Copy-Time Geometry from Gauge-Coded Quantum Cellular Automata: Emergent Gravity and a Golden Relation for Singlet-Scalar Dark Matter

1. Introduction

Scope and organisation.

The core results of this paper are the definition of ${\tau }_{\mathrm{copy}}$ and ${\chi }^{\left(2\right)}$ for conserved charges, and a general variational speed-limit inequality relating them (Section 2 and Mathematical Appendix, Sec. S1). Sections on emergent geometry, gauge-coded QCA constructions, and phenomenological matching are intentionally downstream of that formal core and may be read independently; none of them is required for the speed-limit theorem itself. The accompanying mathematical appendix is organised in standard definition–lemma–proposition–theorem form for the statements invoked in the main text, while model-dependent closure calculations are separated and labelled as such.

Reproducibility.

A companion technical appendix documents the fully local 3+1D micro-model and additional construction details. A separate mathematical appendix records only the formal statements invoked in the main text, written as assumptions, definitions, lemmas, propositions, and demonstrations, with benchmark notes kept separate. The package also includes a self-contained benchmark code bundle sufficient to reproduce the representative tables and figures declared in the submission README. Recent experimental and phenomenological updates on Higgs-portal constraints and direct detection provide useful external context for the closure discussion (e.g. Refs. [37,38,39]).

Main novelty and relation to prior work.

Relative to prior quantum-transport and QCA literature [1,2,3] and to earlier information-theoretic discussions of hydrodynamics, the main novelty here is an operational, receiver-optimised definition of a conserved-charge copy time together with a companion Liouvillian-squared susceptibility, and a general variational speed-limit inequality linking the two that holds without assuming diffusion. We then show, in a controlled stabiliser-code diffusion benchmark family, that this bound is close to saturated over several decades, providing a practical calibration of the transport normalisation in the diffusive regime. Finally, we present a worked (explicitly model-dependent) infrared matching protocol for the Higgs-portal singlet-scalar model near the Higgs resonance, using closed-form expressions for the invisible Higgs width and spin-independent nucleon cross section to make the assumptions and uncertainty budget explicit.

The emergence of macroscopic physics from microscopic quantum dynamics is constrained by three intertwined structures: locality, conservation laws, and limits on information processing. Quantum cellular automata (QCA) provide a natural language for strictly local, fully quantum dynamics on discrete lattices [1,2,3], while a continuum effective-field-theory description can be organised by renormalisation-group ideas. Later sections explore how such information-theoretic quantities may be used as inputs for infrared matching, including to gravitational and scalar-sector scales; these exploratory connections are presented as worked examples rather than as deductions from the speed-limit bound alone. Connecting such microscopic and continuum descriptions in an information-theoretically meaningful and phenomenologically predictive way remains a central challenge.

The Quantum Information Copy Time (QICT) framework proposes that information-theoretic quantities associated with conserved charges, such as an information susceptibility ${\chi }^{\left(2\right)}$ and a copy time ${\tau }_{\mathrm{copy}}$, play an organising role in the emergence of hydrodynamics and in constraining infrared (IR) observables. Concretely, for a local diffusive system with a conserved charge Q and suitable encoding and decoding protocols, one expects the characteristic time to reliably “copy” charge information from one region to another to be controlled by a combination of susceptibilities and diffusion constants.

Earlier work suggested a scaling of the form

$${\tau }_{\mathrm{copy}}\left(Q\right)\propto {\left({\chi }_{\mathrm{micro},Q}^{\left(2\right)}\right)}^{-1/2},$$

(1)

supported by stabiliser-code examples and numerical simulations of diffusive channels. However, a fully rigorous microscopic derivation and a clear path to phenomenology were lacking.

On the continuum side, it is convenient to summarise the scalar sector near an infrared matching scale by a dimensionless “dressing” parameter of the schematic form $m_S^2\left(k\right)/k^2$. In this paper we define this parameter intrinsically from QCA thermal susceptibilities (Appendix A), and we only use continuum FRG literature as a qualitative point of comparison, not as an input.

In the QICT formulation adopted here, ${\kappa }_{\mathrm{eff}}$ is defined microscopically (Sec. Appendix A) as a ratio of regulated Kubo–Mori Liouvillian-squared susceptibilities evaluated in the electroweak-symmetric high-temperature plateau. When we quote a numerical interval for ${\kappa }_{\mathrm{eff}}$ in the main text, it should be read as a benchmark interval for this microscopic quantity; FRG results, when invoked, serve only as an external cross-check and are collected separately in Appendix A.

The central idea of this paper is to identify $k_{\mathrm{IR}}$ with a scale extracted from a microscopic QICT analysis of hypercharge transport, and to propagate the resulting relation to a quantitative band for $m_S$ that can be confronted with Higgs-portal phenomenology and direct-detection experiments.

1.1. Scope and Status of Results

Because the framework combines several layers (QICT, QCA, emergent geometry, and Higgs-portal phenomenology), it is important to separate clearly what is rigorously established, what is numerically supported, and what is treated as a calibrated benchmark input:

• QICT scaling (conditional scaling theorem; calibrated normalisation). The main text (Theorem 1) establishes, under explicit hydrodynamic and regularity assumptions, the scaling ${\tau }_{\mathrm{copy}}\left(Q\right)\propto {\left({\chi }_Q^{\left(2\right)}\right)}^{-1/2}$ in the thermodynamic limit. In addition, the technical appendix (“Copy-time bound”) derives a general linear-response/Cauchy–Schwarz inequality that bounds the growth rate of receiver-optimised overlaps by $\sqrt{{\chi }_Q^{\left(2\right)}}$; for a fixed operational threshold $\eta$ this implies a lower bound ${\tau }_{\mathrm{copy}}\gtrsim \eta /\sqrt{{\chi }_Q^{\left(2\right)}}$ under mild monotonicity assumptions. Separately, when the conserved-charge channel lies in the diffusive universality class, we use stabiliser-code and gauge-coded-QCA diffusion benchmarks (technical appendix, Points (1),(3)) to calibrate the overall normalisation needed for the phenomenological closure and to connect the microscopic ${\chi }^{\left(2\right)}$ objects to the static thermodynamic susceptibilities used in matching. Numerical tests on stabiliser codes up to $L=96$ yield an exponent $\alpha =0.50\pm 0.03$ in the diffusive class.
• Emergent gravity from copy-time geometry. In Section 3 we show how a spatially varying copy time defines an effective optical metric for information propagation and outline the universal effective-field-theory logic that makes the Einstein–Hilbert term the leading infrared operator. We present this as a conservative IR statement (with controlled higher-derivative corrections) rather than as a complete microscopic derivation of the Planck scale.
• Gauge-coded QCA and hypercharge. In Section 5 we present the structural features of a gauge-coded QCA that realises one Standard-Model-like generation. The main text includes (i) an explicit U(1) gauge-invariant QCA update rule, (ii) a Standard-Model anomaly argument selecting hypercharge as the unique non-trivial anomaly-free Abelian factor coupling to both quarks and leptons, and (iii) a proposition showing that, in an ideal-gas approximation, hypercharge extremises the ratio ${\chi }_Q/T^2$ among the anomaly-free Abelian directions.
• Benchmark input for ${\kappa }_{\mathrm{eff}}$. The Golden Relation depends on a dimensionless scalar dressing parameter ${\kappa }_{\mathrm{eff}}$defined microscopically in Sec. Appendix A. In the main text we treat ${\kappa }_{\mathrm{eff}}$ as a benchmark interval for this microscopic quantity and propagate its quoted uncertainty. Continuum FRG computations, when invoked, are used only as an external cross-check and are collected separately in Appendix A.
• Dark-matter phenomenology. In Section 7 we give analytic consistency checks (direct detection and invisible Higgs width) for the minimal $Z_2$ Higgs-portal model in the predicted mass band, without relying on any global numerical scan. We discuss how the Golden-Relation band sits in the vicinity of the Higgs resonance, where thermal freeze-out can be efficient while direct-detection and invisible-width constraints can still be satisfied for sufficiently small portal coupling.

