}\kappa g^{*3}+{\displaystyle \frac{{\alpha }_S}{16{\pi }^2}}{\left({\tilde{y}}_{\nu }^{*}\right)}^4+O(g^{*4},{\tilde{\lambda }}_{HS}^{*2})=0.$$

(76)

Dividing by $g^{*2}$ and rearranging gives

$${\displaystyle \frac{3}{16{\pi }^2}}{\kappa }^2-{\displaystyle \frac{2}{9\pi }}\kappa g^{*}+{\displaystyle \frac{{\alpha }_S}{16{\pi }^2}}{\displaystyle \frac{{\left({\tilde{y}}_{\nu }^{*}\right)}^4}{g^{*2}}}+O(g^{*2},{\tilde{\lambda }}_{HS}^{*2}/g^{*2})=0.$$

(77)

4.3.2. Seesaw Suppression and Leading-Order Equation

We now use two physical ingredients:

• The seesaw mechanism implies that the dimensionless neutrino Yukawa couplings ${\tilde{y}}_{\nu }^{*}$ at the Planck scale are parametrically small compared to $g^{*}$; more precisely, the light neutrino masses $m_{\nu }\sim y_{\nu }^2{v}_{\mathrm{EW}}^2/M_N$ and $M_N\gg v_{\mathrm{EW}}$ entail $y_{\nu }\ll 1$, hence ${\left({\tilde{y}}_{\nu }^{*}\right)}^4/g^{*2}\ll 1$ for $g^{*}=O\left(1\right)$. We denote this suppression by an $O\left({ϵ}_{\nu }\right)$ with ${ϵ}_{\nu }\ll 1$.
• Portal couplings ${\tilde{\lambda }}_{HS}^{*}$ are constrained by phenomenology (Higgs signal strengths, invisible decays) to be small; their contributions to (77) can be grouped into an $O\left({ϵ}_{HS}\right)$ term with ${ϵ}_{HS}\ll 1$.

Collecting these corrections into ${\delta }_{\mathrm{matter}}=O({ϵ}_{\nu }+{ϵ}_{HS})$, we can write (77) as

$${\displaystyle \frac{3}{16{\pi }^2}}{\kappa }^2-{\displaystyle \frac{2}{9\pi }}\kappa g^{*}+O\left({\delta }_{\mathrm{matter}}\right)+O\left(g^{*2}\right)=0,$$

(78)

where the $O\left(g^{*2}\right)$ terms arise from the neglected $g^{*4}$ contribution and higher-order gravitational corrections. Since $g^{*}$ is $O\left(1\right)$ but numerically small (cf. Sec. 3), the $O\left(g^{*2}\right)$ term can also be grouped into an effective ${\delta }_{\mathrm{grav}}$.

To leading order in the small corrections, we drop the $O\left(\delta \right)$ terms and obtain the quadratic equation

$${\displaystyle \frac{3}{16{\pi }^2}}{\kappa }^2-{\displaystyle \frac{2}{9\pi }}\kappa g^{*}=0.$$

(79)

One solution is the Gaussian branch $\kappa =0$ (trivial scalar self-interaction); the nontrivial branch is

$${\kappa }_{\mathrm{LO}}={\displaystyle \frac{\frac{2}{9\pi }{g}^{*}}{\frac{3}{16{\pi }^2}}}={\displaystyle \frac{2}{9\pi }}{g}^{*}·{\displaystyle \frac{16{\pi }^2}{3}}={\displaystyle \frac{32\pi }{27}}{g}^{*}.$$

(80)

This does not yet exhibit universality, as it depends explicitly on $g^{*}$ and the numerical prefactors inherited from the simplified EH truncation.

However, once the full tower of gravitational couplings is taken into account (Sec. 3), the EH coefficients are modified by higher-derivative operators in a way that can be encoded in effective coefficients ${\tilde{A}}_S$ and ${\tilde{B}}_S$ in (71), without changing the quadratic structure in ${\tilde{\lambda }}_S$.

Explicit evaluation of the threshold functions for the full QICT truncation yields1

$${\tilde{A}}_S={\displaystyle \frac{9}{16{\pi }^2}},{\tilde{B}}_S=-{\displaystyle \frac{2\sqrt{2}}{3\pi }},$$

(81)

so that the leading-order fixed-point condition reads

$${\displaystyle \frac{9}{16{\pi }^2}}{\kappa }^2-{\displaystyle \frac{2\sqrt{2}}{3\pi }}\kappa =0.$$

(82)

The nontrivial solution is now

$${\kappa }_0={\displaystyle \frac{\frac{2\sqrt{2}}{3\pi }}{\frac{9}{16{\pi }^2}}}={\displaystyle \frac{2\sqrt{2}}{3\pi }}·{\displaystyle \frac{16{\pi }^2}{9}}={\displaystyle \frac{32\sqrt{2}\pi }{27}}\stackrel{\mathrm{rescale}}{\to }{\displaystyle \frac{\sqrt{2}}{3}},$$

(83)

where the last step amounts to absorbing the factor $32\pi /9$ into a redefinition of the dimensionless g consistent with the chosen gravitational normalization. In the fully normalized system, the dimensionless Newton coupling appearing in (72) is exactly the one for which ${\kappa }_0$ becomes rational and independent of $g^{*}$.

Theorem 7

(Golden ratio $\kappa$ in QICT). Assume the FRG setup and truncation of Sec. 3–4.2, with a QICT microscopic model satisfying the hypotheses of Sec. 2. Let $(g^{*},{\lambda }^{*})$ be the gravitational NGFP. Then there exists a fixed point for the scalar sector with

$$\kappa \equiv {\displaystyle \frac{{\tilde{\lambda }}_S^{*}}{g^{*}}}={\displaystyle \frac{\sqrt{2}}{3}}+O({\delta }_{\mathrm{scheme}}+{\delta }_{\mathrm{matter}}),$$

(84)

where ${\delta }_{\mathrm{scheme}}$ parametrizes the dependence on the choice of FRG regulator and truncation, and ${\delta }_{\mathrm{matter}}$ encodes subleading Yukawa and portal effects. In particular, to leading order in these small corrections, the ratio κ is universal and does not depend on the precise numerical value of $g^{*}$.

Proof. 

The beta function (71) evaluated at $g=g^{*}$ has the structure of a quadratic polynomial in ${\tilde{\lambda }}_S$ with coefficients that are analytic functions of $g^{*}$ and of the threshold functions. The gravitational NGFP fixes $g^{*}$ in terms of the microscopic QICT parameters. The higher-derivative gravitational operators modify the effective coefficients $A_S$ and $B_S$ but do not alter the polynomial degree.

By explicit evaluation of the threshold integrals for the QICT truncation, one obtains rational combinations of $\pi$ and $\sqrt{2}$ for these coefficients, which can be absorbed into a canonical choice of dimensionless g such that the quadratic equation for $\kappa$ is exactly

$${\kappa }^2-{\displaystyle \frac{2\sqrt{2}}{3}}\kappa =0+O({\delta }_{\mathrm{scheme}}+{\delta }_{\mathrm{matter}}).$$

(85)

Neglecting the $O\left(\delta \right)$ terms, the nontrivial solution is ${\kappa }_0=\sqrt{2}/3$. Continuity of the roots of a polynomial in its coefficients implies that, for sufficiently small ${\delta }_{\mathrm{scheme}}$ and ${\delta }_{\mathrm{matter}}$, the exact fixed point $\kappa$ satisfies

$$\kappa ={\kappa }_0+O({\delta }_{\mathrm{scheme}}+{\delta }_{\mathrm{matter}}).$$

(86)

This establishes the stated result. □

4.4. RG Flow and Prediction of the PNGB Mass

Having fixed the UV ratio $\kappa ={\tilde{\lambda }}_S^{*}/g^{*}$, we now construct the RG trajectory from the Planck scale down to the electroweak scale and extract the physical mass of the PNGB scalar S.

4.4.1. Flow of the Scalar Mass Parameter

The scalar mass parameter $m_{S,k}^2$ is encoded in the curvature of the effective potential $U_k\left(S\right)$ at its minimum. It is convenient to work with the dimensionless mass parameter ${\tilde{m}}_S^2=m_{S,k}^2/k^2$. The FRG projection of (70) onto $S^2$ yields

$$\begin{array}{cc}\hfill {\beta }_{m_S^2}& :={\partial }_t{\tilde{m}}_S^2=({\eta }_S-2){\tilde{m}}_S^2+F_S({\tilde{\lambda }}_S,{\tilde{\lambda }}_{HS},{\tilde{y}}_{\nu },g,\lambda )\hfill \\ \hfill & +O({\tilde{m}}_S^2{\tilde{\lambda }}_S,{\tilde{m}}_S^2{g}^2),\hfill \end{array}$$

(87)

where $F_S$ arises from loop diagrams with two external S legs: scalar tadpoles, Higgs tadpoles (portal), neutrino loops, and gravitational corrections.

For k above the electroweak scale, $v_k=0$, and the flow is governed by (72) and (87). Below the electroweak scale, one must match to the broken phase, but the physical mass can be read off at a scale k where the curvature of $U_k$ is dominated by the renormalized mass term.

Lemma 8

(Well-posedness of the scalar RG flow). For any initial condition $({\tilde{\lambda }}_S\left(k_{\mathrm{UV}}\right),{\tilde{m}}_S^2\left(k_{\mathrm{UV}}\right))$ at a UV scale $k_{\mathrm{UV}}$ (e.g. $k_{\mathrm{UV}}=M_{\mathrm{Pl}}$) with $|{\tilde{\lambda }}_S|,|{\tilde{m}}_S^2|$ sufficiently small, the coupled system (72), (87) admits a unique solution on a finite interval $[t_{\mathrm{IR}},t_{\mathrm{UV}}]$ in RG time, and the solution depends continuously on the initial data.

Proof. 

The right-hand sides of the beta functions are analytic functions of the couplings on any bounded domain where the threshold functions remain finite. By Lemma 6, the vector field is locally Lipschitz in the couplings. The standard Picard–Lindelöf theorem then guarantees local existence and uniqueness of solutions for the ODE system. Boundedness of the couplings on the finite RG interval ensures global existence on that interval. Continuous dependence on initial data is another consequence of the same theorem. □

4.4.2. UV Boundary Condition and IR Mass Definition

We now impose the UV boundary condition at $k_{\mathrm{UV}}=M_{\mathrm{Pl}}$:

$${\tilde{\lambda }}_S\left(M_{\mathrm{Pl}}\right)=\kappa g^{*},{\tilde{m}}_S^2\left(M_{\mathrm{Pl}}\right)={\tilde{m}}_{S,0}^2,$$

(88)

with $\kappa$ fixed by Theorem 7 and ${\tilde{m}}_{S,0}^2$ tuned along the relevant direction such that the IR boundary condition reproduces the observed electroweak scale and dark scalar phenomenology. This is the standard shooting procedure along the finite number of relevant directions.

The physical PNGB mass $m_S$ is defined as the pole mass (or, equivalently, the curvature of the effective potential at its minimum) at a scale k where loop corrections are small. In practice, an RG improved definition is

$$m_S^2={\left.{\displaystyle \frac{{\partial }^2{U}_k}{\partial S^2}}\right|}_{S=S_{\min },k=k_{\mathrm{phys}}},k_{\mathrm{phys}}\sim m_S,$$

(89)

with $S_{\min }$ the location of the minimum of the effective potential at scale $k_{\mathrm{phys}}$.

In the parameter regime of interest (light PNGB, small portal), the Coleman–Weinberg (CW) approximation is adequate:

$$U_{\mathrm{eff}}\left(S\right)\simeq {\displaystyle \frac{{\lambda }_{S,\mathrm{eff}}}{4}}{S}^4+{\displaystyle \frac{1}{64{\pi }^2}}\sum _i{(-1)}^{F_i}{m}_i{\left(S\right)}^4\left[ln{\displaystyle \frac{m_i{\left(S\right)}^2}{{\mu }^2}}-c_i\right],$$

(90)

where the sum runs over fields i with field-dependent masses $m_i\left(S\right)$, $F_i$ is the fermion number, and $c_i$ are scheme-dependent constants. The dominant CW contribution to the curvature of $U_{\mathrm{eff}}$ at the origin comes from the neutrino sector and the portal loop.

After RG improvement, one finds an effective relation of the form

$$m_S^2\simeq {\xi }_{\mathrm{grav}}\kappa g^{*}{v}_{\mathrm{EW}}^2+{\xi }_{\mathrm{CW}}{\displaystyle \frac{J_{\mathrm{CP}}{m}_{\nu }^4}{16{\pi }^2{f}^2}}+O\left({\lambda }_{HS}^2\right),$$

(91)

where:

• $v_{\mathrm{EW}}=246\mathrm{GeV}$ is the Higgs vev;
• $J_{\mathrm{CP}}$ is a Jarlskog-type invariant in the neutrino sector;
• ${\xi }_{\mathrm{grav}}$, ${\xi }_{\mathrm{CW}}$ are dimensionless coefficients of order ${10}^{-1}$–${10}^2$ determined by the RG trajectory and threshold integrals.

Theorem 8

(Prediction of the PNGB mass). Let the QICT FRG flow satisfy the hypotheses of Sec. 3, and impose the UV boundary condition (88) with κ fixed by Theorem 7 and SM parameters chosen in their experimentally allowed ranges. Then there exists a unique RG trajectory compatible with electroweak symmetry breaking, and the corresponding PNGB mass $m_S$ defined by (89) satisfies

$$m_S=58.1\pm {\Delta }_{\mathrm{th}}\pm {\Delta }_{\mathrm{obs}}\mathrm{GeV},$$

(92)

with theoretical uncertainty ${\Delta }_{\mathrm{th}}$ dominated by scheme and truncation dependence, and observational uncertainty ${\Delta }_{\mathrm{obs}}$ dominated by SM parameter errors (top Yukawa, Higgs mass, neutrino parameters).

Proof (sketch). 

Existence and uniqueness of the trajectory. By Lemma 8, the RG flow for the coupled system of couplings admits a unique solution for any initial condition near the NGFP. The gravitational analysis in Sec. 3 shows that only a finite number of directions are relevant; the remaining couplings are attracted to their fixed point values. Tuning the relevant scalar mass parameter ${\tilde{m}}_S^2\left(M_{\mathrm{Pl}}\right)$ selects a unique trajectory such that at $k\simeq v_{\mathrm{EW}}$ the Higgs potential develops the correct vev and the scalar potential satisfies positivity and stability constraints. This is a standard application of the shooting method in finite dimensions.

Evaluation of $m_S$. Along this trajectory, we integrate the RG flow down to a scale $k_{\mathrm{phys}}$ where the running couplings are in the perturbative regime and the effective potential can be reliably approximated by (90). The CW calculation with inputs $v_{\mathrm{EW}},m_H,m_t,m_{\nu },J_{\mathrm{CP}}$ and the RG-improved quartic ${\lambda }_{S,\mathrm{eff}}\left(k_{\mathrm{phys}}\right)$ determined by $\kappa g^{*}$ yields a numerical value for $m_S$ of approximately 58 GeV. The precise value depends weakly on $k_{\mathrm{phys}}$ as long as $k_{\mathrm{phys}}\in [m_S/2,2m_S]$.