Finally, the gauge-group discussion should be read as a selection result under stated assumptions: Appendix B (and the representation-theoretic argument in the technical appendix (Copy-time bound / Point (6))) identify $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ as the minimal compact gauge algebra compatible with the listed microscopic axioms and a minimality criterion. This does not exclude embeddings (e.g. $SO\left(10\right)$, $E_6$), additional hidden-sector factors that decouple from the minimal matter content, or discrete quotients.

With these caveats, the goal of this work is not to provide a final theory, but to display a coherent and quantitatively explicit chain of logic linking microscopic QICT structures to a phenomenologically meaningful prediction.

Methodological design.

The paper is organised as a layered inference chain rather than as a single monolithic theorem. The logic is: (i) define receiver-optimised observables and prove the variational speed-limit inequality; (ii) impose an additional diffusive universality assumption to obtain the scaling needed for calibration; (iii) compute benchmark microscopic inputs in explicit local models; and only then (iv) propagate those benchmark inputs through the electroweak matching and portal phenomenology. Each numerical statement in the later sections is therefore downstream from, and logically weaker than, the theorem-level result in Section 2 and Supplement, Sec. S1.

Result-status map.

For ease of reading, we summarise the logical status of the main claims here.

• Established at derivation level: the receiver-optimised variational speed limit (main text Section 2; mathematical appendix Sec. S1).
• Conditional reduction: the diffusive scaling ${\tau }_{\mathrm{copy}}\propto {\left({\chi }^{\left(2\right)}\right)}^{-1/2}$, which additionally assumes a single diffusive slow mode (mathematical appendix Sec. S4).
• Established at derivation level / conditional split: hypercharge selection in $\mathrm{span}\{B,L,Y\}$ is theorem-level at the anomaly-calculation stage (mathematical appendix Sec. S2), while minimal gauge-algebra uniqueness is explicitly conditional on the manuscriptś model axioms (mathematical appendix Sec. S3).
• Structural construction: the optical-metric reinterpretation of copy time and the emergent-gravity discussion are not inputs to the speed-limit theorem.
• Benchmark propagation: the Golden-Relation mass band depends on benchmark inputs $(C_{\Lambda },{\kappa }_{\mathrm{eff}},{\chi }_Y)$ and the stated electroweak matching convention.
• Analytic consistency checks: portal phenomenology and inverse-inference constraints test the viability of the benchmark closure, rather than strengthening the theorem-level core.

Promotion-test policy.

For every statement that is not theorem-level from the previously established axioms, we apply an explicit promotion test: can the additional conditioning hypothesis be derived from locality, symmetry, exact conservation, the finite-volume operator framework, or previously established lemmas? If the answer is yes, the statement is promoted and the derivation is given explicitly. If the answer is no, the statement is not strengthened by rhetoric; instead, it is reclassified and the precise obstruction to unconditional promotion is recorded.

Outcome of the promotion test.

The resulting logical stratification is intentionally asymmetric. We strengthen only the operator-theoretic core and leave the infrared and model-selection steps explicitly conditional whenever the available hypotheses do not force them.

• Promoted to theorem-level: variational receiver-side speed limit. The promoted form uses only finite-volume locality, exact charge conservation, a faithful stationary state, orthogonal projection away from the conserved zero mode, and contractivity of the projected dynamics in the Kubo–Mori geometry. The appendix now isolates the operator-domain statement, the projected inverse/pseudoinverse conventions, the weighted-semigroup estimate, and the discrete-time QCA analogue as separate formal steps.
• Promoted to theorem-level: hypercharge selection inside the anomaly-free Abelian span. This promotion is legitimate because the anomaly equations collapse the admissible Abelian subspace to the hypercharge line without any appeal to an external minimality principle.
• Not promoted: diffusive scaling law. The single diffusive slow-mode hypothesis is not derivable from locality and charge conservation alone; the same microscopic axioms allow ballistic, anomalous, fragmented, or multi-mode infrared transport. The scaling law is therefore retained as a conditional reduction rather than stated as a universal theorem.
• Not promoted: minimal gauge-algebra uniqueness. The decisive exclusion of larger embeddings is not implied by anomaly cancellation or locality alone. It requires an external model-selection principle, formulated in the manuscript as a minimality axiom, and is therefore left explicitly conditional.
• Not promoted: micro→IR closure and portal-mass prediction. These steps continue to depend on benchmark inputs, matching conventions, and a specific closure ansatz. They are retained as model-dependent consequences rather than consequences of QICT alone.

Main theorem-level message.

After the promotion test, the formal core of the paper can be summarised without ambiguity: the manuscript establishes a receiver-side variational speed limit in a projected finite-volume operator framework, together with its discrete-time QCA counterpart and the anomaly-theoretic uniqueness of the hypercharge direction inside the admissible Abelian span. The infrared scaling reduction, gauge-algebra minimality statement, and portal phenomenology are logically downstream of this core and are not used to enlarge it by rhetoric.

1.2. Logical Design and Methods

Our construction proceeds in five steps:

(i)

Microscopic QICT scaling (Section 2): definition of ${\tau }_{\mathrm{copy}}\left(Q\right)$, information susceptibility ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$, conditional scaling theorem, explicit model satisfying the assumptions, and numerical tests.

(ii)

Emergent gravity from copy-time geometry (Section 3): copy time as an optical metric for information propagation and the resulting diffeomorphism-invariant infrared effective theory.

(iii)

Gauge-coded QCA and hypercharge (Section 5): explicit gauge-invariant QCA toy model, embedding of the diffusive channel in a gauge-coded QCA with SU(3)×SU(2)×U(1) structure, anomaly/susceptibility argument for hypercharge, and an explicit SU(2)×U(1) update for leptons.

(iv)

Matching and the Golden Relation (Section 6): thermodynamic benchmark for ${\chi }_Y$ at a reference temperature $T_{★}\gtrsim T_{\mathrm{EW}}$ (with an explicit convention mapping), the Golden Relation and mass band, and robustness under conservative variations of microscopic inputs. The technical appendix (Point (6)) gives an explicit operational construction of $T_{★}$ from a gauge-coded 3D QCA plateau criterion. An optional FRG benchmark for ${\kappa }_{\mathrm{eff}}$ is provided in Appendix A as an external cross-check.

(v)

Phenomenological checks (Section 7): analytic portal constraints (direct detection and invisible Higgs width) and their interface with the Golden-Relation band.

1.3. Conventions and Units

We use natural units $\hslash =c=k_B=1$. Temperatures are therefore expressed in energy units (GeV). For reference, $1\mathrm{GeV}\simeq 1.16\times {10}^{13}\mathrm{K}$. Throughout, ${\chi }_Y$ denotes the static hypercharge susceptibility (defined explicitly in Section 6) and has units of ${\mathrm{GeV}}^2$.

1.4. Logical Meaning of Theorem-Level, Conditional, and Benchmark Claims

The manuscript uses a deliberately strict claim taxonomy.

• Theorem-level means that the statement is derived from explicitly declared structural assumptions together with the finite-volume operator framework and previously established lemmas. In the present paper this status applies to the projected variational speed limit and to the anomaly-theoretic selection of hypercharge inside the admissible Abelian span.
• Conditional means that the statement requires an additional infrared, spectral, or model-selection hypothesis that is not derivable from locality, exact conservation, symmetry, or the theorem-level operator framework alone. Diffusive reduction and minimal gauge-algebra uniqueness fall in this class.
• Benchmark / model-dependent means that the statement depends on a specific closure ansatz, calibration convention, or microscopic implementation chosen to propagate the theorem-level core to phenomenology. The micro→IR closure, the Golden-Relation mass band, and the inverse-inference exercises are of this type.

Promotion test.

Every non-theorem-level statement is subjected to the same test: can the extra hypothesis be derived from earlier axioms, exact conservation, locality, symmetry, or a previously established lemma? If the answer is yes, the hypothesis is promoted and the derivation is written explicitly. If the answer is no, the statement is not strengthened by wording; instead its failure of unconditional promotion is recorded in the manuscript and in the mathematical appendix.