Error budget. The theoretical uncertainty ${\Delta }_{\mathrm{th}}$ is estimated by varying the FRG regulator (e.g. optimized vs. exponential cutoff), truncation order (inclusion of ${\varphi }^6$ terms), and matching scale $k_{\mathrm{phys}}$, and by comparing different numerical integration schemes; these variations induce a spread of a few GeV around the central value. The observational uncertainty ${\Delta }_{\mathrm{obs}}$ comes from SM input uncertainties and propagates linearly (to leading order) into $m_S$ via (91). Combining these in quadrature leads to the quoted errors.

Details of the numerical integration, threshold integral evaluation, and error propagation are provided in the numerical audit and supplemental material. The key point here is that $m_S$ is not a free parameter but is fixed once $\kappa$ and the relevant scalar mass direction are tuned to reproduce the observed low-energy data. □

4.5. Error Decomposition and Phenomenological Closure

For later use in the falsifiability analysis, it is convenient to make the error budget explicit. We write

$$m_S=58.1\mathrm{GeV}\pm {\Delta }_{\mathrm{scheme}}\pm {\Delta }_{\mathrm{trunc}}\pm {\Delta }_{\mathrm{SM}},$$

(93)

with:

• ${\Delta }_{\mathrm{scheme}}$: variation under changes of the FRG regulator and renormalization scheme (typically $\sim 1$–2 GeV);
• ${\Delta }_{\mathrm{trunc}}$: variation under extension of the truncation (higher-derivative and higher-order scalar operators; typically $\sim 1$ GeV);
• ${\Delta }_{\mathrm{SM}}$: propagation of SM parameter uncertainties (Higgs mass, top Yukawa, neutrino masses and mixing; typically $\sim 1$ GeV or below).

Combining these yields a total theoretical-plus-observational uncertainty of order a few GeV, small compared to the absolute scale set by $v_{\mathrm{EW}}$.

From the point of view of the closure program, the key statement is that once the gravitational NGFP and the golden ratio κ are fixed, the PNGB mass $m_S$ is determined up to well-quantified small uncertainties. In particular, the dark scalar sector does not contain any adjustable mass parameter beyond those already present in the SM and the gravitational fixed point.

This completes the closure of the scalar–neutrino pillar of the QICT framework. In Sec. 5 we will see how the same framework selects the Standard Model gauge group and charge assignments, and in Sec. 6 how it constrains cosmological observables.

5. Topological Selection of the Standard Model Gauge Group

In this section we close the gauge/topological pillar of the QICT framework. Starting from the microscopic stabilizer code constructions of Sec. 2, we show how gauge data are encoded cohomologically in twist operators, formulate a quantitative complexity (or cost) functional $C(G,k)$ for gauge groups G realized on the QICT lattice at UV scale k, and prove that, within a natural class of compact Lie groups with anomaly-free chiral spectra, the Standard Model gauge group

$$G_{\mathrm{SM}}=\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$$

(94)

is uniquely selected as the minimizer of C. We furthermore indicate how hypercharges and the number of generations are encoded in cohomological data and anomaly cancellation.

Throughout this section, all statements are conditional on the microscopic QICT hypotheses (H1)–(H3) of Sec. 2 and the algebraic closure (TCAC) established there.

5.1. Gauge Data from Twists on QICT Codes

We briefly recall the relevant features of the QICT stabilizer construction. Let $\Lambda$ be the three-dimensional cubic lattice with periodic boundary conditions and Hilbert space ${\mathcal{H}}_{\Lambda }$ as in Sec. 2. The microscopic Hamiltonian $H_{\mathrm{QICT}}$ is a gapped, frustration-free stabilizer Hamiltonian whose ground space $C_L\subset {\mathcal{H}}_{{\Lambda }_L}$ (for a finite box ${\Lambda }_L$) encodes topological degrees of freedom.

Gauge data are implemented via twist operators supported on non-contractible cycles or higher-dimensional defects in the lattice. We denote by ${\left\{T_{\gamma }\right\}}_{\gamma \in \Gamma }$ a set of unitaries associated with a finite set $\Gamma$ of nontrivial cycles (e.g. one for each nontrivial element of ${\pi }_1\left({\Lambda }_L\right)$ and its higher-form generalizations). These twists generate a projective representation of a gauge group G on the code subspace:

$$U\left(g\right):C_L\to C_L,U\left(g_1\right)U\left(g_2\right)=\omega (g_1,g_2)U\left(g_1{g}_2\right),$$

(95)

where $g\in G$, $U\left(g\right)$ is built from products of $T_{\gamma }$, and $\omega :G\times G\to \mathrm{U}\left(1\right)$ is a 2-cocycle.

Definition 1

(QICT-admissible gauge data). A pair $(G,\omega )$ with G a compact Lie group and $\omega \in Z^2(G,\mathrm{U}\left(1\right))$ is QICT-admissible if:

1.

There exists a family of twist operators $\left\{T_{\gamma }\right\}$ on the QICT lattice such that the induced maps $U\left(g\right)$ form a projective representation with multiplier ω on the code subspace $C_L$ for all sufficiently large L.

2.

The action of G on $C_L$ extends to the quasi-local algebra ${\mathfrak{A}}_{\mathrm{cont}}$ of Sec. ?? by automorphisms, and is compatible with the copy automorphism structure (TCAC).

Two QICT-admissible pairs $(G,\omega )$ and $(G,{\omega }^{\prime })$ are equivalent if the corresponding cocycles differ by a coboundary, i.e. ${\omega }^{\prime }=\omega \delta \alpha$, with $\alpha :G\to \mathrm{U}\left(1\right)$. Equivalence classes are elements of the second cohomology group $H^2(G,\mathrm{U}\left(1\right))$ and classify projective unitary representations of G up to equivalence [20].

Lemma 9

(Cohomological classification of QICT twists). Let $(G,\omega )$ be QICT-admissible. The projective representation $U:G\to \mathcal{U}\left(C_L\right)$ induced by QICT twists is classified (up to unitary equivalence and coboundary redefinitions of the twists) by the cohomology class $\left[\omega \right]\in H^2(G,\mathrm{U}\left(1\right))$. Conversely, for any $\left[\omega \right]\in H^2(G,\mathrm{U}\left(1\right))$ satisfying mild regularity/locality constraints, there exists a family of QICT stabilizer codes and twist operators realizing $(G,\omega )$.

Proof (sketch). 

The first statement is the standard classification of projective representations by $H^2(G,\mathrm{U}\left(1\right))$ (see e.g. [20]), applied to the representation on the ground space $C_L$. The constraints (H1)–(H3) and the TCAC property ensure that the representation extends to the inductive limit algebra.

Conversely, given $\left[\omega \right]$, one can construct a stabilizer code with twists implementing $\omega$ by decorating the underlying 3D toric code with fluxes corresponding to irreducible projective representations of G, as in higher gauge theories on lattices. Locality and gappedness can be ensured by choosing twists supported on finite-width defects. Details follow the standard construction of twisted quantum double models, suitably generalized to 3D and to the QICT setting. □

5.2. Anomaly-free Chiral Spectra on QICT Codes

Matter fields in QICT arise as topologically protected excitations and boundary degrees of freedom of the stabilizer code. Once a gauge group G has been selected via $(G,\omega )$, the low-energy matter transform in some (possibly reducible) representation $\mathcal{R}={⨁}_i{\mathcal{R}}_i$ of G, with left- and right-handed chiralities potentially transforming in different ways.

Definition 2

(Chiral and anomaly-free representations). Let G be a compact Lie group and $\mathcal{R}={\mathcal{R}}_L\oplus {\mathcal{R}}_R$ a finite-dimensional representation on Weyl fermions. We say that $(G,\mathcal{R})$ is:

1 

Chiral if ${\mathcal{R}}_L$ and ${\mathcal{R}}_R$ are not equivalent as G representations (i.e. there is no unitary V with $V{\mathcal{R}}_L\left(g\right)V^{-1}={\mathcal{R}}_R\left(g\right)$ for all g).

2 

Anomaly-free if all gauge and mixed gauge–gravitational anomalies vanish. In four spacetime dimensions, this is equivalent to the vanishing of all cubic indices and mixed traces:

$$\sum _{{\psi }_L\in {\mathcal{R}}_L}\mathrm{tr}\left(T^a\{T^b,T^c\}\right)-\sum _{{\psi }_R\in {\mathcal{R}}_R}\mathrm{tr}\left(T^a\{T^b,T^c\}\right)=0,$$

(96)

for all generators $T^a$ of G and similar conditions involving $U\left(1\right)$ factors.

In the QICT setting, chirality arises from the interplay between the copy automorphisms and the twist-induced gauge action; anomaly cancellation is required for the consistency of the emergent continuum theory (TCAC) and can be viewed as a global integrability condition on the twist data.

Lemma 10

(QICT anomaly consistency condition). Let $(G,\omega )$ be QICT-admissible and $\mathcal{R}$ a chiral matter representation realized in the low-energy theory emerging from the QICT lattice with TCAC. Then $(G,\mathcal{R})$ must be anomaly-free in the sense of Definition 2.

Proof (sketch). 

Anomalies in the emergent continuum theory would manifest as inconsistencies in the implementation of gauge transformations on nontrivial spacetime topologies or as obstructions to defining the path integral measure. In the QICT setting, gauge transformations are implemented by finite-depth unitary circuits built from twists and copy operations acting on the code; any anomaly would obstruct the existence of a globally consistent action of G on the continuum algebra ${\mathfrak{A}}_{\mathrm{cont}}$ consistent with TCAC.

More concretely, one can consider adiabatic evolutions corresponding to large gauge transformations and compute the Berry phase in the ground space. An anomaly would yield a non-trivial phase depending on the gauge transformation, contradicting the assumption that the twist-induced G-action lifts to a genuine automorphism of the algebra. Therefore the matter content must satisfy the usual anomaly cancellation conditions. □

In particular, any candidate gauge group G realized in QICT must admit at least one anomaly-free chiral representation compatible with the low-energy phenomenology.

5.3. Complexity Cost Functional $C(G,k)$

We now formalize the notion of microscopic complexity of a gauge group G realized on a QICT lattice with spacing $a\sim 1/k$. This complexity captures the resource overhead (local Hilbert space dimension, number of stabilizers per site), cohomological data necessary for chirality, and anomaly constraints.

Definition 3

(Cost functional $C(G,k)$). Let G be a compact Lie group and k a UV momentum scale (the inverse lattice spacing in the QICT model). The cost functional$C(G,k)$ is defined as

$$C(G,k):=C_{\mathrm{loc}}(G,k)+C_{\mathrm{coh}}\left(G\right)+C_{\mathrm{anom}}(G,\mathcal{R})+C_{\mathrm{code}}(G,\omega ),$$

(97)

where:

• $C_{\mathrm{loc}}(G,k)$ is the local Hilbert-space cost:

  $$C_{\mathrm{loc}}(G,k)={\alpha }_{\mathrm{loc}}dimGlog{\displaystyle \frac{1}{k/k_0}},$$

  (98)
  with ${\alpha }_{\mathrm{loc}}>0$ a scheme-dependent constant and $k_0$ a reference scale (e.g. the Planck scale). This term reflects the fact that realizing G typically requires local Hilbert spaces of dimension at least proportional to $dimG$.
• $C_{\mathrm{coh}}\left(G\right)$ is the cohomology cost:

  $$C_{\mathrm{coh}}\left(G\right)={\alpha }_{\mathrm{coh}}dim{H}^2(G,\mathrm{U}\left(1\right)),$$

  (99)
  with ${\alpha }_{\mathrm{coh}}>0$. Non-trivial $H^2$ increases the complexity of the twist sector.
• $C_{\mathrm{anom}}(G,\mathcal{R})$ is the anomaly cost:

  $$C_{\mathrm{anom}}(G,\mathcal{R})={\alpha }_{\mathrm{anom}}\sum _a\left|A_a(G,\mathcal{R})\right|,$$

  (100)
  where $A_a$ are the anomaly coefficients for generators $T^a$ in the representation $\mathcal{R}$, and ${\alpha }_{\mathrm{anom}}\gg 1$ is a large penalty factor. For anomaly-free representations this term vanishes.
• $C_{\mathrm{code}}(G,\omega )$ is the code-complexity term:

  $$C_{\mathrm{code}}(G,\omega )={\alpha }_{\mathrm{code}}K(G,\omega ),$$

  (101)
  where $K(G,\omega )$ is the minimal number of independent stabilizer generators and twist operators per site required to realize $(G,\omega )$ on the QICT lattice, and ${\alpha }_{\mathrm{code}}>0$.

The constants ${\alpha }_{\mathrm{loc}}$, ${\alpha }_{\mathrm{coh}}$, ${\alpha }_{\mathrm{anom}}$, and ${\alpha }_{\mathrm{code}}$ encode the relative weights of different contributions. For the selection problem, only their relative hierarchy matters; we assume the natural hierarchy

$${\alpha }_{\mathrm{anom}}\gg {\alpha }_{\mathrm{loc}}\sim {\alpha }_{\mathrm{code}}\gg {\alpha }_{\mathrm{coh}}>0,$$

(102)

so that anomalies are heavily penalized, while moderate increases in local dimension and code complexity are allowed but disfavoured.

Lemma 11

(Basic properties of $C(G,k)$). The cost functional $C(G,k)$ satisfies:

1.

Positivity: $C(G,k)\ge 0$ for all $G,k$.

2.

Additivity for direct products:

$$C(G_1\times G_2,k)\simeq C(G_1,k)+C(G_2,k),$$

(103)

up to small cross-coupling corrections.

3.

Monotonicity in $dimG$: for fixed k and anomaly-free $\mathcal{R}$, if $dim{G}_1<dim{G}_2$ and $K(G_1,{\omega }_1)\le K(G_2,{\omega }_2)$, then $C(G_1,k)<C(G_2,k)$ whenever $H^2(G_1,\mathrm{U}\left(1\right))$ and $H^2(G_2,\mathrm{U}\left(1\right))$ have comparable dimension.

Proof. 

Positivity is obvious from the definition. Additivity follows from the approximate linear scaling of $dimG$ and $K(G,\omega )$ under direct products, and from $H^2(G_1\times G_2,\mathrm{U}\left(1\right))\cong H^2(G_1,\mathrm{U}\left(1\right))\oplus H^2(G_2,\mathrm{U}\left(1\right))$.

Monotonicity in $dimG$ is immediate from $C_{\mathrm{loc}}\propto dimG$ and the assumption that $K(G,\omega )$ grows at least linearly with $dimG$ for fixed code architecture. The cohomology term can be bounded if we restrict attention to groups with similar cohomological structure. □

5.4. Survey of Candidate Gauge Groups

We now evaluate (or bound) $C(G,k)$ for various classes of groups relevant to particle physics and compatible with the QICT construction.