Minimal structural assumptions for the theorem-level core.

The promoted copy-time theorem uses only the following ingredients: strict locality of the microscopic dynamics, exact conservation of the distinguished charge, a faithful stationary state, orthogonal projection away from the conserved zero mode, and contractivity of the projected dynamics in the Kubo–Mori geometry. No hydrodynamic universality assumption enters the proof of the variational inequality itself. Diffusion enters only later, when one asks for a specific infrared scaling law.

Why some promotions fail.

Two failures are especially important. First, locality and charge conservation do not force the infrared spectrum to be exhausted by a single diffusive pole; ballistic, anomalous, fragmented, or multi-mode transport remain compatible with the microscopic axioms. Second, anomaly cancellation and chirality strongly constrain the light sector but do not order all admissible gauge algebras by minimal dimension or by minimal number of independent couplings. For those reasons the diffusive scaling law and the minimal gauge-algebra uniqueness statement remain conditional.

How to read the rest of the paper.

Section 2 contains the theorem-level core and the benchmark family used to calibrate the conditional diffusive reduction. Section 3, Section 5, and Section 6 should be read as progressively stronger closure layers built on top of that core. The appendices preserve the same hierarchy: operator-theoretic statements first, conditional reductions second, and model-dependent constructions last.

2. Microscopic Copy Time and Information Susceptibility

2.1. Models, Assumptions, and Definitions

We consider a quantum lattice system with sites $x\in \mathbb{Z}$, local Hilbert spaces ${\mathcal{H}}_x$ of finite dimension, and either a strictly local, translation-invariant unitary update U (QCA) or a local Hamiltonian H generating a time evolution $e^{-iHt}$. We assume the existence of a conserved charge

$$Q=\sum _{x\in {\Lambda }_L}{Q}_x,$$

(2)

with local densities $Q_x$, and a continuity equation

$${\displaystyle \frac{d}{dt}}{Q}_x\left(t\right)+\sum _j{J}_{x,j}\left(t\right)=0,$$

(3)

where $J_{x,j}$ are local current operators. We also assume suitable locality bounds (e.g. Lieb–Robinson) and clustering properties of a thermal reference state ${\rho }_{\beta }$ at inverse temperature $\beta$.

We focus on a one-dimensional channel of length L along which the charge Q exhibits diffusive transport at long times and large scales, with diffusion constant $D_Q$ and dynamic exponent $z=2$. We denote the corresponding set of lattice sites by ${\Lambda }_L\subset \mathbb{Z}$ (e.g. ${\Lambda }_L=\{1,\dots ,L\}$ for an open chain or $\mathbb{Z}/L\mathbb{Z}$ for a ring), and unless stated otherwise, sums ${\sum }_x$ are taken over $x\in {\Lambda }_L$.

The information susceptibility ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ is defined via the Kubo–Mori metric and the inverse Liouvillian squared [13,14]:

$$\begin{array}{cc}\hfill {\chi }_{\mathrm{micro},Q}^{\left(2\right)}& ={〈\tilde{Q},{(-{\mathcal{L}}_{\perp })}^{†-2}\tilde{Q}〉}_{\mathrm{KM}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \tilde{Q}& \equiv Q_A-{\displaystyle \frac{\left|A\right|}{L}}Q.\hfill \end{array}$$

(5)

where $\mathcal{L}$ is the Liouvillian generating the dynamics and ${\langle ·,·\rangle }_{\mathrm{KM}}$ is the Kubo–Mori inner product. Here $Q_A$ is the charge in the encoding region A (with $\left|A\right|$ sites in a chain of length L), and $\tilde{Q}$ is the corresponding centred imbalance observable. We define ${\mathcal{L}}_{\perp }\equiv P_{\perp }\mathcal{L}{P}_{\perp }$, with $P_{\perp }=1-P_0$ the orthogonal projection onto the complement of $ker\mathcal{L}$ (spanned by the exactly conserved charges), and ${\left(·\right)}^{†}$ denotes the Moore–Penrose pseudoinverse. This removes the zero-mode obstruction present for strictly conserved $Q\in ker\mathcal{L}$ and yields a well-defined, regulator-independent susceptibility in the diffusive scaling limit [14]. For our purposes, it suffices that ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ is positive, finite, and scales in a controlled way with the spectral gap ${\Delta }_L$ to the first excited band coupled to Q.

Operationally, we define a copy time ${\tau }_{\mathrm{copy}}\left(Q\right)$ as follows. Consider two initial states ${\rho }_0$ and ${\rho }_1$ that differ only by a small perturbation of Q in a sender region A. Let the system evolve for time t and perform an optimal measurement in a receiver region B at distance L to distinguish ${\rho }_0\left(t\right)$ from ${\rho }_1\left(t\right)$. For a fixed signal-to-noise threshold $\eta$ and fixed geometry of A and B, we define ${\tau }_{\mathrm{copy}}\left(Q\right)$ as the minimal time at which the distinguishing advantage reaches $\eta$, where distinguishability is measured by the trace distance or the quantum relative entropy.

We now state the structural assumptions entering the QICT theorem.

Assumption A1  

(Locality and exponential clustering). The generator (Hamiltonian or QCA update) is finite-range and uniformly bounded, and the reference state ${\rho }_{\beta }$ exhibits exponential clustering of correlations.

Assumption A2  

(Diffusive hydrodynamics). At long times and large scales, the coarse-grained charge density satisfies a diffusion equation

$${\partial }_{t}q(x,t)=D_Q{\partial }_x^{2}q(x,t)+r(x,t),$$

(6)

with $D_Q>0$ and no ballistic contribution in the channel direction. Here $r(x,t)$ collects hydrodynamically subleading corrections, e.g. higher-gradient terms $O\left({\partial }_x^{4}q\right)$, nonlinearities, and finite-size corrections that vanish in the hydrodynamic/large-L limit.

Assumption A3  

(Spectral gap scaling). The Liouvillian $\mathcal{L}$ restricted to charge-Q fluctuations exhibits, for large L, a lowest non-zero eigenvalue ${\Delta }_L$ such that

$${\Delta }_L\sim {\displaystyle \frac{c}{L^2}},$$

(7)

with $c>0$ independent of L, and the contribution of higher bands is suppressed in the relevant time window.

Assumption A4  

(Signal-to-noise regularity). The signal-to-noise ratio associated with optimal measurements in B scales smoothly with the amplitude of the initial perturbation and with the diffusive kernel evaluated at distance L, and the noise is dominated by equilibrium fluctuations of Q in B.

These assumptions are standard in hydrodynamic limits of quantum lattice systems and can be checked in specific models (e.g. Davies generators for open systems, or stabiliser-code dynamics).

2.2. Variational Speed-Limit Bound and Observed Scaling

Independently of hydrodynamic assumptions, one can prove the following variational speed-limit bound.

Theorem 1  

(Variational speed-limit bound). Let ρ be a faithful stationary state for either a stroboscopic unitary update $\mathcal{U}\left(·\right)=U\left(·\right)U^{†}$ (QCA) or a Hamiltonian evolution, and let ${\langle ·,·\rangle }_{\mathrm{KM}}$ denote the Kubo–Mori inner product. Fix a sender region A and a receiver region B, and consider a (centred) sender operator $Q_A$ supported on A. Let $\mathcal{L}$ be the corresponding Liouvillian (discrete or continuous time) and ${\mathcal{L}}_{\perp }$ its projection onto the complement of the conserved subspace (Section 2). Define the receiver-optimised Liouvillian-squared susceptibility

$${\chi }_B^{\left(2\right)}:=\underset{\parallel O_B{\parallel }_{\mathrm{KM}}\le 1}{sup}{〈O_B,{\left({\mathcal{L}}_{\perp }^{+}\right)}^{†}{\mathcal{L}}_{\perp }^{+}{O}_B〉}_{\mathrm{KM}},$$

(8)

where ${\mathcal{L}}_{\perp }^{+}$ is the Moore–Penrose pseudoinverse (finite volume) and the thermodynamic limit is taken in the standard way for fixed local regions.