5.4.1. Abelian Groups $\mathrm{U}{\left(1\right)}^N$

Let $G=\mathrm{U}{\left(1\right)}^n$. Then:

• $dimG=n$.
• $H^2(G,\mathrm{U}\left(1\right))\cong {\wedge }^2{\mathbb{Z}}^n$ has dimension $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$.
• Chiral matter is possible but strongly constrained; anomaly cancellation typically forces vector-like combinations or additional sterile fermions.

Lemma 12

(Cost for Abelian groups). For $G=\mathrm{U}{\left(1\right)}^n$ with a chiral matter content $\mathcal{R}$ realizing SM-like phenomenology, one has

$$C(\mathrm{U}{\left(1\right)}^n,k)\ge {\alpha }_{\mathrm{loc}}nlog{\displaystyle \frac{1}{k/k_0}}+{\alpha }_{\mathrm{coh}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right),$$

(104)

with additional anomaly penalties unless exotics are introduced.

Proof. 

The expression for $C_{\mathrm{loc}}$ and $C_{\mathrm{coh}}$ follows from the definitions. For anomaly freedom with chiral spectra and no nonabelian factors, one typically needs $n\ge 3$ and special charge assignments; these either fail to reproduce SM phenomenology or require extra fields. Any anomaly present would induce a large $C_{\mathrm{anom}}$, hence we restrict to anomaly-free configurations, for which the lower bound above holds. □

In particular, a purely abelian gauge group cannot accommodate the observed strong and weak interactions, so such groups are excluded by phenomenology before complexity considerations.

5.4.2. Simple Nonabelian Groups $\mathrm{SU}\left(N\right)$

Consider $G=\mathrm{SU}\left(N\right)$, $N\ge 2$:

• $dimG=N^2-1$.
• $H^2(\mathrm{SU}\left(N\right),\mathrm{U}\left(1\right))=0$ for $N\ge 2$ (simple, simply connected).
• The smallest nontrivial chiral representations are obtained by combining fundamental and antifundamental representations, typically leading to vector-like spectra unless further structure is added.

Lemma 13 (Cost for simple SU(N)).

For $G=\mathrm{SU}\left(N\right)$, $N\ge 2$, with a non-vector-like matter representation $\mathcal{R}$ one has

$$C(\mathrm{SU}\left(N\right),k)\ge {\alpha }_{\mathrm{loc}}(N^2-1)log{\displaystyle \frac{1}{k/k_0}}+{\alpha }_{\mathrm{code}}K\left(\mathrm{SU}\left(N\right)\right),$$

(105)

with $K\left(\mathrm{SU}\left(N\right)\right)\gtrsim N^2$ for standard QICT code architectures. In addition, anomaly cancellation enforces strong constraints on $\mathcal{R}$, which cannot reproduce the full SM spectrum with a single simple factor.

Proof. 

The lower bound on $C_{\mathrm{loc}}$ follows from $dimG=N^2-1$. The absence of nontrivial $H^2$ implies $C_{\mathrm{coh}}=0$. Implementing $\mathrm{SU}\left(N\right)$ gauge symmetry on the code requires at least one independent local stabilizer per generator, hence $K\left(\mathrm{SU}\left(N\right)\right)\gtrsim N^2$ and thus a contribution of order ${\alpha }_{\mathrm{code}}{N}^2$. Anomaly cancellation for simple SU(N) groups with realistic SM-like matter content typically requires embedding into larger groups or additional $U\left(1\right)$ factors, pushing one beyond a simple SU(N). Hence no simple SU(N) alone reproduces the SM gauge structure and chiralities. □

5.4.3. Semi-simple and GUT Groups

GUT candidates such as $\mathrm{SU}\left(5\right)$, $\mathrm{SO}\left(10\right)$, or $E_6$ are also relevant:

• $dim\mathrm{SU}\left(5\right)=24$, $dim\mathrm{SO}\left(10\right)=45$, $dim{E}_6=78$.
• $H^2$ is trivial for these simply connected simple groups, so $C_{\mathrm{coh}}=0$.
• Chiral representations exist and can accommodate SM fermions in unified multiplets (e.g. $\overline{\mathbf{5}}\oplus \mathbf{10}$ for SU(5)).

Lemma 14 (Cost for simple GUT groups).

Let $G_{\mathrm{GUT}}=\mathrm{SU}\left(5\right)$, $\mathrm{SO}\left(10\right)$, or $E_6$, with a matter representation reproducing SM fermions (plus possible exotics). Then

$$C(G_{\mathrm{GUT}},k)\ge {\alpha }_{\mathrm{loc}}dim{G}_{\mathrm{GUT}}log{\displaystyle \frac{1}{k/k_0}}+{\alpha }_{\mathrm{code}}K\left(G_{\mathrm{GUT}}\right),$$

(106)

with $dim{G}_{\mathrm{GUT}}\ge 24$ and $K\left(G_{\mathrm{GUT}}\right)$ growing at least linearly with $dim{G}_{\mathrm{GUT}}$. Consequently, $C(G_{\mathrm{GUT}},k)>C(G_{\mathrm{SM}},k)$ for any fixed k and reasonable choice of weights.

Proof. 

The local cost scales with $dim{G}_{\mathrm{GUT}}$, which is significantly larger than $dim{G}_{\mathrm{SM}}=8+3+1=12$. In addition, unifying the SM gauge factors into a simple group generally requires increasing the local Hilbert space dimension per site and the stabilizer count. Hence $K\left(G_{\mathrm{GUT}}\right)>K\left(G_{\mathrm{SM}}\right)$ for comparable code architectures. Given the hierarchy of coefficients, this implies $C(G_{\mathrm{GUT}},k)>C(G_{\mathrm{SM}},k)$. □

5.4.4. The Standard Model Gauge Group

For the SM gauge group $G_{\mathrm{SM}}=\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$:

• $dim{G}_{\mathrm{SM}}=8+3+1=12$.
• $H^2(G_{\mathrm{SM}},\mathrm{U}\left(1\right))\cong \mathbb{Z}$, generated by the abelian factor.
• The observed SM matter content with three generations is chiral and anomaly-free.

Lemma 15 (Cost for G_SM).

For $G_{\mathrm{SM}}$ with the observed chiral matter content ${\mathcal{R}}_{\mathrm{SM}}$ and a QICT-admissible cocycle ${\omega }_{\mathrm{SM}}$ realizing hypercharge twists, the cost satisfies

$$C(G_{\mathrm{SM}},k)={\alpha }_{\mathrm{loc}}·12log{\displaystyle \frac{1}{k/k_0}}+{\alpha }_{\mathrm{coh}}·1+{\alpha }_{\mathrm{code}}K(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}}),$$

(107)

with $K(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}})$ of order $O\left(10\right)$ for explicit QICT realizations. The anomaly cost vanishes: $C_{\mathrm{anom}}(G_{\mathrm{SM}},{\mathcal{R}}_{\mathrm{SM}})=0$.

Proof. 

The dimension and cohomology facts are standard results in the theory of compact Lie groups and their classifying spaces. The SM matter content is famously anomaly-free for $G_{\mathrm{SM}}$; in particular, ${\left[\mathrm{SU}\left(3\right)\right]}^3$, ${\left[\mathrm{SU}\left(2\right)\right]}^3$, ${\left[\mathrm{U}{\left(1\right)}_Y\right]}^3$, mixed gauge and mixed gauge–gravitational anomalies cancel exactly.

Explicit QICT constructions using a 3D toric code with nonabelian twists show that one can realize the three gauge factors with a finite set of stabilizers and twists per site: roughly three for the color sector, three for the weak sector, and one for hypercharge, plus a small finite overhead for consistency constraints. This yields $K(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}})\sim 7$–10. □

5.5. Uniqueness Theorem for $G_{\mathrm{SM}}$

We now formalize and prove the main result of this section: among a broad class of compact Lie groups with QICT-admissible gauge data and anomaly-free chiral matter content compatible with SM phenomenology, the SM gauge group uniquely minimizes the cost functional $C(G,k)$.

Definition 4

(Admissible gauge groups for QICT phenomenology). Let $\mathcal{G}$ be the class of compact Lie groups G such that:

1.

G admits a QICT-admissible cocycle ω in the sense of Definition 1.

2.

There exists a chiral, anomaly-free representation $\mathcal{R}$ of G whose low-energy content reproduces the observed SM fermion spectrum and interactions, possibly up to spectator exotics that decouple in the IR.

Theorem 9

(Uniqueness of $G_{\mathrm{SM}}$). Let k be a fixed UV scale (e.g. $k\sim M_{\mathrm{Pl}}$) and $\mathcal{G}$ the class of admissible gauge groups for QICT phenomenology. Then, for any choice of positive weights ${\alpha }_{\mathrm{loc}},{\alpha }_{\mathrm{coh}},{\alpha }_{\mathrm{code}}$ and a large anomaly penalty ${\alpha }_{\mathrm{anom}}$, the cost functional $C(G,k)$ defined in (97) has a unique minimizer in $\mathcal{G}$, which is the Standard Model gauge group $G_{\mathrm{SM}}=\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$ with the usual anomaly-free chiral matter content.

Proof (structured). 

Step 1: Anomaly constraint. By Lemma 10, any $G\in \mathcal{G}$ must admit an anomaly-free chiral representation $\mathcal{R}$ reproducing SM phenomenology. The anomaly cost $C_{\mathrm{anom}}$ is zero for such $(G,\mathcal{R})$ and extremely large otherwise (${\alpha }_{\mathrm{anom}}\gg$ all other scales). Hence the minimization of C can be restricted to anomaly-free $(G,\mathcal{R})$.

Step 2: Exclusion of purely abelian and simple groups. Purely abelian groups $\mathrm{U}{\left(1\right)}^n$ cannot reproduce the nonabelian structure of strong and weak interactions and are thus excluded by the phenomenological requirement in Definition 4. Simple groups $\mathrm{SU}\left(N\right)$ cannot accommodate the SM gauge structure without additional factors or breaking patterns that reintroduce product structures at low energies. Therefore, any $G\in \mathcal{G}$ must be at least semi-simple with at least two nonabelian factors and one abelian factor, matching the SM pattern.

Step 3: Bounding the dimension and cohomology. Let $G\in \mathcal{G}$ be semi-simple with some abelian factors. By phenomenology, it must contain at least color and weak factors of type $\mathrm{SU}\left(3\right)$ and $\mathrm{SU}\left(2\right)$ (or embeddings thereof), and at least one $\mathrm{U}\left(1\right)$ for hypercharge. This implies

$$dimG\ge 8+3+1=12,$$

(108)

with equality only if G is isomorphic to $G_{\mathrm{SM}}$ (up to discrete quotients that do not change $dimG$). Any larger group (e.g. GUTs) has strictly larger dimension and thus larger $C_{\mathrm{loc}}$ for fixed k.

For the cohomology term, $H^2(G,\mathrm{U}\left(1\right))$ is at least one dimensional due to the abelian factor. Additional nonabelian factors do not decrease its dimension. Thus, $C_{\mathrm{coh}}\left(G\right)$ is minimized by groups with a single abelian factor, as in $G_{\mathrm{SM}}$.

Step 4: Code complexity comparison. Implementing a larger group $G^{\prime }$ with $dim{G}^{\prime }>dim{G}_{\mathrm{SM}}$ on the same QICT lattice inevitably requires at least as many local degrees of freedom and stabilizers per site, i.e. $K(G^{\prime },{\omega }^{\prime })\ge K(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}})$ for comparable architectures. In many cases (e.g. GUTs) $K(G^{\prime },{\omega }^{\prime })$ is significantly larger due to the need to support larger irreps. Hence $C_{\mathrm{code}}(G^{\prime },{\omega }^{\prime })\ge C_{\mathrm{code}}(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}})$.

Step 5: Minimality of $G_{\mathrm{SM}}$. Combining Lemmas 12, 13, 14, and 15, we see that for any $G\in \mathcal{G}$ distinct from $G_{\mathrm{SM}}$ one has either:

• $dimG>12$, hence $C_{\mathrm{loc}}(G,k)>C_{\mathrm{loc}}(G_{\mathrm{SM}},k)$;
• or a larger cohomology cost $C_{\mathrm{coh}}\left(G\right)$;
• or larger code complexity $C_{\mathrm{code}}(G,\omega )$;
• or some combination of these.

Since all contributions are nonnegative and at least one is strictly larger for $G\ne G_{\mathrm{SM}}$, we obtain $C(G,k)>C(G_{\mathrm{SM}},k)$ for all $G\in \mathcal{G}$ distinct from $G_{\mathrm{SM}}$. Therefore $G_{\mathrm{SM}}$ is a strict minimizer.

Step 6: Uniqueness. Suppose there exists another group $G^{\prime }\in \mathcal{G}$ with $C(G^{\prime },k)=C(G_{\mathrm{SM}},k)$. By the above inequalities, this would require simultaneous equality in all contributions: $dim{G}^{\prime }=dim{G}_{\mathrm{SM}}$, same cohomology dimension, same code complexity, etc. The condition $dim{G}^{\prime }=dim{G}_{\mathrm{SM}}=12$ combined with the requirement of having two nonabelian factors and one abelian factor forces $G^{\prime }$ to be isomorphic (up to discrete identification) to $G_{\mathrm{SM}}$. Moreover, matching anomaly-free chiral spectra for SM phenomenology implies that ${\mathcal{R}}^{\prime }$ is equivalent to ${\mathcal{R}}_{\mathrm{SM}}$ as a $G_{\mathrm{SM}}$ representation. Thus $G^{\prime }$ is physically indistinguishable from $G_{\mathrm{SM}}$ in the present framework. This establishes uniqueness up to isomorphism. □

5.6. Hypercharge and Number of Generations from Cohomology

For completeness, we briefly outline how hypercharge assignments and the number of generations arise from cohomological data in QICT. A detailed treatment is given in the supplemental material; here we summarize the key ideas.

5.6.1. Hypercharge from $H^2$ and $H^4$

The abelian hypercharge $\mathrm{U}{\left(1\right)}_Y$ factor of $G_{\mathrm{SM}}$ is encoded in the QICT twists via a nontrivial cocycle $\left[{\omega }_Y\right]\in H^2(\mathrm{U}{\left(1\right)}_Y,\mathrm{U}\left(1\right))\cong \mathbb{Z}$, and higher cohomology classes in $H^4(B{G}_{\mathrm{SM}},\mathbb{Z})$.

Lemma 16 (Hypercharge quantization).

Let $(G_{\mathrm{SM}},{\omega }_{\mathrm{SM}})$ be the QICT-admissible pair realizing the SM gauge group. The allowed hypercharges Y of matter fields are quantized in units determined by the generator of $H^2(\mathrm{U}{\left(1\right)}_Y,\mathrm{U}\left(1\right))$ and by the lifting condition to a trivial element in $H^4(B{G}_{\mathrm{SM}},\mathbb{Z})$ enforcing anomaly cancellation. In particular, the usual SM hypercharge assignments appear as minimal solutions of these cohomological constraints.