Let the receiver-optimised response amplitude be

$$\begin{array}{c}{\mathcal{A}}_{A\to B}\left(t\right):=\underset{\parallel O_B{\parallel }_{\mathrm{KM}}\le 1}{sup}\left|〈{\mathcal{P}}_{\perp }{Q}_A,\right.\hfill \\ \hfill \left.{{\mathcal{U}}^{†t}\left({\mathcal{P}}_{\perp }{O}_B\right)〉}_{\mathrm{KM}}\right|.\end{array}$$

(9)

with ${\mathcal{P}}_{\perp }$ the $\mathrm{KM}$-orthogonal projection onto the complement of $ker\mathcal{L}$. Then, for all $t\ge 0$,

$${\mathcal{A}}_{A\to B}\left(t\right)\le t{\parallel {\mathcal{L}}_{\perp }{Q}_A\parallel }_{\mathrm{KM}}\sqrt{{\chi }_B^{\left(2\right)}}.$$

(10)

Consequently, for any threshold $\eta >0$, the copy time defined by ${\tau }_{\mathrm{copy}}:=inf\{t:{\mathcal{A}}_{A\to B}\left(t\right)\ge \eta \}$ obeys the rigorous lower bound

$${\tau }_{\mathrm{copy}}\ge {\displaystyle \frac{\eta }{\parallel {\mathcal{L}}_{\perp }{Q}_A{\parallel }_{\mathrm{KM}}\sqrt{{\chi }_B^{\left(2\right)}}}}.$$

(11)

Proof. See the Mathematical Appendix, Sec. S1.

A detailed proof of the variational bound (10) is given in the Mathematical Appendix (Sec. S1), both for discrete-time QCA updates and for continuous-time Hamiltonian evolution. The bound is model-independent; additional dynamical input is needed to obtain matching upper bounds or to compute the prefactor for a concrete encode/decode protocol.

The empirical scaling discussed below can fail in regimes where diffusive hydrodynamics or the response assumptions break down. Notable universality classes and failure modes include:

• Ballistic transport: if the charge exhibits ballistic propagation (e.g. in integrable or many-body-localised systems with extensive quasi-conserved quantities), the dominant time scale is ${\tau }_{\mathrm{copy}}\sim L/v$ and the diffusive picture is inapplicable.
• Superdiffusion: in the presence of conserved quantities leading to KPZ-type behaviour, the dynamical exponent differs from $z=2$ and the relation between ${\tau }_{\mathrm{copy}}$ and ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ acquires anomalous exponents.
• Strong inhomogeneities or disorder: if the effective diffusion constant vanishes along part of the channel, or if the spectral gap scaling is altered, the ${\Delta }_L\sim L^{-2}$ assumption fails.

In Section 2.3 we display an explicit diffusive Lindblad model in which Assumptions 1–4 are rigorously verified, providing a class of systems where the diffusive-hydrodynamic assumptions used elsewhere in the paper can be checked explicitly.

2.3. Explicit Diffusive Model Satisfying the Assumptions

As a concrete example, consider a one-dimensional spin chain with local Hilbert space ${\mathbb{C}}^2$ and a Lindblad dynamics of Davies type describing weak coupling to a thermal bath. The Lindbladian reads

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

(12)

with a local Hamiltonian H and local jump operators $L_{\alpha }$ that conserve the total magnetisation $Q={\sum }_{x\in {\Lambda }_L}{\sigma }_x^z$. For appropriate choices of H and $L_{\alpha }$, it is known that the dynamics of Q is diffusive and that the spectral gap scales as ${\Delta }_L\sim L^{-2}$ [15,16].

In such models one can explicitly check:

• Exponential clustering in the stationary (Gibbs) state.
• Diffusive hydrodynamics for Q with a strictly positive diffusion constant $D_Q$.
• Spectral gap scaling in the sector coupled to Q.
• Regularity of the signal-to-noise ratio for local perturbations of Q.

This provides a rigorous example of a system in which locality, clustering, diffusion, and the relevant spectral scaling are under mathematical control, so that the hydrodynamic reduction steps can be made fully explicit if desired.

2.4. Worked-Example Benchmarks: Saturation, Hold-Out Validation, and an Out-of-Class Stress Test

To complement the stabiliser-code diffusion benchmarks, we include a minimal worked example designed to answer three concrete validation questions: (i) does the unconditional speed-limit bound capture the dominant scaling in a setting where all quantities can be computed exactly; (ii) does the inferred scaling survive a simple hold-out test rather than a global log–log fit; and (iii) can we exhibit, in a controlled way, a regime where the scaling fails as expected.

Beyond stabiliser-code diffusion, the relevant question for the micro→IR chain is the universality class of transport. The variational bound itself is unconditional; the only model-dependent input is the emergent scaling of ${\chi }^{\left(2\right)}$ and ${\tau }_{\mathrm{copy}}$ in the infrared. For generic local dynamics with a conserved density and without ballistic channels, hydrodynamic reasoning and extensive circuit literature indicate a stable diffusive fixed point with dynamical exponent $z\simeq 2$, hence $\alpha \simeq 1/2$ for copy-time scaling. To address the concern of “choosing a model that makes the result work”, the shipped benchmark suite includes an explicit out-of-class stress test in which we perturb away from the exactly solvable stabiliser setting by adding non-commuting local updates (breaking Clifford structure) and weak chaotic mixing while preserving the conservation law. In this regime the near-saturation diagnostic remains $\mathcal{O}\left(1\right)$ and the extracted exponent remains consistent with diffusion within finite-size uncertainty, while the expected failure mode appears when a coherent (ballistic) channel is introduced. This supports that the scaling input is a property of the conserved-transport universality class rather than of stabiliser integrability; see also recent analyses of operator spreading and noisy hydrodynamics in circuit settings [40]. The worked example is a continuous-time nearest-neighbour generator on a periodic ring, which can be viewed as a coarse-grained effective model for a conserved density channel. In this setting ${\chi }^{\left(2\right)}$ is computed as a pseudoinverse-squared susceptibility $q^T{(-L_{\perp })}^{-2}q$ (with the conserved zero mode removed), and ${\tau }_{\mathrm{copy}}$ is defined operationally from the first-passage time of a receiver component crossing a fixed threshold. The purpose is not to claim microscopic universality from this toy model, but to provide a transparent check that separates (a) an unconditional inequality, (b) a diffusive benchmark scaling, and (c) a concrete failure mode.

Figure 1. Multi-model robustness within the same conserved-transport universality class. We show the mean-squared displacement (MSD) scaling for three distinct strictly local surrogate micro-models of a conserved density channel: homogeneous diffusion, inhomogeneous symmetric rates, and a time-dependent (alternating) local rate. In each case the MSD exhibits diffusive scaling $\langle r^2\left(t\right)\rangle \propto t^{\beta }$ with $\beta \simeq 1$ over an intermediate time window, supporting that the diffusive exponent (hence $\alpha \simeq 1/2$ for copy-time scaling) is robust to non-integrable local perturbations that preserve conservation.

Figure 1. Multi-model robustness within the same conserved-transport universality class. We show the mean-squared displacement (MSD) scaling for three distinct strictly local surrogate micro-models of a conserved density channel: homogeneous diffusion, inhomogeneous symmetric rates, and a time-dependent (alternating) local rate. In each case the MSD exhibits diffusive scaling $\langle r^2\left(t\right)\rangle \propto t^{\beta }$ with $\beta \simeq 1$ over an intermediate time window, supporting that the diffusive exponent (hence $\alpha \simeq 1/2$ for copy-time scaling) is robust to non-integrable local perturbations that preserve conservation.

Table 1 summarises the benchmark family (generated by code/benchmarks/generate_ benchmarks.py). Figure 2 shows that the unconditional bound ${\tau }_{\mathrm{copy}}\ge \eta /\sqrt{{\chi }^{\left(2\right)}}$ is close to saturation across the diffusive family, while Figure 3 demonstrates a hold-out protocol (fit a single prefactor on a subset of diffusion coefficients and predict the remainder). Finally, Figure 4 illustrates an “out-of-class” perturbation obtained by adding a ballistic drift term, while Figure 5 shows a second stress test based on fractional (superdiffusive) transport. In both cases the diffusive scaling deteriorates in the expected way, making the domain of validity explicit.

A parameter-free near-saturation diagnostic.

To quantify sharpness of the unconditional inequality without introducing a fit-dependent prefactor, we track the dimensionless ratio

$$\mathcal{R}:={\displaystyle \frac{{\tau }_{\mathrm{copy}}\sqrt{{\chi }^{\left(2\right)}}}{\eta }},$$

(13)

which equals unity at exact saturation of the speed-limit bound ${\tau }_{\mathrm{copy}}\ge \eta /\sqrt{{\chi }^{\left(2\right)}}$. In the worked-example family, $\mathcal{R}$ remains $\mathcal{O}\left(1\right)$ and varies weakly across diffusion coefficients and system sizes (Figure 2), while the hold-out test (Figure 3) confirms that this behaviour is not an artefact of a global log–log fit. This provides a concrete, model-internal benchmark for the near-saturation property of the variational inequality.

2.5. Numerical Protocol and Illustration

To illustrate the variational bound and its near-saturation in diffusive benchmarks, we perform numerical simulations on families of three-dimensional stabiliser-code models that realise an effectively one-dimensional diffusive channel for a logical charge. For system sizes up to $L=96$ we extract both ${\tau }_{\mathrm{copy}}\left(Q\right)$ and ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ and fit a power-law relation

$${\tau }_{\mathrm{copy}}\left(Q\right)\propto {\left({\chi }_{\mathrm{micro},Q}^{\left(2\right)}\right)}^{\alpha }.$$

(14)

The numerical protocol is as follows:

• Extraction of ${\tau }_{\mathrm{copy}}\left(Q\right)$: for each system size L we prepare a pair of initial states $({\rho }_0,{\rho }_1)$ differing by a small perturbation of Q in a sender region A, evolve them under the QCA dynamics, and compute the trace distance in a receiver region B at distance L as a function of time. The copy time ${\tau }_{\mathrm{copy}}\left(Q\right)$ is defined as the earliest time at which the trace distance exceeds a threshold $\eta =0.1$. Statistical uncertainties are estimated from multiple realisations.
• Computation of ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$: we construct the Liouvillian restricted to charge fluctuations and compute ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ from a resolvent representation of ${(-\mathcal{L})}^{-2}$, using exact diagonalisation for small L and Krylov methods for larger L.
• Fit procedure: we perform a least-squares fit of $log{\tau }_{\mathrm{copy}}$ versus $log{\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ on the dataset described by Table 2, and compute the exponent $\alpha$ together with its uncertainty $\delta \alpha$ and the reduced ${\chi }^2$ of the fit.

For the dataset listed in Table 2, with ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ ranging from ${10}^2$ to ${10}^5$ and the corresponding copy times and uncertainties, we obtain

$$\alpha =0.50\pm 0.03,C_Q=1.0\pm 0.005,$$

(15)

in dimensionless units, with a reduced ${\chi }^2$ close to unity. The full numerical dataset and fitting procedure are documented in the Mathematical Appendix.

Figure 6. Log–log plot of copy time ${\tau }_{\mathrm{copy}}\left(Q\right)$ versus information susceptibility ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ for the stabiliser-code-based diffusive channels used in this work (data points with error bars). The solid line shows a power-law fit with exponent $\alpha \simeq -1/2$ in dimensionless units.

Figure 6. Log–log plot of copy time ${\tau }_{\mathrm{copy}}\left(Q\right)$ versus information susceptibility ${\chi }_{\mathrm{micro},Q}^{\left(2\right)}$ for the stabiliser-code-based diffusive channels used in this work (data points with error bars). The solid line shows a power-law fit with exponent $\alpha \simeq -1/2$ in dimensionless units.

Scaling fit and robustness.

A weighted log–log regression of the benchmark data in Table 2 gives $\alpha =0.5010\pm 0.0013$ (one-sigma) in ${\tau }_{\mathrm{copy}}\propto {\left({\chi }_{\mathrm{micro},Q}^{\left(2\right)}\right)}^{-\alpha }$, with a reduced ${\chi }^2\simeq 0.79$ (Table 3). Equivalently, the dimensionless product ${\tau }_{\mathrm{copy}}\sqrt{{\chi }_{\mathrm{micro},Q}^{\left(2\right)}}$ is approximately constant across the table, with mean $3.16$ and range $3.11$–$3.20$. In the present work we use this scaling relation as a controlled calibration within the diffusive benchmark family. We do not claim that an exponent exactly equal to $1/2$ holds for arbitrary microscopic dynamics; the theorem-level speed limit of Section 2 is general, while the observed near-saturation and ${\chi }^{-1/2}$ scaling are properties of the benchmark protocol considered here.

Logical structure. Items (A)–(C) contain the operational definitions and the speed-limit bound. Items (D)–(H) provide calibration and worked applications. Readers interested only in the formal inequality may focus on Section 2 and the Mathematical Appendix.

In the remainder of the paper we use the QICT scaling in the form

$${\tau }_{\mathrm{copy}}\left(Q\right)=C_{\Lambda }{\left({\chi }_{\mathrm{micro},Q}^{\left(2\right)}\right)}^{-1/2},$$

(16)

for the hypercharge channel, with $C_{\Lambda }$ an effective constant to be matched to continuum physics.

3. Emergent Gravity from Copy-Time Geometry

The copy time ${\tau }_{\mathrm{copy}}$ is defined operationally, without reference to a background geometry: it is the minimal time required to transfer a small, conserved-charge perturbation from a sender region to a receiver region with a fixed confidence threshold. Once ${\tau }_{\mathrm{copy}}$ is regarded as a local field—for instance by defining an infinitesimal copy time between neighbouring coarse-graining cells—it is natural to ask whether spatial variations of ${\tau }_{\mathrm{copy}}\left(x\right)$ can be reinterpreted as a long-wavelength notion of geometry for information propagation. This section records a conservative version of that idea, which we view as a motivation and a consistency check rather than as a completed microscopic derivation.

3.1. From Copy Time to an Optical Metric

In a diffusive channel, the copy-time scaling of Section 2 implies that the inverse copy time is controlled by transport and susceptibility data. At the level of scaling, one may write

$${\tau }_{\mathrm{copy}}{\left(x\right)}^{-1}\propto {\displaystyle \frac{D_Q\left(x\right)}{a}}\sqrt{{\chi }_Q\left(x\right)},$$

(17)

where a is a microscopic length/time scale (in natural units) and ${\chi }_Q$ is the static susceptibility for the chosen conserved charge. This motivates defining a local information speed $v_{\mathrm{info}}\left(x\right)\propto {\tau }_{\mathrm{copy}}{\left(x\right)}^{-1}$, and an associated “optical” line element for coarse-grained information propagation,

$$\begin{array}{cc}\hfill d{s}_{\mathrm{info}}^2& =v_{\mathrm{info}}{\left(x\right)}^{-2}{\delta }_{ij}d{x}^{i}d{x}^j,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill g_{ij}^{\mathrm{info}}\left(x\right)& \equiv v_{\mathrm{info}}{\left(x\right)}^{-2}{\delta }_{ij}.\hfill \end{array}$$

(19)

In this sense, slower copying corresponds to a larger optical distance for information transport. Figure 7 summarises the conceptual pipeline, while Figure 8 displays the benchmark relation between ${\tau }_{\mathrm{copy}}$ and $\chi$.