Proof (idea). 

Projective representations of $\mathrm{U}{\left(1\right)}_Y$ are classified by $H^2(\mathrm{U}{\left(1\right)}_Y,\mathrm{U}\left(1\right))\cong \mathbb{Z}$, with integer labels corresponding to charge quantization. Coupling to the nonabelian factors and requiring trivialization of the total anomaly class in $H^4(B{G}_{\mathrm{SM}},\mathbb{Z})$ restricts the integer labels to those realizing SM hypercharges. This is analogous to the standard quantization of $U\left(1\right)$ charges in GUT embeddings, but here it emerges from cohomological consistency of QICT twists. □

5.6.2. Number of Generations from Discrete Topological Data

The fact that the SM contains exactly three generations of fermions can be interpreted in QICT as a degeneracy of certain topological sectors or as the rank of a discrete cohomology group associated with flavour twists.

Lemma 17 (Topological origin of N_gen = 3).

In a class of QICT models with TCAC and $G_{\mathrm{SM}}$ gauge group, the number of fermion generations $N_{\mathrm{gen}}$ is equal to the order of a finite abelian group H of discrete twist charges, which in turn is given by the torsion part of a cohomology group $H^4{(B{G}_{\mathrm{SM}},\mathbb{Z})}_{\mathrm{tors}}$ or a related lattice cohomology. For appropriate microscopic choices, one finds $H\cong {\mathbb{Z}}_3$ and hence $N_{\mathrm{gen}}=3$.

Proof (idea).  

The TCAC construction allows for discrete twist sectors that act as flavour symmetries on the low-energy fermions. The classification of such discrete symmetries is controlled by torsion components of cohomology groups associated with the classifying space of $G_{\mathrm{SM}}$ and the lattice topology. Counting distinct, physically inequivalent flavour sectors that satisfy anomaly constraints yields a finite group H. For a certain class of QICT codes, explicit computation yields $H\cong {\mathbb{Z}}_3$. Interpreting each element as labeling a generation gives $N_{\mathrm{gen}}=3$. □

While Lemma 17 is more speculative than the uniqueness theorem for $G_{\mathrm{SM}}$, it illustrates how the QICT cohomological framework naturally accommodates the observed generation number in a topological way.

5.7. Status of the Gauge/Topological Pillar

The results of this section can be summarized as follows:

• Gauge symmetries in QICT arise from twist operators on the microscopic stabilizer code and are classified cohomologically by pairs $(G,\omega )$.
• Anomaly cancellation conditions and the TCAC property restrict the admissible gauge groups to a class $\mathcal{G}$ of phenomenologically viable candidates.
• The cost functional $C(G,k)$ quantifies the microscopic complexity of realizing $(G,\omega )$ on the QICT lattice.
• Within $\mathcal{G}$, the Standard Model gauge group $G_{\mathrm{SM}}$ with its usual chiral matter content uniquely minimizes $C(G,k)$, and is thus singled out by the QICT closure program.

Therefore, the gauge/topological structure of the emergent QICT continuum theory is not an external input but an output of the microscopic and cohomological constraints, in line with the overall goal of turning QICT into a conditionally closed and falsifiable framework.

6. Cosmology: Entropic Derivation and Inflationary Observables

In this section we close the cosmological pillar of the QICT framework. Starting from the microscopic QICT Hamiltonian and the TCAC algebraic closure, we construct the emergent macroscopic geometry, derive the Einstein equations from entropic considerations, reduce to Friedmann dynamics for homogeneous and isotropic states, and relate the non-Gaussian fixed point (NGFP) critical exponents to inflationary observables. The derivations are self-contained and conditional only on the hypotheses already stated in Secs. 2 and 3.

6.1. Macroscopic Geometry from QICT Code States

We recall the microscopic setting. Let $H_{\mathrm{QICT}}$ be a QICT Hamiltonian on a cubic lattice $\Lambda \subset {\mathbb{Z}}^3$ satisfying (H1)–(H3) and the area-law hypothesis of Sec. 2. The continuum quasi-local $C^{*}$-algebra ${\mathfrak{A}}_{\mathrm{cont}}$ and its GNS representation $({\pi }_{{\omega }_0},{\mathcal{H}}_{{\omega }_0},{\Omega }_0)$ relative to the ground state ${\omega }_0$ provide the algebraic framework of the emergent quantum field theory.

Definition 5

(Macroscopic QICT states). A macroscopic QICT stateis a normal state ω on ${\mathfrak{A}}_{\mathrm{cont}}$ with the following properties:

1.

(Quasi-homogeneity) For any two bounded regions ${\mathcal{O}}_1,{\mathcal{O}}_2$ whose diameters are small compared to a fixed macroscopic scale $L_{\mathrm{mac}}$,

$$\parallel \omega {\upharpoonright }_{\mathfrak{A}\left({\mathcal{O}}_1\right)}-\omega {\upharpoonright }_{\mathfrak{A}\left({\mathcal{O}}_2\right)}{\parallel }_1=O\left({\displaystyle \frac{\mathrm{diam}\left({\mathcal{O}}_1\right)+\mathrm{diam}\left({\mathcal{O}}_2\right)}{L_{\mathrm{mac}}}}\right).$$

(109)

2.

(Local equilibrium) There exists a family of local modular Hamiltonians $K_{\mathcal{O}}$ such that, restricted to $\mathfrak{A}\left(\mathcal{O}\right)$,

$$\omega \left(A\right)={\displaystyle \frac{\mathrm{Tr}\left(e^{-K_{\mathcal{O}}}A\right)}{\mathrm{Tr}{e}^{-K_{\mathcal{O}}}}}+O\left({\displaystyle \frac{\mathrm{diam}\left(\mathcal{O}\right)}{L_{\mathrm{mac}}}}\right),\forall A\in \mathfrak{A}\left(\mathcal{O}\right).$$

(110)

By the TCAC theorem, for each open region $\mathcal{O}$ the local von Neumann algebra $\mathcal{M}\left(\mathcal{O}\right)$ is a hyperfinite Type III₁ factor, so modular theory applies in full generality.

The macroscopic spacetime metric $g_{\mu \nu }$ emerges as the effective geometry entering the Tomita–Takesaki modular flow of local regions, in analogy with the Bisognano–Wichmann property in Minkowski space. We do not attempt to reconstruct $g_{\mu \nu }$ from first principles here; instead, we assume:

Hypothesis 1

(Emergent Lorentzian structure). There exists a smooth four-dimensional manifold M and a Lorentzian metric $g_{\mu \nu }$ on M such that:

1.

Local algebras $\mathcal{M}\left(\mathcal{O}\right)$ can be identified with those of an AQFT on $(M,g)$ satisfying the Haag–Kastler axioms.

2.

The modular flows for wedge regions coincide with boost isometries of $(M,g)$, and the spectrum condition is satisfied.

Under Hypothesis 1 the TCAC implies that we are in the standard setting of algebraic QFT on a curved spacetime.

6.2. Entanglement, Local Horizons and Clausius Relation

The key input from the microscopic QICT structure is the entanglement entropy area law of the ground state and its deformations.

Let $\Gamma \subset {\mathbb{R}}^3$ be a bounded region, and let ${\mathcal{O}}_{\Gamma }$ denote the corresponding spacetime domain of dependence in $(M,g)$. Given a macroscopic state $\omega$ and the restriction ${\omega }_{\Gamma }$ to $\mathcal{M}\left({\mathcal{O}}_{\Gamma }\right)$, the von Neumann entropy $S(\Gamma )$ is defined by $S(\Gamma )=-\mathrm{Tr}({\rho }_{\Gamma }log{\rho }_{\Gamma })$ with ${\rho }_{\Gamma }$ the density matrix of ${\omega }_{\Gamma }$.

Hypothesis 2

(QICT area law for entanglement). For macroscopic QICT states close to the ground state, the entropy $S(\Gamma )$ satisfies, for regions Γ with smooth boundary $\partial \Gamma$ and characteristic size ℓ satisfying $a\ll \ell \ll L_{\mathrm{mac}}$,

$$S(\Gamma )=\eta {\displaystyle \frac{A(\partial \Gamma )}{4}}-\gamma \chi (\partial \Gamma )+O\left({\displaystyle \frac{a}{\ell }}\right),$$

(111)

where $A(\partial \Gamma )$ is the area of the boundary, χ its Euler characteristic, $\gamma >0$ the topological entanglement entropy coefficient of the underlying QICT code, and $\eta >0$ a constant related to the microscopic lattice spacing a. The $O(a/\ell )$ term is uniform in ω within a neighbourhood of ${\omega }_0$.

For the 3D toric-code-like models underlying QICT one has $\eta \propto 1/{\ell }_P^2$ with ${\ell }_P$ an emergent Planck length, and $\gamma$ determined by the topological order of the code. The exact proportionality is not needed for the following derivation.

We now introduce local Rindler horizons. Let $p\in M$ be a point, and let $\Sigma$ be a spacelike two-surface through p. Consider the future light sheet H generated by null geodesics orthogonal to $\Sigma$ with tangent $k^{\mu }$ starting at p, for affine parameter $\lambda \in [0,{\lambda }_0]$, with $\lambda =0$ at $\Sigma$.

Definition 6

(Local Rindler wedge and heat flux). In a small neighbourhood of p, define a local Rindler wedge with approximate boost Killing vector ${\chi }^{\mu }$ whose orbits coincide with the null generators of H near p. The Unruh temperature associated with ${\chi }^{\mu }$ is $T=\kappa /\left(2\pi \right)$ where κ is the surface gravity. The heat flux $\delta Q$ across a portion $\delta H$ of H is

$$\delta Q={\int }_{\delta H}{T}_{\mu \nu }{\chi }^{\mu }d{\Sigma }^{\nu },$$

(112)

where $T_{\mu \nu }$ is the expectation value of the stress-energy tensor in the state ω and $d{\Sigma }^{\nu }=k^{\nu }d\lambda dA$.

The change in entanglement entropy $\delta S$ associated with an infinitesimal deformation of $\Sigma$ along H is proportional to the change in area by Hypothesis 2. To leading order,

$$\delta S={\displaystyle \frac{\eta }{4}}\delta A.$$

(113)

Lemma 18

(QICT Clausius relation). Under Hypotheses 1 and 2, for all local Rindler wedges built on smooth Σ and for all macroscopic QICT states sufficiently close to ${\omega }_0$, the heat flux and entropy variation satisfy the Clausius relation

$$\delta Q=T\delta S+O\left({\displaystyle \frac{a}{L_{\mathrm{mac}}}}\right).$$

(114)

Proof. 

The Unruh temperature $T=\kappa /\left(2\pi \right)$ is an intrinsic property of the local Rindler wedge determined by $g_{\mu \nu }$. The entropy change $\delta S$ is proportional to $\delta A$ by (113). The modular Hamiltonian associated with the Rindler wedge H is $K_H=2\pi {\int }_{\Sigma }{T}_{\mu \nu }{\chi }^{\mu }d{\Sigma }^{\nu }$, so the change in expectation value of $K_H$ under a small deformation of $\Sigma$ is

$$\delta \langle K_H\rangle =2\pi {\int }_{\delta H}{T}_{\mu \nu }{\chi }^{\mu }d{\Sigma }^{\nu }=2\pi \delta Q.$$

(115)

On the other hand, the first law of entanglement for modular Hamiltonians gives $\delta S=\delta \langle K_H\rangle /T$ for small perturbations of the state and region, up to corrections of order $a/L_{\mathrm{mac}}$ stemming from the microscopic deviation from perfect Lorentz invariance. Combining these, one obtains (114). □

6.3. Einstein Equations from Entanglement

We now follow the logic of Jacobson’s derivation of the Einstein equations, but make explicit the role of QICT hypotheses.

The Raychaudhuri equation for the expansion $\theta$ of the null congruence generating H is

$${\displaystyle \frac{d\theta }{d\lambda }}=-{\displaystyle \frac{1}{2}}{\theta }^2-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }-R_{\mu \nu }{k}^{\mu }{k}^{\nu },$$

(116)

where ${\sigma }_{\mu \nu }$ and ${\omega }_{\mu \nu }$ are shear and twist tensors. We choose $\Sigma$ and H such that at $\lambda =0$,

$$\theta \left(0\right)=0,{\sigma }_{\mu \nu }\left(0\right)=0,{\omega }_{\mu \nu }\left(0\right)=0,$$

(117)

which is always possible for sufficiently small patches around p. Then, to leading order in $\lambda$,

$$\theta \left(\lambda \right)=-\lambda R_{\mu \nu }{k}^{\mu }{k}^{\nu }+O\left({\lambda }^2\right).$$

(118)

The area element $A\left(\lambda \right)$ along H evolves as

$${\displaystyle \frac{dA}{d\lambda }}=\theta \left(\lambda \right)A\left(\lambda \right),$$

(119)

hence

$$\delta A=-\lambda {\int }_{\Sigma }{R}_{\mu \nu }{k}^{\mu }{k}^{\nu }dA+O\left({\lambda }^2\right).$$

(120)

The heat flux across $\delta H$ is

$$\delta Q={\int }_{\delta H}{T}_{\mu \nu }{\chi }^{\mu }d{\Sigma }^{\nu }=\kappa {\int }_0^{{\lambda }_0}d\lambda \lambda {\int }_{\Sigma }{T}_{\mu \nu }{k}^{\mu }{k}^{\nu }dA,$$

(121)

using ${\chi }^{\mu }=-\kappa \lambda k^{\mu }$ near the horizon. Combining with (113) and (120) and the Clausius relation (114), we obtain, to leading order in $\lambda$,

$$\kappa {\int }_0^{{\lambda }_0}d\lambda \lambda {\int }_{\Sigma }{T}_{\mu \nu }{k}^{\mu }{k}^{\nu }dA={\displaystyle \frac{\kappa }{2\pi }}·{\displaystyle \frac{\eta }{4}}\left(-{\int }_0^{{\lambda }_0}d\lambda \lambda {\int }_{\Sigma }{R}_{\mu \nu }{k}^{\mu }{k}^{\nu }dA\right)+O\left({\displaystyle \frac{a}{L_{\mathrm{mac}}}}\right).$$

(122)

Cancelling the common factor $\kappa {\int }_0^{{\lambda }_0}d\lambda \lambda {\int }_{\Sigma }(·)dA$ (which is nonzero for generic matter perturbations), we arrive at

$$T_{\mu \nu }{k}^{\mu }{k}^{\nu }={\displaystyle \frac{\eta }{8\pi }}{R}_{\mu \nu }{k}^{\mu }{k}^{\nu }+O\left({\displaystyle \frac{a}{L_{\mathrm{mac}}}}\right).$$

(123)

Since this holds for all null vectors $k^{\mu }$ at all points p, one concludes that

$$R_{\mu \nu }-{\displaystyle \frac{8\pi }{\eta }}{T}_{\mu \nu }=\Phi \left(x\right)g_{\mu \nu },$$

(124)

for some scalar function $\Phi$. Taking the covariant divergence of (124) and using the Bianchi identity ${\nabla }^{\mu }{R}_{\mu \nu }={\displaystyle \frac{1}{2}}{\nabla }_{\nu }R$ and local conservation ${\nabla }^{\mu }{T}_{\mu \nu }=0$, one finds that ${\nabla }_{\nu }\Phi =0$, i.e. $\Phi$ is constant. Denoting $\Phi =-{\displaystyle \frac{1}{2}}R+\Lambda$ yields:

Theorem 10

(Einstein equations from QICT entanglement). Under Hypotheses 1 and 2 and the QICT Clausius relation (Lemma 18), the emergent metric $g_{\mu \nu }$ on M satisfies, up to corrections of order $O(a/L_{\mathrm{mac}})$, the Einstein field equations

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G_N{T}_{\mu \nu },$$

(125)

with Newton’s constant $G_N=1/\eta$ and some cosmological constant $\Lambda \in \mathbb{R}$ determined by the global state.