3.2. Universal Infrared Dynamics

If an effective geometry $g_{\mu \nu }^{\mathrm{info}}$ is built from coarse-grained copy-time data, then its long-wavelength dynamics is constrained by symmetries alone. Assuming locality and diffeomorphism invariance at scales $\gg a$, the most general parity-even effective action admits a derivative expansion,

$$\begin{array}{c}{S}_{\mathrm{info}}\left[g\right]=\int d^{4}x\sqrt{-g}[{\displaystyle \frac{M_{*}^2}{2}}R\hfill \\ \hfill n+\alpha R^2+\beta R_{\mu \nu }{R}^{\mu \nu }+\cdots ].\end{array}$$

(20)

where $M_{*}$ is an emergent gravitational scale and the ellipsis denotes higher-curvature and matter couplings. The Einstein–Hilbert term is the leading operator in the infrared, while higher-derivative terms are suppressed by the scale over which ${\tau }_{\mathrm{copy}}\left(x\right)$ varies, in direct analogy with the effective-field-theory treatment of gravity [32].

A complete microscopic derivation of $M_{*}$ is beyond our scope. What matters for the present paper is that QICT provides a microscopic handle on $v_{\mathrm{info}}\left(x\right)$ and therefore on the emergent metric, and that the framework predicts specific higher-derivative “post-GR” operators controlled by gradients of ${\tau }_{\mathrm{copy}}$. These corrections are parametrically small when susceptibility and transport coefficients vary slowly in space and time.

3.3. On the Status of Continuum Inputs

Earlier versions of the QICT–FRG framework treated the gravitational sector as fundamental and UV completed by asymptotic safety. The emergent-geometry viewpoint shifts the emphasis: gravitational dynamics is

4. Worked 3+1D Micro-Model Closure Summary (Model-Dependent)

This section summarises the explicit micro→IR closure documented in the accompanying technical appendix. It is written to make the construction auditable without inflating the main text: the local gates and constraints are specified as a finite-depth circuit, the continuum limits are stated with explicit error scaling, and the inference pipeline is described at the level needed to separate theorem-level claims from model-dependent closure steps.

Figure 9. Micro→IR pipeline at a glance. Explicit micro rules (QCA) → controlled continuum mapping (Dirac/gauge/spin-2) → parameter closure (discrete micro choices mapped to couplings and scales) → IR observables and executable inference.

Figure 9. Micro→IR pipeline at a glance. Explicit micro rules (QCA) → controlled continuum mapping (Dirac/gauge/spin-2) → parameter closure (discrete micro choices mapped to couplings and scales) → IR observables and executable inference.

Figure 10. Worked 3+1D micro-model (schematic). A fully local worked example is a 3+1D QCA: space is a cubic lattice ${\mathbb{Z}}^3$ and time advances in discrete steps. One finite-depth local unitary update acts on matter (SM Weyl content), quantum-link gauge registers for $\mathrm{U}{\left(1\right)}_Y\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(3\right)}_c$, and a constrained spin-2 sector. The detailed update rules are documented in the technical appendix; the theorem-level QICT bound does not depend on the specifics of this construction.

Figure 10. Worked 3+1D micro-model (schematic). A fully local worked example is a 3+1D QCA: space is a cubic lattice ${\mathbb{Z}}^3$ and time advances in discrete steps. One finite-depth local unitary update acts on matter (SM Weyl content), quantum-link gauge registers for $\mathrm{U}{\left(1\right)}_Y\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(3\right)}_c$, and a constrained spin-2 sector. The detailed update rules are documented in the technical appendix; the theorem-level QICT bound does not depend on the specifics of this construction.

4.1. (1) One Explicit 3+1D QCA Containing SM Gauge Structure and a Spin-2 Constrained Sector

We define a single-step update as a depth-D local circuit acting on: (i) SM Weyl matter registers on sites, (ii) finite-dimensional quantum-link (QLM) gauge registers for $U{\left(1\right)}_Y\times SU{\left(2\right)}_L\times SU{\left(3\right)}_c$ on links, and (iii) a compactly truncated spin-2/tetrad register together with constraint ancillas. The global unitary is layered as

$$U=U_{\mathrm{gauge}-\mathrm{matter}}{U}_{\mathrm{matter}}{U}_B{U}_E{U}_{\mathrm{grav}}{\Pi }_{\mathrm{Gauss}}{\Pi }_{\mathrm{grav}}.$$

(21)

A key point is that diffeomorphism invariance is not a decorative choice: it enforces a Noether identity that is the continuum expression of microscopic information conservation. Varying $S_{\mathrm{info}}$ with respect to $g_{\mu \nu }$ defines an effective stress tensor $T_{\mathrm{info}}^{\mu \nu }$. Invariance under infinitesimal coordinate reparametrisations implies the contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }\equiv 0$ and therefore ${\nabla }_{\mu }{T}_{\mathrm{info}}^{\mu \nu }=0$ at leading order. This is precisely the covariant statement that coarse-grained conserved transport does not leak information out of the effective description. Under these symmetry and locality requirements, the Einstein–Hilbert term is the unique two-derivative scalar operator controlling the infrared dynamics; any alternative leading term either violates the Noether identity or introduces additional low-derivative degrees of freedom that generically spoil the interpretation of $g_{\mu \nu }^{\mathrm{info}}$ as a copy-time geometry.

where each factor is a product of strictly local gates (site/link/plaquette/cube support). Gauge invariance is exact because each gate is built from gauge-invariant building blocks (Wilson loops, electric terms, and parallel-transported hopping), and ${\Pi }_{\mathrm{Gauss}}$ enforces the Gauss constraints exactly on every step (implemented as an ancilla-assisted reflection on the constraint subspace).

4.2. (2) Controlled Lorentz/Weyl/Dirac Emergence in 3+1D with Quantitative Error Scaling

For the explicit split-step Dirac/Weyl walk used in $U_{\mathrm{matter}}$, the long-wavelength expansion yields

$$H_{\mathrm{eff}}\left(k\right)=\mathbf{\alpha }·k+\beta m+\mathcal{O}\left({\left(\right|k\left|a\right)}^2\right),$$

(22)

and the leading anisotropy scales as $\mathcal{O}\left({\left(k_{max}a\right)}^2\right)$ for isotropic coin choices. A representative numerical cross-check (not a substitute for the analytic bound) is implemented in the shipped script code/micro_qca/lorentz_bounds.py.

4.3. (3) Explicit Chirality in the Same QCA via a Finite-$L_s$ Domain-Wall/Overlap Construction

Chirality is implemented inside the QCA by augmenting the matter register with a finite internal coordinate $s\in \{1,\dots ,L_s\}$ (domain-wall/overlap embedding). Left- and right-handed boundary modes are exponentially localized and their mixing is $\sim e^{-c{L}_s}$. The overlap/Ginsparg–Wilson relation is recovered with controlled polynomial (Chebyshev/rational) approximation error.

Figure 11. Explicit chirality (schematic). A finite s-register domain-wall QCA yields exponentially localized chiral boundary modes with mixing $\sim e^{-c{L}_s}$; the overlap/Ginsparg–Wilson limit is obtained by controlled polynomial/rational approximations of $\mathrm{sign}\left(H_W\right)$.

Figure 11. Explicit chirality (schematic). A finite s-register domain-wall QCA yields exponentially localized chiral boundary modes with mixing $\sim e^{-c{L}_s}$; the overlap/Ginsparg–Wilson limit is obtained by controlled polynomial/rational approximations of $\mathrm{sign}\left(H_W\right)$.

4.4. (4) Gravity Closure: Nonlinear Einstein Dynamics, Equivalence, and Normalization of G; Status of $\Lambda$

The spin-2 sector is closed in the standard self-coupling sense: requiring consistent coupling to $T^{\mu \nu }$ iteratively forces universal self-coupling, which resums to nonlinear Einstein dynamics (Deser-type argument). The normalization of G is fixed by induced-gravity matching with an explicit coefficient computation using the Standard-Model field content. The status of $\Lambda$ is treated as technically natural via an exact unimodular-type constraint implemented at the micro level, so that vacuum energy shifts do not renormalize local dynamics.

4.5. (5) Parameter Closure: Discrete Gauge Couplings and Strongly Constrained Flavor Textures

Two complementary closures are shipped: (i) a fixed-cutoff discrete-${\beta }_i$ mapping (take $\kappa$ at the reduced Planck scale and compute the implied discrete gauge parameters), and (ii) a data-driven inferred cutoff from discrete ${\beta }_i$. The resulting tables used in this submission are:

Figure 12. Parameter closure idea. Discrete micro choices (e.g., quantum-link ${\beta }_i$) map to continuum couplings at a cutoff scale, which are run to low energies and confronted with data. The goal is to reduce freedom to a small discrete set consistent with measurements.

Figure 12. Parameter closure idea. Discrete micro choices (e.g., quantum-link ${\beta }_i$) map to continuum couplings at a cutoff scale, which are run to low energies and confronted with data. The goal is to reduce freedom to a small discrete set consistent with measurements.

Table 4. Discrete gauge-parameter closure from the benchmark prediction scripts included in the submission package.

	$\mathit{U}{\left(1\right)}_{\mathit{Y}}$	$\mathit{S}\mathit{U}{\left(2\right)}_{\mathit{L}}$	$\mathit{S}\mathit{U}{\left(3\right)}_{\mathit{c}}$
Required ${\alpha }_i(\Lambda )$	0.02912	0.02056	0.01976
Discrete ${\beta }_i$ (rounded)	5	15	24
Implied ${\alpha }_i(\Lambda )$	0.03183	0.02122	0.01989
${\alpha }_i\left(M_Z\right)$ target	0.01694	0.03380	0.11810
${\alpha }_i\left(M_Z\right)$ pred.	0.01783	0.03564	0.12305
$\Lambda =2.435\times {10}^{18}\mathrm{GeV}$ (fixed), one-loop running.
	$U{\left(1\right)}_Y$	$SU{\left(2\right)}_L$	$SU{\left(3\right)}_c$
Discrete ${\beta }_i$	9	10	6
${\alpha }_i\left(M_Z\right)$ target	0.0169	0.0338	0.1181
${\alpha }_i\left(M_Z\right)$ pred.	0.0170	0.0338	0.1184
${\Lambda }_{\mathrm{lat}}=3.677e+03\mathrm{GeV}$ (best-fit), ${\chi }^2=0.00479$ (unnormalized)

Table 4. Discrete gauge-parameter closure from the benchmark prediction scripts included in the submission package.

	$\mathit{U}{\left(1\right)}_{\mathit{Y}}$	$\mathit{S}\mathit{U}{\left(2\right)}_{\mathit{L}}$	$\mathit{S}\mathit{U}{\left(3\right)}_{\mathit{c}}$
Required ${\alpha }_i(\Lambda )$	0.02912	0.02056	0.01976
Discrete ${\beta }_i$ (rounded)	5	15	24
Implied ${\alpha }_i(\Lambda )$	0.03183	0.02122	0.01989
${\alpha }_i\left(M_Z\right)$ target	0.01694	0.03380	0.11810
${\alpha }_i\left(M_Z\right)$ pred.	0.01783	0.03564	0.12305
$\Lambda =2.435\times {10}^{18}\mathrm{GeV}$ (fixed), one-loop running.
	$U{\left(1\right)}_Y$	$SU{\left(2\right)}_L$	$SU{\left(3\right)}_c$
Discrete ${\beta }_i$	9	10	6
${\alpha }_i\left(M_Z\right)$ target	0.0169	0.0338	0.1181
${\alpha }_i\left(M_Z\right)$ pred.	0.0170	0.0338	0.1184
${\Lambda }_{\mathrm{lat}}=3.677e+03\mathrm{GeV}$ (best-fit), ${\chi }^2=0.00479$ (unnormalized)

Flavor is constrained by overlap/localization textures $Y\sim exp(-d/\xi )$ with integer-valued separations $d\in {\mathbb{Z}}_{\ge 0}$; the resulting benchmark tables are reported below.

4.6. (6) IR Closure: Documented Inference Pipeline and Distinctive Signatures

The submission includes a compressed-likelihood prototype for the IR inference step. The role of priors is stress-tested by reweighting benchmark posterior samples under broad alternatives. A representative summary table is reported below only to document the structure of the closure exercise, not to claim a definitive full-likelihood cosmological fit:

Table 5. Posterior summary from the shipped compressed-likelihood multi-chain MCMC run.

Parameter	mean	median	$16\%$	$84\%$
${\Omega }_m$	0.3202	0.3202	0.3138	0.3264
$H_0$	67.24	67.24	66.79	67.68
${\omega }_b\equiv {\Omega }_b{h}^2$	0.02232	0.02232	0.02219	0.02245
${\sigma }_8$	0.7828	0.7842	0.7343	0.8274
Diagnostics: acceptance 0.110; $\widehat{R}$=(1.009,1.010,1.008,1.039).

Table 5. Posterior summary from the shipped compressed-likelihood multi-chain MCMC run.

Parameter	mean	median	$16\%$	$84\%$
${\Omega }_m$	0.3202	0.3202	0.3138	0.3264
$H_0$	67.24	67.24	66.79	67.68
${\omega }_b\equiv {\Omega }_b{h}^2$	0.02232	0.02232	0.02219	0.02245
${\sigma }_8$	0.7828	0.7842	0.7343	0.8274
Diagnostics: acceptance 0.110; $\widehat{R}$=(1.009,1.010,1.008,1.039).

For full-likelihood precision cosmology, a dedicated archival release would still be required. Distinctive signatures beyond standard $\Lambda$CDM include log-periodic (DSI-like) modulations of ${\mathcal{P}}_{\mathcal{R}}\left(k\right)$ and correlated deviations in lensing and $P\left(k\right)$ tied to the same micro-parameters (illustration in Figure ??).

interpreted as an infrared effective description of the copy-time geometry of an underlying quantum system. From this perspective, FRG fixed-point results are best regarded as a useful benchmark calibration for dimensionless parameters (such as ${\kappa }_{\mathrm{eff}}$) rather than as an additional foundational postulate.

5. Gauge-Coded QCA and Hypercharge

Pedagogical note. This section contains a 1D $U\left(1\right)$ toy QCA to make gauge-coding and Gauss-law enforcement fully explicit at minimal algebraic cost. The actual micro-model used for closure is the explicit 3+1D $U{\left(1\right)}_Y\times SU{\left(2\right)}_L\times SU{\left(3\right)}_c$ quantum-link QCA with a constrained spin-2 sector described in the micro-to-IR construction appendix (Sec. 1) and summarised in Section 4.

5.1. A Minimal Gauge-Invariant QCA Toy Model

Before turning to the full SU(3)×SU(2)×U(1) structure, we present a simple gauge-invariant QCA update in a U(1) toy setting, which serves as a concrete example of gauge coding.

Consider a one-dimensional lattice with staggered fermions ${\psi }_x$ of charge $+1$ on sites and gauge links $U_{x+1/2}=e^{i{A}_{x+1/2}}$ on edges. The local Hilbert space is

$$\mathcal{H}=\underset{x}{⨂}{\mathcal{H}}_x^{\mathrm{matter}}\otimes {\mathcal{H}}_{x+1/2}^{\mathrm{gauge}},$$

(23)

with Gauss-law constraint

$$G_x=E_{x+1/2}-E_{x-1/2}-{\psi }_x^{†}{\psi }_x\approx 0,$$

(24)

where $E_{x+1/2}$ is the electric-field operator conjugate to $A_{x+1/2}$.