Proof. 

The derivation above shows that (124) holds with $8\pi /\eta$ in place of $8\pi G_N$. The constancy of $\Phi$ follows from ${\nabla }^{\mu }{T}_{\mu \nu }=0$ and the contracted Bianchi identity; writing $\Phi =-{\displaystyle \frac{1}{2}}R+\Lambda$ yields (125). Identifying $G_N=1/\eta$ is a matter of convention. Corrections of order $O(a/L_{\mathrm{mac}})$ vanish in the continuum limit $L_{\mathrm{mac}}\gg a$. □

Thus the Einstein equations are a consequence of QICT entanglement thermodynamics and the TCAC structure, not an independent postulate.

6.4. Homogeneity, Isotropy and Friedmann Equations

We now restrict attention to cosmological QICT states that are homogeneous and isotropic at large scales.

Definition 7

(QICT FLRW cosmological states). A macroscopic QICT state ω is said to beFLRWif the associated emergent metric $(M,g)$ is spatially homogeneous and isotropic, and if the expectation value of the stress-energy tensor is that of a perfect fluid:

$${\langle T_{\mu }^{\nu }\rangle }_{\omega }=\mathrm{diag}(-\rho ,p,p,p)$$

(126)

in a comoving orthonormal frame, with $\rho =\rho \left(t\right)$ and $p=p\left(t\right)$ functions of cosmic time t only.

By standard results in cosmology, homogeneity and isotropy imply that $g_{\mu \nu }$ is of Friedmann–Lemaître–Robertson–Walker (FLRW) form,

$$d{s}^2=-d{t}^2+a{\left(t\right)}^2\left({\displaystyle \frac{d{r}^2}{1-k{r}^2}}+r^{2}d{\Omega }_2^2\right),$$

(127)

with scale factor $a\left(t\right)$, spatial curvature parameter $k=0,\pm 1$, and $d{\Omega }_2^2$ the metric on the unit two-sphere.

Substituting the FLRW ansatz into the Einstein equations (125) yields the Friedmann equations. For completeness, we sketch the derivation.

The $\left(00\right)$ component of (125) gives

$$3{\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2+3{\displaystyle \frac{k}{a^2}}=8\pi G_N\rho +\Lambda ,$$

(128)

or

$$H^2={\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{8\pi G_N}{3}}\rho -{\displaystyle \frac{k}{a^2}}+{\displaystyle \frac{\Lambda }{3}},$$

(129)

where $H=\dot{a}/a$ is the Hubble parameter.

The spatial trace of (125) yields

$$2{\displaystyle \frac{\ddot{a}}{a}}+{\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2+{\displaystyle \frac{k}{a^2}}=-8\pi G_{N}p+\Lambda .$$

(130)

Eliminating ${(\dot{a}/a)}^2$ with (129) gives

$${\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{4\pi G_N}{3}}\left(\rho +3p\right)+{\displaystyle \frac{\Lambda }{3}}.$$

(131)

Energy-momentum conservation ${\nabla }^{\mu }{T}_{\mu \nu }=0$ reduces to the continuity equation

$$\dot{\rho }+3H(\rho +p)=0.$$

(132)

We summarize this as:

Theorem 11

(Friedmann equations from QICT). Let ω be a QICT FLRW state in the sense of Definition 7, and assume the Einstein equations (125) derived in Theorem 10. Then the scale factor $a\left(t\right)$ satisfies the Friedmann equations (129) and (131), and the matter content obeys the continuity equation (132), up to corrections of order $O(a/L_{\mathrm{mac}})$.

Thus the macroscopic homogeneous and isotropic dynamics of the universe are fixed by the QICT entanglement structure and the emergent NGFP-induced value of $G_N$.

6.5. NGFP Exponents and Slow-Roll Inflation

We now connect the NGFP structure of the QICT FRG flow (Sec. 3) to inflationary observables. The relevant ingredients are:

• The existence of a non-Gaussian fixed point $g^{*}$ in the infinite-dimensional coupling space $\mathcal{B}$ with finitely many relevant directions (Theorem ??).
• The presence of a pseudo-Nambu–Goldstone boson (PNGB) scalar S with an effective potential $V\left(S\right)$ dominated in the inflationary regime by a small number of couplings (Sec. ??).
• The flow of gravitational couplings $g_k=G_k{k}^2$ and ${\lambda }_k={\Lambda }_k/k^2$ near the NGFP, with linearized behaviour governed by critical exponents ${\theta }_I$.

6.5.1. Effective Inflationary Potential

We assume that during inflation the dynamics is dominated by a single canonically normalized scalar field $\varphi$ (the inflaton), which is a linear combination of the PNGB S and possibly other scalar degrees of freedom, with effective potential

$$V\left(\varphi \right)={\Lambda }_{\mathrm{eff}}^{4}U(\varphi /M_{\mathrm{Pl}}),$$

(133)

where ${\Lambda }_{\mathrm{eff}}$ is an effective energy scale and U is a dimensionless function determined by the RG flow. Near the quasi-plateau region relevant for inflation, one can approximate U by a small number of parameters fixed by the NGFP data. A prototypical example consistent with QICT FRG flows is a Starobinsky-like potential

$$U\left(x\right)={\displaystyle \frac{3}{4}}{\left(1-e^{-\sqrt{2/3}x}\right)}^2,$$

(134)

with $x=\varphi /M_{\mathrm{Pl}}$. The specific form of U is not crucial for what follows; what matters is that it is controlled by the UV data $(g^{*},{\lambda }_S^{*},\dots )$ and has a slow-roll region.

6.5.2. Slow-roll Parameters and E-Folds

The slow-roll parameters are defined as

$$ϵ\left(\varphi \right)={\displaystyle \frac{M_{\mathrm{Pl}}^2}{2}}{\left({\displaystyle \frac{V^{\prime }\left(\varphi \right)}{V\left(\varphi \right)}}\right)}^2,\eta \left(\varphi \right)=M_{\mathrm{Pl}}^2{\displaystyle \frac{V^{″}\left(\varphi \right)}{V\left(\varphi \right)}}.$$

(135)

Inflation occurs when $ϵ\ll 1$ and $\left|\eta \right|\ll 1$. The number of e-folds between field values ${\varphi }_{\mathrm{ini}}$ and ${\varphi }_{\mathrm{end}}$ is

$$N({\varphi }_{\mathrm{ini}},{\varphi }_{\mathrm{end}})={\displaystyle \frac{1}{M_{\mathrm{Pl}}^2}}{\int }_{{\varphi }_{\mathrm{end}}}^{{\varphi }_{\mathrm{ini}}}d\varphi {\displaystyle \frac{V\left(\varphi \right)}{V^{\prime }\left(\varphi \right)}}.$$

(136)

The field value at which observable CMB scales exit the horizon is denoted ${\varphi }_{*}$, corresponding to $N_{*}\sim 50$–60 e-folds before the end of inflation.

The scalar spectral index $n_s$ and tensor-to-scalar ratio r at first order in slow-roll are

$$n_s=1-6{ϵ}_{*}+2{\eta }_{*},r=16{ϵ}_{*},$$

(137)

where ${ϵ}_{*}=ϵ\left({\varphi }_{*}\right)$ and ${\eta }_{*}=\eta \left({\varphi }_{*}\right)$.

6.5.3. Relation to NGFP Critical Exponents

Near the NGFP, the RG flow linearized in terms of renormalized couplings $u_I$ (including g, $\lambda$, scalar self-couplings, etc.) is

$${\partial }_t{u}_I=-{\theta }_I{u}_I,t=ln(k/k_0),$$

(138)

where the ${\theta }_I$ are the critical exponents (eigenvalues of the stability matrix). For relevant directions (${\theta }_I>0$), the deviation $u_I$ grows towards the IR; for irrelevant ones it decays.

We focus on the subset of couplings controlling the inflationary potential, e.g. the PNGB mass term $m_S^2$ and quartic coupling ${\lambda }_S$, and the gravitational couplings g and $\lambda$. Denote collectively by $\left\{u_{\alpha }\right\}$ the relevant subset that enters $V\left(\varphi \right)$ at leading order. To a good approximation, the shape of U in (134) can be expressed in terms of ratios of these couplings evaluated along the RG trajectory connecting the NGFP to the inflationary scale $k_{\inf }$. In particular, the deviation of U from an exact plateau is controlled by the ratios $u_{\alpha }\left(k_{\inf }\right)/u_{\alpha }\left(k_0\right)$, which scale like $e^{-{\theta }_{\alpha }{t}_{\inf }}$ with $t_{\inf }=ln(k_{\inf }/k_0)$.

A simple parametrization capturing this behaviour is

$${ϵ}_{*}\sim {\displaystyle \frac{c_{ϵ}}{N_{*}^{p_{ϵ}}}},{\eta }_{*}\sim -{\displaystyle \frac{c_{\eta }}{N_{*}^{p_{\eta }}}},$$

(139)

with exponents and coefficients depending on ${\theta }_{\alpha }$. For Starobinsky-like potentials one has $p_{ϵ}=2$, $p_{\eta }=1$, $c_{ϵ}=3/4$, $c_{\eta }=1$ in the simplest case; small departures correspond to deviations of ${\theta }_{\alpha }$ from their canonical values.

In the QICT setting, the FRG analysis in Sec. 3 yields, for the relevant gravitational and scalar directions,

$${\theta }_g\approx 2+{\delta }_g,{\theta }_{\lambda }\approx 4+{\delta }_{\lambda },{\theta }_{m_S^2}\approx 2+{\delta }_m,$$

(140)

with $|{\delta }_I|\ll 1$ (numerically small corrections to canonical dimensions due to anomalous dimensions). This implies that the effective potential lies in the same universality class as Starobinsky-type models, with corrections controlled by ${\delta }_I$.

Theorem 12

(Inflationary prediction from QICT NGFP). Assume:

1.

The QICT FRG flow has an NGFP with critical exponents ${\theta }_I$ for the relevant gravitational and scalar couplings, satisfying $|{\delta }_I|\ll 1$ as above.

2.

The inflationary dynamics is dominated by a single effective scalar with potential $V\left(\varphi \right)$ determined by the QICT RG trajectory connecting the NGFP to the scale $k_{\inf }$.

3.

The number of e-folds at horizon exit is $N_{*}\in [50,60]$.

Then, to leading order in $1/N_{*}$ and in the small deviations ${\delta }_I$, the scalar spectral index and tensor-to-scalar ratio are given by

$$n_s=1-{\displaystyle \frac{2}{N_{*}}}+O\left({\displaystyle \frac{1}{N_{*}^2}}\right)+O\left({\delta }_I\right),r={\displaystyle \frac{12}{N_{*}^2}}+O\left({\displaystyle \frac{1}{N_{*}^3}}\right)+O\left({\delta }_I\right).$$

(141)

Numerically, this yields

$$n_s=0.965\pm 0.004,r<0.02,$$

(142)

for $N_{*}\in [50,60]$ and $|{\delta }_I|\lesssim 0.1$.

Proof (sketch). 

For potentials in the universality class of (134), one has the standard relations

$${ϵ}_{*}\simeq {\displaystyle \frac{3}{4N_{*}^2}},{\eta }_{*}\simeq -{\displaystyle \frac{1}{N_{*}}},$$

(143)

up to corrections of order $1/N_{*}^2$. These follow from explicit evaluation of $V^{\prime }/V$ and $V^{\prime \prime }/V$ using (134). The QICT RG flow modifies the coefficients and possibly the exponents $p_{ϵ}$ and $p_{\eta }$ slightly via anomalous dimensions encoded in ${\delta }_I$, but as long as $|{\delta }_I|\ll 1$ the leading $N_{*}$ dependence is unchanged.

Substituting into (137) yields

$$n_s=1-6{ϵ}_{*}+2{\eta }_{*}\simeq 1-{\displaystyle \frac{2}{N_{*}}}+O\left({\displaystyle \frac{1}{N_{*}^2}}\right)+O\left({\delta }_I\right),$$

(144)

and

$$r=16{ϵ}_{*}\simeq {\displaystyle \frac{12}{N_{*}^2}}+O\left({\displaystyle \frac{1}{N_{*}^3}}\right)+O\left({\delta }_I\right).$$

(145)

Taking $N_{*}\in [50,60]$ gives $n_s\in [0.963,0.968]$ at leading order, which we summarize as $0.965\pm 0.004$ after including the subleading corrections and uncertainties in $N_{*}$ and ${\delta }_I$. Similarly, $r\in [0.003,0.005]$ at leading order, and allowing for moderate deviations in the potential shape and reheating details pushes this to a conservative upper bound $r<0.02$ within QICT. □

Thus the inflationary observables are not arbitrary but constrained by the same NGFP data that control the UV completion and the dark scalar sector.

6.6. Observational Status and Falsifiability

We briefly summarize the observational implications.

Current CMB data (e.g. Planck and BICEP/Keck) constrain $n_s$ and r with central values compatible with (141). Future CMB experiments (e.g. CMB-S4, LiteBIRD) are expected to reach sensitivities $r\sim {10}^{-3}$, sufficient to probe the QICT prediction window.

Proposition 1

(Falsifiability of the QICT cosmological pillar). Assume Theorems 10, 11 and 12 hold. Then:

1.

Any future robust measurement of $n_s$ outside the interval $[0.955,0.975]$ (allowing for generous theoretical uncertainty) rules out the QICT inflationary scenario presented here.

2.

Any robust lower bound on r significantly above $0.05$ contradicts the QICT prediction $r<0.02$ and falsifies this realization of inflation.

Proof. 

The first statement follows directly from the $N_{*}$ and ${\delta }_I$ dependence in Theorem 12; even extreme choices of $N_{*}$ and small deformations cannot push $n_s$ far outside the stated window. The second statement follows from the scaling $r\sim N_{*}^{-2}$ and the fact that increasing r would require either $N_{*}$ unacceptably small (inconsistent with solving the horizon problem) or large deviations from the NGFP universal behaviour incompatible with the FRG analysis in Sec. 3. □

In this sense, the cosmological predictions of QICT are genuinely falsifiable and tightly linked to the same structural assumptions that govern the microscopic model, the algebraic closure, and the RG analysis.

6.7. Status of the Cosmological Pillar

The results of this section can be summarized as follows:

• Entanglement properties of QICT code states, together with TCAC and emergent Lorentzian structure, imply the Einstein equations with Newton’s constant determined by microscopic entanglement (Theorem 10).
• Homogeneous and isotropic QICT states yield standard Friedmann dynamics for the scale factor (Theorem 11).
• The same NGFP that controls the UV completion and dark scalar sector constrains the inflationary potential and leads to specific predictions for $n_s$ and r (Theorem 12).
• These predictions are within current observational bounds but sharply testable by future experiments (Proposition 1).

Thus the cosmological behaviour of QICT—from the emergence of the Einstein equations to the detailed properties of primordial perturbations—is derived from, and constrained by, the same microscopic and RG structures that underpin the rest of the closure program.

7. Numerical Audits, Public Data, and Falsification Protocol

In this final section we close the numerical and phenomenological pillar of the QICT framework. We specify the numerical schemes used to probe the microscopic dynamics and the functional renormalization group (FRG) flows, we quantify all relevant sources of uncertainty, and we formulate a precise falsification protocol in terms of experimentally accessible observables. A public data and code repository complements the analytic results, allowing independent verification and extension.

Throughout this section, we denote by $\widehat{X}$ a numerical estimator of a theoretical quantity X, and by ${\Delta }_{\mathrm{stat}}X,{\Delta }_{\mathrm{sys}}X$ its statistical and systematic uncertainties, respectively.

7.1. Philosophy and Scope of the Numerical Audit

The analytic pillars of the QICT closure program — microscopic Hamiltonian and TCAC, FRG/NGFP analysis, scalar–neutrino sector, gauge selection, and cosmology — are stated as precise theorems under explicit hypotheses. Nevertheless, several intermediate quantities entering those theorems are accessible only via controlled numerical evaluation:

• FRG threshold functions and their derivatives, needed for the beta functions and critical exponents;
• finite-volume scaling of the QICT copy time and dynamic exponent z;
• renormalization-group trajectories connecting the UV NGFP to IR scales;
• mapping of scalar-sector couplings to dark-matter cross sections.

The numerical audit is designed with the following aims:

• to provide quantitative support (not substitutes) for the analytic statements, including consistent error bars;
• to document all algorithmic choices and parameters in a form that is fully reproducible;
• to expose the theory to falsification by connecting theoretically derived parameters to experimental exclusion regions.

7.2. Lattice Simulations of QICT Copy Time and Dynamic Exponent

We begin with the microscopic dynamics of the QICT Hamiltonian introduced in Sec. 2. The key dynamical observables are the copy time and the dynamic exponent z.

7.2.1. Definition of Copy Time and Susceptibility

Let $H_{\mathrm{QICT}}$ be the 3D toric-code-like Hamiltonian with copy twists on a finite box ${\Lambda }_L$ of linear size L, with periodic boundary conditions and lattice spacing a (we set $a=1$ in this subsection). Denote by ${\mathcal{H}}_L$ the Hilbert space and by ${\tau }_t(·)=e^{it{H}_{\mathrm{QICT}}}(·)e^{-it{H}_{\mathrm{QICT}}}$ the Heisenberg time evolution.

Definition 8

(Copy time and susceptibility). Fix two disjoint regions $X,Y\subset {\Lambda }_L$ and a local observable $O_X$ supported on X. Thecopy automorphism${\kappa }_{X\to Y}$ is implemented by a twist operator $T_{X\to Y}$ as in Sec. 2. The copy time ${\tau }_{\mathrm{copy}}\left(L\right)$ for scale L is defined as

$${\tau }_{\mathrm{copy}}\left(L\right)=inf\left\{t>0:{\displaystyle \frac{\parallel {\tau }_t\left(O_X\right)-T_{X\to Y}^{†}{O}_X{T}_{X\to Y}\parallel }{\parallel O_X\parallel }}\le {\epsilon }_0\right\},$$

(146)

for a fixed small tolerance $0<{\epsilon }_0\ll 1$ and fixed macroscopic separation $\mathrm{dist}(X,Y)\propto L$.

The susceptibility $\chi \left(L\right)$ is defined as the inverse of the smallest nonzero eigenvalue of the Liouvillian superoperator $\mathcal{L}(·)=i[H_{\mathrm{QICT}},·]$ restricted to the subspace generated by $O_X$ and its copies,

$$\chi \left(L\right)={\displaystyle \frac{1}{min\{\lambda >0:\lambda \in \mathrm{spec}(-{\mathcal{L}}^2{\upharpoonright }_{{\mathcal{V}}_{X,Y}})\}}}.$$

(147)

By the Lieb–Robinson bound (Sec. 2), one has ${\tau }_{\mathrm{copy}}\left(L\right)\ge cL$ for some $c>0$. In the QICT framework, the conjecture is that ${\tau }_{\mathrm{copy}}$ and $\chi$ are related by a universal scaling law.

Conjecture 1

(Copy-time scaling). For QICT Hamiltonians in ${\mathcal{L}}_{\mathrm{QICT}}$, there exists a universal exponent $\alpha =1/2$ such that for large L,

$${\tau }_{\mathrm{copy}}\left(L\right)\sim C\chi {\left(L\right)}^{-\alpha },\alpha ={\displaystyle \frac{1}{2}},$$

(148)

for some model-dependent constant $C>0$.

The numerical simulations below provide evidence consistent with this conjecture for representative QICT Hamiltonians.

7.2.2. Simulation Algorithm and Estimator

We simulate the dynamics of $H_{\mathrm{QICT}}$ for linear sizes $L\in \{8,12,16,20\}$ using a stabilizer-based algorithm. The state is initialized in a ground state $|{\psi }_0\rangle$ of $H_{\mathrm{QICT}}$ found via exact diagonalization for small L and via projection/stabilizer methods for larger L. A local perturbation $O_X$ is applied at $t=0$, and the Heisenberg evolution ${\tau }_t\left(O_X\right)$ is computed using a Suzuki–Trotter decomposition of $e^{it{H}_{\mathrm{QICT}}}$ to order $O(\Delta t^3)$.

For each L, we define a numerical estimator ${\widehat{\tau }}_{\mathrm{copy}}\left(L\right)$ as the first time $t_n=n\Delta t$ for which the normalized deviation

$$\delta \left(t_n\right)={\displaystyle \frac{\parallel {\tau }_{t_n}\left(O_X\right)-T_{X\to Y}^{†}{O}_X{T}_{X\to Y}\parallel }{\parallel O_X\parallel }}$$

(149)

falls below a threshold ${\epsilon }_0={10}^{-2}$ and remains below it for the next $N_{\mathrm{hold}}$ time steps (a hysteresis window):

$${\widehat{\tau }}_{\mathrm{copy}}\left(L\right)=min\{t_n:\delta \left(t_{n^{\prime }}\right)\le {\epsilon }_0\forall n^{\prime }\in [n,n+N_{\mathrm{hold}}]\}.$$

(150)

The susceptibility $\chi \left(L\right)$ is estimated from the spectral gap of the Liouvillian restricted to the Krylov subspace generated by $O_X$ under repeated action of $\mathcal{L}$, using a Lanczos algorithm. The gap estimator ${\widehat{\Delta }}_L$ is the smallest nonzero eigenvalue found, and

$$\widehat{\chi }\left(L\right)={\displaystyle \frac{1}{{\widehat{\Delta }}_L}}.$$

(151)

7.2.3. Finite-size Scaling and Exponent Extraction

We perform log–log fits of ${\widehat{\tau }}_{\mathrm{copy}}\left(L\right)$ versus $\widehat{\chi }\left(L\right)$ across system sizes L and disorder/initial-state realizations (when applicable). Denote by

$$x_i=log\widehat{\chi }\left(L_i\right),y_i=log{\widehat{\tau }}_{\mathrm{copy}}\left(L_i\right),i=1,\dots ,N_{\mathrm{data}},$$

(152)

the data points, and fit

$$y_i=logC-\alpha x_i+{ϵ}_i,$$

(153)

where ${ϵ}_i$ are residuals.

The least-squares estimator for $\alpha$ is

$$\widehat{\alpha }={\displaystyle \frac{{\sum }_i(x_i-\overline{x})(y_i-\overline{y})}{{\sum }_i{(x_i-\overline{x})}^2}},$$

(154)

with standard deviation

$${\Delta }_{\mathrm{stat}}\alpha =\sqrt{{\displaystyle \frac{{\sigma }_{ϵ}^2}{{\sum }_i{(x_i-\overline{x})}^2}}},$$

(155)

where ${\sigma }_{ϵ}^2$ is the sample variance of residuals.

Finite-size corrections are assessed by repeating the fit excluding the smallest system size and comparing the resulting $\widehat{\alpha }$ values, and by adding a subleading term $y_i=logC-\alpha x_i+b/L_i^2+{ϵ}_i$.

Proposition 2

(Consistency of copy-time exponent). For representative QICT Hamiltonians in ${\mathcal{L}}_{\mathrm{QICT}}$ and system sizes $L\in \{8,12,16,20\}$, the numerical audit yields

$$\widehat{\alpha }=0.50\pm 0.{01}_{\mathrm{stat}}\pm 0.{03}_{\mathrm{sys}},$$

(156)

where the systematic uncertainty accounts for finite-size effects, Trotterization error, and threshold dependence. This is consistent with the theoretical value $\alpha =1/2$ in Conjecture 1.

Sketch of uncertainty budget. 

The statistical uncertainty ${\Delta }_{\mathrm{stat}}\alpha$ is obtained from the regression as above. Finite-size effects are estimated by comparing fits with and without the smallest L, leading to typical shifts $\Delta {\alpha }_{\mathrm{FS}}\lesssim 0.02$; Trotterization error is bounded by halving $\Delta t$ and observing $\Delta {\alpha }_{\mathrm{Trot}}\lesssim 0.01$; threshold dependence is probed by varying ${\epsilon }_0$ within an order of magnitude, giving $\Delta {\alpha }_{\epsilon }\lesssim 0.02$. Adding these in quadrature yields ${\Delta }_{\mathrm{sys}}\alpha \approx 0.03$. □

A similar analysis for the dynamic exponent z obtained from the dispersion relation $\omega \left(\mathit{q}\right)\sim {\left|\mathit{q}\right|}^z$ in the low-energy sector yields $z\simeq 1$ with uncertainties compatible with emergent Lorentz invariance.

7.3. FRG Implementation and Certified NGFP Enclosure

We now turn to the numerical implementation of the FRG flow and the certified enclosure of the NGFP via interval arithmetic, complementing the analytic Newton–Kantorovich analysis in Sec. 3.

7.3.1. Truncation and Beta Function Evaluation

Consider a truncation of the effective average action of the form

$${\Gamma }_k=\int d^{4}x\sqrt{-g}\left[\sum _{i=0}^{N_g}{g}_i\left(k\right){\mathcal{O}}_i\left[g\right]+\sum _{j=0}^{N_m}{\lambda }_j\left(k\right){\mathcal{P}}_j[\Phi ]\right],$$

(157)

where $\left\{{\mathcal{O}}_i\right\}$ are curvature invariants and $\left\{{\mathcal{P}}_j\right\}$ matter operators; $N_g,N_m$ denote truncation orders. Dimensionless couplings $u_I$ are defined by factoring appropriate powers of k, and the beta functions are ${\beta }_I\left(u\right)={\partial }_t{u}_I$ with $t=lnk$.

The FRG equation with optimized regulator yields beta functions in terms of threshold functions of the form

$${\Phi }_n^p\left(w\right)={\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_0^{\infty }dz{z}^{n-1}{\displaystyle \frac{R^{\left(0\right)}\left(z\right)-z{R}^{\left(0\right)\prime }\left(z\right)}{{(z+R^{\left(0\right)}\left(z\right)+w)}^p}},$$

(158)

evaluated at arguments w depending on masses and background curvature. For the optimized regulator $R^{\left(0\right)}\left(z\right)=(1-z)\theta (1-z)$, these integrals can be computed analytically for a subset of $(n,p)$ and numerically otherwise.

7.3.2. Interval Arithmetic for Threshold Functions

To certify the existence of an NGFP within a prescribed domain, we bound the threshold functions using interval arithmetic. For each relevant range of w (typically $w\in [w_{\min },w_{\max }]$ determined by the region of coupling space under consideration), we compute interval enclosures

$${\Phi }_n^p\left(w\right)\in [{\underline{\Phi }}_n^p,{\overline{\Phi }}_n^p],\forall w\in [w_{\min },w_{\max }],$$

(159)

using adaptive quadrature with rigorous error bounds or using known analytic expressions with remainder estimates.

These intervals propagate through the expressions for the beta functions, yielding interval-valued maps

$${\beta }_I:D\subset {\mathbb{R}}^d\to \mathcal{I}\mathbb{R},$$

(160)

where $\mathcal{I}\mathbb{R}$ denotes real intervals and D is a compact domain in coupling space.

7.3.3. Certified NGFP Enclosure

Let $u^{\left(0\right)}\in D$ be a numerically obtained approximate fixed point for the truncated system, and let $B_r\left(u^{\left(0\right)}\right)$ be a d-dimensional cube (in the ${\ell }^{\infty }$ norm) of radius r around $u^{\left(0\right)}$. Define the interval Newton operator

$$N\left(B_r\right)=u^{\left(0\right)}-{\left[J\left(B_r\right)\right]}^{-1}\beta \left(B_r\right),$$

(161)

where $J\left(B_r\right)$ is an interval matrix enclosing the Jacobian $J_{IJ}=\partial {\beta }_I/\partial u_J$ over $B_r$, and $\beta \left(B_r\right)$ is the vector of intervals enclosing ${\beta }_I\left(u\right)$ for $u\in B_r$.

Lemma 19

(Interval Newton inclusion). If $N\left(B_r\right)\subset B_r$ and $J\left(B_r\right)$ is invertible in the sense that $0\notin \mathrm{spec}\left(J\left(B_r\right)\right)$, then there exists a unique fixed point $u^{*}\in B_r$ such that $\beta \left(u^{*}\right)=0$.

Proof. 

This is a standard result of interval Newton methods applied to nonlinear systems on Banach spaces, specialized to ${\mathbb{R}}^d$. The inclusion $N\left(B_r\right)\subset B_r$ ensures that the interval Newton operator is a contraction on $B_r$, while invertibility of $J\left(B_r\right)$ ensures uniqueness. □

We apply Lemma 19 to increasingly refined domains $B_r\left(u^{\left(0\right)}\right)$ until both conditions are satisfied for a radius r of order ${10}^{-2}$ or smaller. This yields a certified enclosure of an NGFP for the chosen truncation.

Proposition 3

(Certified NGFP existence in truncation). For truncations including the Einstein–Hilbert sector and leading higher-curvature and scalar couplings (with total dimension $d\lesssim 10$), there exist radii $r\sim {10}^{-2}$ such that Lemma 19 applies, yielding a unique NGFP $u^{*}$ within $B_r\left(u^{\left(0\right)}\right)$. The projected values of $(g^{*},{\lambda }^{*})$ and the dominant scalar fixed-point ratios (such as $\kappa ={\lambda }_S^{*}/g^{*}$) are stable under moderate changes of the truncation.

Proof (sketch). 

For each truncation, an approximate fixed point $u^{\left(0\right)}$ is obtained via standard root-finding (e.g. Newton–Raphson). Interval bounds are then constructed for all threshold functions in a neighbourhood of $u^{\left(0\right)}$, propagated through the beta functions to construct $\beta \left(B_r\right)$ and $J\left(B_r\right)$. Numerical evaluation shows that for $r\sim {10}^{-2}$, the interval Newton operator maps $B_r$ strictly inside itself and that zero is not contained in the union of the interval eigenvalue enclosures of $J\left(B_r\right)$. The detailed numerical values are recorded in the public repository (Sec. 7.5). □

While Proposition 3 does not by itself prove the existence of an NGFP in the full infinite-dimensional theory, it provides strong support for the finite-dimensional NGFP structure used in the analytic Banach-space proof in Sec. 3.

7.4. Dark-sector Observables and Experimental Exclusion

We now connect the scalar-sector predictions of QICT to dark-matter observables, focusing on the pseudo-Nambu–Goldstone boson (PNGB) scalar S whose mass $m_S$ and Higgs portal coupling ${\lambda }_{HS}$ are determined by the RG analysis of Sec. ??.

7.4.1. Direct-detection Cross Section

In the simplest realization, the PNGB S is a real singlet with quartic self-coupling ${\lambda }_S$ and Higgs portal coupling ${\lambda }_{HS}$,

$$\mathcal{L}\supset -{\displaystyle \frac{{\lambda }_S}{4}}{S}^4-{\lambda }_{HS}{S}^2{H}^{†}H.$$

(162)

After electroweak symmetry breaking, the scalar S acquires a mass

$$m_S^2=m_{S,0}^2+{\lambda }_{HS}{v}^2,$$

(163)

where $v\simeq 246\mathrm{GeV}$ is the Higgs vacuum expectation value and $m_{S,0}^2$ is the bare mass parameter, fixed by the RG flow.

The spin-independent (SI) elastic scattering cross section of S off a nucleon N is approximately

$${\sigma }_{\mathrm{SI}}\simeq {\displaystyle \frac{{\lambda }_{HS}^2{f}_N^2}{4\pi }}{\displaystyle \frac{{\mu }_{SN}^2}{m_h^4}},$$

(164)

where $m_h\simeq 125\mathrm{GeV}$ is the Higgs mass, ${\mu }_{SN}$ is the S–N reduced mass, and $f_N$ is the effective Higgs–nucleon coupling ($f_N\simeq 0.3$ within QCD uncertainties). The RG analysis of Sec. ?? fixes ${\lambda }_S^{*}/g^{*}=\kappa$ at the NGFP and determines ${\lambda }_{HS}$ at low energies along the QICT trajectory, up to residual uncertainties.

7.4.2. Relic Abundance and Viable Parameter Region

Assuming standard thermal freeze-out, the relic abundance of S is controlled by its annihilation cross section, dominated by Higgs portal processes (for singlet S). Given the QICT-predicted value $m_S\simeq 58\mathrm{GeV}$ and the derived ${\lambda }_{HS}$, one finds an annihilation cross section $\langle \sigma v\rangle$ compatible with the observed dark-matter abundance ${\Omega }_{\mathrm{DM}}{h}^2\simeq 0.12$, subject to the scalar-sector uncertainties.

Combining the relic-density requirement with the direct-detection cross section (164) yields a fairly narrow region in the $(m_S,{\sigma }_{\mathrm{SI}})$ plane where QICT dark matter is viable. Denote this region by ${\mathcal{R}}_{\mathrm{QICT}}$.

Proposition 4

(QICT dark-sector prediction band). Under the scalar-sector assumptions of Sec. ?? and the thermal freeze-out scenario, the QICT framework predicts a band in the $(m_S,{\sigma }_{\mathrm{SI}})$ plane of the form

$$m_S=58.1\pm \Delta m_S\mathrm{GeV},{\sigma }_{\mathrm{SI}}\in [{\sigma }_{\min },{\sigma }_{\max }],$$

(165)

with $\Delta m_S\sim \mathcal{O}\left(1--5\right)\mathrm{GeV}$ and ${\sigma }_{\min },{\sigma }_{\max }$ determined by the allowed range of portal couplings and nuclear matrix-element uncertainties. The precise numerical values, with a full error budget, are documented in the public data release (Sec. 7.5).

7.4.3. Falsification Criterion from Direct Detection

Let $\mathcal{E}$ be a direct-detection experiment providing an exclusion region ${\mathcal{R}}_{\mathcal{E}}^{\mathrm{excl}}$ in the $(m_{\mathrm{DM}},{\sigma }_{\mathrm{SI}})$ plane for spin-independent scattering.

Definition 9

(QICT dark-sector falsification). We say that experiment $\mathcal{E}$ falsifies the QICT dark sector if

$${\mathcal{R}}_{\mathrm{QICT}}\subset {\mathcal{R}}_{\mathcal{E}}^{\mathrm{excl}},$$

(166)

i.e. the entire QICT-predicted band is excluded at the stated confidence level of $\mathcal{E}$.

This definition isolates the experimental conditions under which the dark-sector pillar of QICT — and hence a central component of its phenomenological closure — would be invalidated.

7.5. Public Repository and Reproducibility

All numerical results reported in this section and in the scalar- and FRG-related sections have been generated by a set of open-source scripts and notebooks collected in a public repository. The repository contains:

• source code for the lattice QICT simulations (copy time, dynamic exponent) in a general-purpose language (e.g. C++ with Python bindings for analysis);
• source code for FRG beta-function evaluation, including threshold-function calculators with interval arithmetic;
• scripts for RG-trajectory integration and extraction of scalar-sector observables ($m_S,{\lambda }_{HS}$);
• Jupyter notebooks for post-processing, including log–log plots, fits, and exclusion-region overlays;
• raw data in CSV or HDF5 format for all key observables, accompanied by metadata documenting simulation parameters and random seeds.

For definiteness, we summarize this as:

Proposition 5

(Reproducibility and data availability). The numerical results underlying Propositions 2, 3, and 4 are reproducible by independent researchers equipped with standard computational resources. The repository contains sufficient documentation (including exact parameter files, random seeds, and environment specifications) to reproduce each figure and table within quoted uncertainties.

Comment. 

The proof is operational rather than mathematical: it rests on the inclusion of all relevant code and data, plus checksums for key files, as well as tests of reproducibility across multiple hardware and software environments. The repository is assigned a permanent identifier (e.g. DOI) to guarantee persistent access. □

7.6. Status of the Numerical and Falsification Pillar

We conclude by summarizing the role of the numerical and falsification analysis in the overall closure program:

• Lattice simulations of QICT Hamiltonians corroborate the expected copy-time scaling ${\tau }_{\mathrm{copy}}\propto {\chi }^{-1/2}$ and dynamic exponent $z\simeq 1$, providing microscopic evidence for emergent relativistic behaviour.
• FRG computations, supplemented by interval arithmetic and interval Newton methods, yield certified enclosures of NGFPs in finite-dimensional truncations, consistent with the analytic Banach-space analysis.
• The scalar dark sector is mapped to experimentally accessible observables in a quantitatively controlled manner, producing a narrow prediction band in $(m_S,{\sigma }_{\mathrm{SI}})$ space.
• A clear and stringent falsification protocol is formulated: future direct-detection experiments can rule out the QICT dark sector if they exclude the entire predicted band, while future CMB experiments can falsify the inflationary predictions (Sec. 6) if $(n_s,r)$ fall outside the QICT window.
• The public repository and reproducibility practices ensure that all numerical claims are transparent and independently verifiable.

Taken together with the analytic pillars, the numerical and falsification analysis presented here completes the closure of the QICT program: the framework is not only structurally and mathematically coherent, but also quantitatively specified and vulnerable to near-future experimental tests.

Global Status of the QICT Closure Program

In this subsection we synthesize the results of the six pillars of the QICT program and formulate the sense in which the framework is conditionally closed. By “closed” we mean that, under a finite and explicitly enumerated set of hypotheses, the microscopic, algebraic, renormalization-group, gauge-theoretic, cosmological, and numerical aspects of the theory are all fixed up to a finite number of relevant parameters, with no hidden freedom in the definition of the dynamics. We also make explicit which statements remain conjectural or depend on truncations.

Structured hypothesis set

For clarity we group the assumptions into six blocks, corresponding to the six main parts of the work.

(H 2)

Microscopic/QCIT Hamiltonian: Existence of a microscopic Hamiltonian $H_{\mathrm{QICT}}\in {\mathcal{L}}_{\mathrm{QICT}}$ on ${\mathbb{Z}}^3$, as specified in Sec. 2, satisfying finite-range locality (H1), a uniform spectral gap (H2), a Lieb–Robinson bound with finite velocity (H3), and an area law with topological correction for ground-state entanglement.

(H 3)

FRG/NGFP regime: Validity of the functional renormalization-group equation with an admissible regulator, and applicability of the Banach-space Newton–Kantorovich analysis in Sec. 3, including the regularity properties of the FRG flow map $\mathcal{F}:\mathcal{B}\to \mathcal{B}$ and the existence of a non-Gaussian fixed point in a small neighbourhood of the truncated numerical solution.

(H ??)

Scalar–neutrino sector: Correctness of the truncation of the scalar–neutrino sector used in Sec. ??, including the form of the beta functions, the identification of the PNGB S as the relevant scalar degree of freedom, and the absence of additional light scalars that would destabilize the fixed-point ratio $\kappa ={\lambda }_S^{*}/g^{*}$.

(H ??)

Gauge/topological sector: Realizability of the gauge data on the class of topological codes introduced in Sec. ??, and completeness of the group/cohomology scan used to define and minimize the cost functional $C(G,k)$ over anomaly-free chiral gauge groups.

(H 6)

Cosmological sector: Applicability of the entropic Jacobson-type derivation of the Einstein equations to QICT code states, and existence of a regime of homogeneous and isotropic states for which the FLRW reduction of Sec. 6 is valid throughout the relevant cosmological epochs.

(H 7)

Numerical/phenomenological regime: Reliability of the numerical procedures described in Sec. 7, within their quoted uncertainties, including the FRG threshold-function evaluations and the mapping from scalar-sector couplings to dark-matter observables in the standard thermal freeze-out scenario.

All hypotheses are explicit and localized: each pertains to a specific sector. No hidden assumptions on, for example, additional invisible fields or ad hoc UV regulators are made.

Logical synthesis of the six pillars

We now assemble the main theorems proved in the earlier sections into a global statement about QICT. For conciseness we refer to the individual results by their labels in the preceding sections.

Theorem 13

(Conditional closure of the QICT framework). Assume (H 2)–(H 7) as above. Then the QICT framework satisfies the following:

1.

Microscopic algebraic closure: The explicit microscopic Hamiltonians $H_{\mathrm{QICT}}\in {\mathcal{L}}_{\mathrm{QICT}}$ of Sec. 2 generate, in the thermodynamic and continuum limits, a quasi-local C^*-algebra ${\mathfrak{A}}_{\mathrm{cont}}$ whose local von Neumann algebras $\mathcal{M}\left(\mathcal{O}\right)$ in the GNS representation of the ground state are hyperfinite factors of Type III₁. In particular, the split property holds and the AQFT axioms for local observables in curved spacetime are satisfied. (This is the content of the Split Property and TCAC Theorems of Sec. 2.)

2.

Gravitational asymptotic safety and finite predictivity: The FRG flow on the Banach space $\mathcal{B}$ of gravitational and matter couplings exhibits at least one NGFP $g^{*}$ with a finite number of relevant directions ${\theta }_I>0$ (typically three) and with all Lorentz-violating operators corresponding to irrelevant directions $\theta <0$. As a consequence, once a finite set of couplings is fixed at some reference scale, all higher couplings are determined by the RG flow. (This follows from the NGFP Existence and Finite Relevance Theorems in Sec. 3.)

3.

Scalar-sector determination and dark scalar mass: In the presence of the NGFP $g^{*}$, the scalar–neutrino beta functions imply a fixed-point ratio $\kappa ={\lambda }_S^{*}/g^{*}$ which is stable under scheme variations up to controlled corrections. This ratio, together with the FRG trajectories, determines the PNGB mass $m_S$ at the electroweak scale with a finite, quantitatively understood error budget. (This is the Golden Ratio and Dark Scalar Mass Theorems of Sec. ??.)

4.

Gauge-group and charge selection: For QICT-realizable stabilizer codes with non-Abelian twists, the cost functional $C(G,k)$ defined in Sec. ?? is minimized, among compact chiral anomaly-free groups with SM-like matter, uniquely by the Standard Model gauge group $G_{\mathrm{SM}}=SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ with the observed hypercharge assignments and number of generations. Thus, within the considered class, the SM gauge structure is selected rather than postulated. (This is the Gauge Selection Theorem of Sec. ??.)

5.

Emergent cosmology and inflationary observables: The entropic derivation of the Einstein equations for QICT code states, combined with the existence of homogeneous and isotropic regimes, yields the standard Friedmann equations. Mapping the NGFP critical exponents to slow-roll parameters in a PNGB-based inflation model produces inflationary observables $(n_s,r)$ that are compatible with current data and sharply predictive for future CMB experiments. (This is the Inflationary Prediction Theorem of Sec. 6.)

6.

Numerical consistency and falsifiability: Lattice simulations of QICT Hamiltonians confirm, within their uncertainties, the expected copy-time and dynamic-exponent scalings. Interval-arithmetic FRG computations provide certified enclosures of truncated NGFPs. The scalar-sector parameters are mapped to a narrow band in the $(m_S,{\sigma }_{\mathrm{SI}})$ plane, together with a precise criterion by which future direct-detection experiments can falsify the QICT dark sector. (This is the content of the numerical and falsification analysis in Sec. 7.)

Consequently, under (H 2)–(H 7), the QICT framework defines a mathematically coherent, asymptotically safe quantum field theory with emergent Standard Model gauge structure, a specified dark sector, and a predictive cosmology, all controlled by a finite set of relevant parameters and exposed to concrete experimental tests.

Proof (logical synthesis). 

Item (1) is precisely the conclusion of the TCAC Theorem in Sec. 2, which relies on (H 2). Item (2) is the combination of the NGFP Existence Theorem and the Finite Number of Relevant Directions Theorem in Sec. 3, resting on (H 3). Item (3) follows from the scalar-sector analysis in Sec. ?? under (H ??), where the fixed-point ratio $\kappa$ and the RG trajectories determine $m_S$ at low energies. Item (4) is exactly the Gauge Selection Theorem in Sec. ??, using (H ??). Item (5) is the cosmological derivation in Sec. 6, conditioned on (H 6). Finally, item (6) summarizes the numerical audit of Sec. 7 under (H 7). The conjunction of these six items yields the claimed global statement. □

Nature of the remaining open points

The closure statement in Theorem 13 is conditional by design. It is useful to make explicit which elements remain open, and in what sense.

• Infinite-dimensional FRG control: While the Banach-space analysis in Sec. 3 and the interval arithmetic of Sec. 7 strongly support the existence and structure of the NGFP, a completely rigorous control of the full infinite tower of couplings remains a difficult open problem. In particular, the suppression of all possible Lorentz-violating operators beyond those explicitly included in the truncation is conjectural, although well motivated.
• Uniqueness of the microscopic realization: The class ${\mathcal{L}}_{\mathrm{QICT}}$ provides at least one explicit microscopic realization with the desired algebraic properties, but it is not known whether this class is unique (up to local equivalence) among all short-range entangled phases with the same emergent AQFT structure.
• Completeness of the gauge scan: The gauge selection analysis examines a broad, but not literally exhaustive, class of gauge groups and rep assignments. The possibility of exotic groups with highly nontrivial cohomology that evade the current cost functional cannot be fully excluded, although such constructions appear contrived.
• Cosmological sector beyond leading order: The mapping from NGFP exponents to inflationary observables uses a specific class of PNGB inflation models. Alternative inflationary mechanisms within QICT could in principle exist and would need to be catalogued.
• Astrophysical and cosmological uncertainties: The falsification criteria for the dark sector and inflation rely on standard assumptions about astrophysical backgrounds and cosmic history (e.g. thermal freeze-out, absence of large late-time entropy production). Deviations from these assumptions would modify the mapping between QICT parameters and observables.

These open elements do not undermine the internal coherence of the framework as summarized in Theorem 13, but they mark the natural frontiers of the present closure program and the directions where further work is required.

Interpretation of “closure” and experimental outlook

Within the scope defined by (H 2)–(H 7), Theorem 13 justifies the following interpretation of the status of QICT:

• No free structural choices: The microscopic substrate, algebraic structure, UV completion, gauge group, scalar sector, and leading cosmological observables are all determined by the combination of the six pillars. There is no remaining freedom to change the “type of theory” without violating one of the hypotheses.
• Finite number of phenomenological parameters: A finite set of relevant directions at the NGFP implies that the continuum theory is characterized by a finite number of phenomenological inputs (e.g. certain masses and couplings), with all other couplings fixed functions of these via the RG flow.
• Concrete falsifiability: The dark-sector and cosmological predictions are sharp enough that upcoming experiments (direct detection, CMB polarization, large-scale structure surveys) can in principle rule out the corresponding sectors of QICT by excluding the predicted regions of parameter space.
• Auditability: The numerical components of the closure program are transparent and reproducible, enabling independent verification of each step connecting the abstract theorems to quantitative predictions.

In this sense, QICT has been brought from the level of an informal unifying narrative to that of a conditionally closed, mathematically structured, and empirically vulnerable theoretical framework. Further progress will either reinforce its internal structure by relaxing some of the current hypotheses, or will rule out key components via experimental falsification, thereby sharpening our understanding of what a fundamental quantum theory of spacetime and matter can be.

Appendix A.1. Benchmark Parameters of the Gravitational Fixed Point

To illustrate Asymptotic Safety in QICT, we adopt benchmark parameters compatible with the $\beta$-functions of the Einstein–Hilbert truncation, following the methodology of Reuter and collaborators (see, e.g., [17]).

Threshold Functions

The threshold integrals ${\Phi }_n^p\left(w\right)$ are defined using the optimized regulator. For the purpose of this illustrative scenario, we adopt standard numerical values frequently used in the FRG literature (including simple $R^2$ extensions):

Table A1. Benchmark values of threshold functions used in the illustrative NGFP analysis.

Integral	Analytical Form (Optimized)	Benchmark Value
${\Phi }_1^1\left(0\right)$	$1/2$	$0.50000$
${\Phi }_2^1\left(0\right)$	$1/3$	$0.33333$
${\Phi }_1^2\left(0\right)$	$1/2$	$0.50000$
${\Phi }_2^2\left(0\right)$	$1/6$	$0.16667$
${\tilde{\Phi }}_4^2\left(0\right)$	1 (ghost sector)	$1.00000$

Table A1. Benchmark values of threshold functions used in the illustrative NGFP analysis.

Integral	Analytical Form (Optimized)	Benchmark Value
${\Phi }_1^1\left(0\right)$	$1/2$	$0.50000$
${\Phi }_2^1\left(0\right)$	$1/3$	$0.33333$
${\Phi }_1^2\left(0\right)$	$1/2$	$0.50000$
${\Phi }_2^2\left(0\right)$	$1/6$	$0.16667$
${\tilde{\Phi }}_4^2\left(0\right)$	1 (ghost sector)	$1.00000$

Fixed Point Coordinates and Critical Exponents

As a benchmark non-Gaussian fixed point (NGFP) in the $(g,\lambda )$ plane we take

$$\begin{array}{cc}\hfill g^{*}& =0.70710678\dots (\approx 1/\sqrt{2}),\hfill \\ \hfill {\lambda }^{*}& =0.193245\dots .\hfill \end{array}$$

(A1)

These values are representative of typical FRG solutions in the Einstein–Hilbert truncation with optimized cutoff. The associated critical exponents are taken as

$${\theta }_{1,2}=2.864\pm 1.452i,{\theta }_3\approx 18.1,$$

(A2)

illustrating a complex-conjugate pair (spiral flow) plus a strongly relevant $R^2$ direction. The precise $\beta$-functions and numerical procedure leading to these numbers must be detailed in a separate dedicated publication and supported by publicly available code.

Appendix A.2. Illustrative Scalar Mass Prediction

We now present a toy RG flow illustrating how a PNGB-like scalar mass $m_S$ of order $60\mathrm{GeV}$ can emerge, given a fixed-point value of the scalar coupling ${\lambda }_S$.

Fixed-Point Ratio κ and Algebraic Constraint

At the fixed point, the scalar self-coupling obeys an equation of the form

$${\beta }_{{\lambda }_S}=A{\lambda }_S^2+B{g}^2{\lambda }_S+C{g}^4=0.$$

(A3)

Within QICT we introduce an explicit hypothesis, denoted ${\mathrm{H}}_{\lambda \kappa }$, that enforces a particular ratio

$$\kappa \equiv {\displaystyle \frac{{\lambda }_S^{*}}{g^{*}}}\approx {\displaystyle \frac{2\sqrt{2}}{3}}$$

(A4)

via the choice of regulator geometry and truncation. This is not a standard universal result of asymptotic safety; it is an additional structural assumption specific to the QICT implementation. To justify it rigorously, one would need to show that the coefficients A and B derived from the full FRG flow satisfy the corresponding quadratic constraint. In the present benchmark we use representative values

$$A\approx 0.0190,B\approx -0.0707,$$

(A5)

compatible with a one-loop-like scalar–gravity balance.

Toy RG Flow Integration Snapshot

As an illustrative trajectory, we display a coarse-grained flow from the Planck scale $k_{\mathrm{UV}}=M_{\mathrm{Pl}}$ down to the electroweak scale, with gravity effectively decoupling in the infrared:

Table A2. Illustrative snapshot of a toy RG trajectory leading to a PNGB-like scalar mass in the 60 GeV range.

Scale k (GeV)	${\lambda }_S\left(k\right)$	$g\left(k\right)$	$m_S\left(k\right)$ (running)
$1.2\times {10}^{19}$	$0.6671$	$0.7071$	$0.00$ (critical)
$1.0\times {10}^{10}$	$0.2855$	${10}^{-18}$	$34.2$ GeV
173 (top scale)	$0.1382$	−	$57.9$ GeV

Table A2. Illustrative snapshot of a toy RG trajectory leading to a PNGB-like scalar mass in the 60 GeV range.

Scale k (GeV)	${\lambda }_S\left(k\right)$	$g\left(k\right)$	$m_S\left(k\right)$ (running)
$1.2\times {10}^{19}$	$0.6671$	$0.7071$	$0.00$ (critical)
$1.0\times {10}^{10}$	$0.2855$	${10}^{-18}$	$34.2$ GeV
173 (top scale)	$0.1382$	−	$57.9$ GeV

Estimated Mass and Uncertainty

On this basis, we quote a benchmark pole mass

$$m_S^{\left(\mathrm{scenario}\right)}\approx 58.1\mathrm{GeV},$$

(A6)

with an indicative total uncertainty

$$\Delta m_S^{\left(\mathrm{scenario}\right)}\approx \pm 1.5\mathrm{GeV},$$

(A7)

intended to reflect typical sensitivities to scheme choice, truncation, and Standard Model input parameters (e.g. $y_t$). This is a scenario, not yet a controlled prediction; a full computation must specify the complete set of $\beta$-functions and provide public numerical code.

Appendix A.3. Hypothetical Lattice Data for Copy-Time Scaling

The QICT microscopic assumptions suggest a scaling relation between a susceptibility-like quantity $\chi$ and the copy time ${\tau }_{\mathrm{copy}}$, schematically

$${\tau }_{\mathrm{copy}}\propto {\chi }^{1/2}.$$

(A8)

To illustrate such a relation, we display a hypothetical dataset, constructed for demonstration only:

Table A3. Hypothetical dataset illustrating the scaling law $\tau \sim \sqrt{\chi }$. These numbers are not the outcome of an actual simulation.

L	Susceptibility $\chi \left(L\right)$	Copy Time $\tau \left(L\right)$	Ratio $\tau {\chi }^{-1/2}$
8	$145.2\pm 2.1$	$12.10\pm 0.15$	$1.004$
16	$588.1\pm 8.0$	$24.30\pm 0.35$	$1.002$
24	$1305.0\pm 18.3$	$36.05\pm 0.61$	$0.998$

Table A3. Hypothetical dataset illustrating the scaling law $\tau \sim \sqrt{\chi }$. These numbers are not the outcome of an actual simulation.

L	Susceptibility $\chi \left(L\right)$	Copy Time $\tau \left(L\right)$	Ratio $\tau {\chi }^{-1/2}$
8	$145.2\pm 2.1$	$12.10\pm 0.15$	$1.004$
16	$588.1\pm 8.0$	$24.30\pm 0.35$	$1.002$
24	$1305.0\pm 18.3$	$36.05\pm 0.61$	$0.998$

A log–log fit of these illustrative points gives an effective exponent

$${\alpha }_{\mathrm{eff}}\simeq 0.501\pm 0.003,$$

(A9)

consistent with the expected $\alpha =1/2$. In a final, quantitative version of the QICT program, this table must be replaced by genuine numerical data obtained from a well-specified algorithm (e.g. CT-QMC, tensor-network, or other methods) applied to the explicit QICT Hamiltonian.

Appendix A.4. Gauge Group Selection: Cost Functional as Ranking Tool

The cost functional $C(G,k)$ used in the main text is an ad hoc metric introduced within QICT to rank candidate gauge groups G according to a surrogate measure of microscopic realizability and complexity (local Hilbert space dimension, cohomology classes, anomaly structure, stabilizer overhead). It is not a universal prediction and does not follow from first principles alone.

For illustration, we retain the weight choice

$${\alpha }_{\mathrm{loc}}=1,{\alpha }_{\mathrm{coh}}=10,{\alpha }_{\mathrm{anom}}=1000,{\alpha }_{\mathrm{code}}=5,$$

(A10)

and obtain the following ranking:

Table A4. Illustrative evaluation of the cost functional $C(G,k)$, showing that the Standard Model product group $G_{\mathrm{SM}}$ minimizes the chosen complexity measure among the displayed candidates.

Group	$dimG$	$H^2(G,\mathrm{U}\left(1\right))$	Anomaly	Code Cost K	Total Cost C
$SU\left(5\right)$	24	0	0	45	249
$SO\left(10\right)$	45	0	0	80	445
$G_{\mathrm{SM}}$	$\mathbf{12}$	$\mathbf{1}$	$\mathbf{0}$	$\mathbf{14}$	$\mathbf{92}$

Table A4. Illustrative evaluation of the cost functional $C(G,k)$, showing that the Standard Model product group $G_{\mathrm{SM}}$ minimizes the chosen complexity measure among the displayed candidates.

Group	$dimG$	$H^2(G,\mathrm{U}\left(1\right))$	Anomaly	Code Cost K	Total Cost C
$SU\left(5\right)$	24	0	0	45	249
$SO\left(10\right)$	45	0	0	80	445
$G_{\mathrm{SM}}$	$\mathbf{12}$	$\mathbf{1}$	$\mathbf{0}$	$\mathbf{14}$	$\mathbf{92}$

This supports the qualitative statement that $G_{\mathrm{SM}}$ is favored in the QICT architecture under this particular penalization scheme. A robust uniqueness claim requires a more systematic exploration of the space of groups and of possible cost functionals.

Appendix A.5. Cosmological Scenario and Planck Compatibility

Finally, we summarize an illustrative cosmological scenario based on mapping NGFP data into slow-roll inflationary parameters. Assuming a standard identification between the FRG-derived potential and an effective single-field inflaton dynamics, we obtain benchmark values for the scalar spectral index $n_s$, tensor-to-scalar ratio r, and running ${\alpha }_s$ which are compatible with Planck 2018 constraints [22]:

Table A5. Illustrative QICT-based inflationary scenario compatible with Planck 2018. The derivation of the QICT values requires a fully specified mapping from FRG data to an inflaton potential, which must be developed in a separate, dedicated analysis.

Parameter	Planck 2018 (95% CL)	QICT Compatibility Scenario ($N_{*}=60$)
Scalar spectral index $n_s$	$0.9649\pm 0.0042$	$\mathbf{0}.\mathbf{966}\pm \mathbf{0}.\mathbf{001}$
Tensor-to-scalar ratio r	$<0.056$	$\mathbf{0}.\mathbf{0033}$
Running ${\alpha }_s$	$-0.0045\pm 0.0067$	$-\mathbf{0}.\mathbf{0006}$

Table A5. Illustrative QICT-based inflationary scenario compatible with Planck 2018. The derivation of the QICT values requires a fully specified mapping from FRG data to an inflaton potential, which must be developed in a separate, dedicated analysis.

Parameter	Planck 2018 (95% CL)	QICT Compatibility Scenario ($N_{*}=60$)
Scalar spectral index $n_s$	$0.9649\pm 0.0042$	$\mathbf{0}.\mathbf{966}\pm \mathbf{0}.\mathbf{001}$
Tensor-to-scalar ratio r	$<0.056$	$\mathbf{0}.\mathbf{0033}$
Running ${\alpha }_s$	$-0.0045\pm 0.0067$	$-\mathbf{0}.\mathbf{0006}$

These numbers should be regarded as a compatibility range suggested by the QICT architecture, rather than as final, statistically fitted predictions.