A gauge-invariant QCA update can be built as a product of local unitaries

$$U=\prod _x{U}_{x+1/2}^{\mathrm{link}}{U}_x^{\mathrm{matter}},$$

(25)

where $U_{x+1/2}^{\mathrm{link}}$ acts on $({\psi }_x,U_{x+1/2},{\psi }_{x+1})$ and implements a gauge-covariant hopping, while $U_x^{\mathrm{matter}}$ acts only on ${\psi }_x$ and respects the Gauss law. For example,

$$U_{x+1/2}^{\mathrm{link}}=exp\left[-i\theta \left({\psi }_{x+1}^{†}{U}_{x+1/2}{\psi }_x+\mathrm{h}.\mathrm{c}.\right)\right]$$

(26)

is manifestly gauge-invariant under

$$\begin{array}{cc}\hfill {\psi }_x& \to e^{i{\alpha }_x}{\psi }_x,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill U_{x+1/2}& \to e^{i({\alpha }_x-{\alpha }_{x+1})}{U}_{x+1/2}.\hfill \end{array}$$

(28)

Such constructions can be generalised to non-Abelian gauge groups and extended local Hilbert spaces, as discussed in the quantum link-model literature [4,5,6]. In the Mathematical Appendix we sketch an analogous construction for an SU(3)×SU(2)×U(1) gauge-coded QCA that realises one Standard-Model-like generation.

5.2. Diffusive Hydrodynamics of the Gauge-Coded Charge

We embed an effectively one-dimensional channel for a gauge-coded charge $Q_Y$ (to be identified with hypercharge) into the QCA. The technical appendix (Points (1),(3)) provides independent minimal gauge-coded cellular-automaton benchmarks in both 1D and 3D, with explicit exponent extraction and a direct statistical test that disfavors ballistic growth. Numerically, we also verify within the QCA channel that the two-point function of the local charge density $q_Y(x,t)$ exhibits diffusive behaviour,

$$\begin{array}{cc}\hfill C_Y(x,t)& =\langle q_Y(x,t)q_Y(0,0)\rangle \hfill \\ \hfill & \sim {\displaystyle \frac{1}{\sqrt{4\pi D_{Y}t}}}exp\left(-{\displaystyle \frac{x^2}{4D_{Y}t}}\right),\hfill \end{array}$$

(29)

for times t in an intermediate window where finite-size and ultraviolet effects are negligible. Fitting $C_Y(x,t)$ across several system sizes yields a diffusion constant

$$D_Y\simeq 0.10{\mathrm{GeV}}^{-1},$$

(30)

with an estimated relative uncertainty of order $20\%$. This provides an explicit realisation of the “Diffusive hydrodynamics” assumption for the charge used in the QICT analysis.

5.3. Hypercharge as Anomaly-Free Abelian Direction

We consider one chiral generation of Standard-Model fermions without right-handed neutrinos. The relevant left- and right-handed Weyl fermions and their global charges $(B,L,Y)$ are listed in Table 6, with multiplicities from colour and weak isospin.

In the continuum Standard Model, it is a textbook result that hypercharge Y is the unique non-trivial Abelian factor in the gauge group SU(3)×SU(2)×U(1)_Y for which all gauge anomalies cancel with the observed fermion content. In particular, baryon number B and lepton number L are anomalous, whereas Y is anomaly-free.

We consider a generic Abelian charge

$$Q(\beta ,\gamma ,\delta )=\beta B+\gamma L+\delta Y.$$

(31)

Demanding cancellation of all gauge and mixed anomalies (SU(2)²U(1)_Q, SU(3)²U(1)_Q, gravity²U(1)_Q, and U(1${)}_Q^3$) yields a homogeneous linear system for $(\beta ,\gamma ,\delta )$. Solving this system with the charges in Table 6 shows that, in the absence of additional fermions, the anomaly-free subspace is one-dimensional and spanned by hypercharge:

$$(\beta ,\gamma ,\delta )\propto (0,0,1).$$

(32)

Remark (scope). This statement is within the assumed fermion content of Table 6; it establishes a one-dimensional anomaly-free subspace for Abelian charges but does not, by itself, exclude embeddings, extra generations, or additional spectator fermions.

Within the gauge-coded QCA, the matter content and charge assignments are chosen to reproduce this Standard-Model pattern at low energies. The anomaly analysis can be recast in terms of discrete charge operators acting on the QCA Hilbert space, with the same conclusion: the only non-trivial Abelian direction in the $(B,L,Y)$ space that is anomaly-free and couples to both quark and lepton sectors is proportional to Y. The explicit anomaly sums in the QCA representation are presented in the Mathematical Appendix.

Theorem 2  

(Hypercharge as distinguished Abelian direction). In the space of Abelian charges spanned by $(B,L,Y)$, for one Standard-Model-like generation without right-handed neutrinos and no additional fermions, the only non-trivial direction that is anomaly-free with respect to the non-Abelian gauge group and gravitational anomalies and couples to both quark and lepton sectors is proportional to hypercharge Y.

Proof. See the Mathematical Appendix, Sec. S2.

5.4. Susceptibility Extremisation

We complement the anomaly analysis with an information-theoretic criterion. Let $\Omega (T,{\mu }_B,{\mu }_L,{\mu }_Y)$ denote the thermodynamic potential in the electroweak-symmetric phase, coupled to chemical potentials $({\mu }_B,{\mu }_L,{\mu }_Y)$. The $3\times 3$ susceptibility matrix is

$${\Xi }_{ab}\left(T\right)={\displaystyle \frac{{\partial }^2\Omega }{\partial {\mu }_a\partial {\mu }_b}}{|}_{\mu =0},a,b\in \{B,L,Y\},$$

(33)

assumed positive-definite in the regime of interest. For a unit-norm vector $\overrightarrow{q}$ in $(B,L,Y)$ space, the quadratic form

$$\mathcal{S}[\overrightarrow{q};T]={\overrightarrow{q}}^T\Xi \left(T\right)\overrightarrow{q}$$

(34)

measures the susceptibility associated with the corresponding charge.

Proposition 1.  

In an ideal-gas approximation to the electroweak-symmetric phase with one Standard-Model generation, and restricting to the anomaly-free subspace in $(B,L,Y)$ space, the quadratic form $\mathcal{S}[\overrightarrow{q};T]$ has an extremum along the hypercharge direction $\overrightarrow{q}\propto (0,0,1)$.

The proof is a straightforward eigenvalue analysis of $\Xi$ subject to the anomaly constraints and is given in the Mathematical Appendix. It provides an information-theoretic justification for focusing on hypercharge in the QICT analysis.

5.5. Explicit SU(2)×U(1) QCA Update for a Lepton Doublet

To make the SU(2)×U(1) structure fully explicit, we now construct a gauge-invariant QCA update for a single left-handed lepton doublet

$$L_x=\left(\begin{array}{c}{u}_{L,x}\\ e_{L,x}\end{array}\right),Y_L=-\frac{1}{2},$$

coupled to SU(2) link variables $W_{x+1/2}\in \mathrm{SU}\left(2\right)$ and U(1)_Y link variables $U_{x+1/2}=e^{i{Y}_L{B}_{x+1/2}}$ on the edges.

The local Hilbert space on one edge consists of the matter field $L_x$ at site x, the link $(W_{x+1/2},U_{x+1/2})$, and the matter field $L_{x+1}$ at site $x+1$. We define the gauge-covariant hopping unitary

$$\begin{array}{cc}\hfill U_{x+1/2}^{\mathrm{lep}}=& exp[-i\theta \hfill \\ \hfill & \times \left(L_{x+1}^{†}{W}_{x+1/2}{U}_{x+1/2}{L}_x+\mathrm{h}.\mathrm{c}.\right)].\hfill \end{array}$$

(35)

which acts only on this edge Hilbert space. Under a local gauge transformation with parameters $G_x\in \mathrm{SU}\left(2\right)$ and ${\alpha }_x\in \mathbb{R}$,

$$\begin{array}{cc}\hfill L_x& \to e^{i{Y}_L{\alpha }_x}{G}_x{L}_x,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill W_{x+1/2}& \to G_{x+1}{W}_{x+1/2}{G}_x^{†},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill U_{x+1/2}& \to e^{i{Y}_L({\alpha }_x-{\alpha }_{x+1})}{U}_{x+1/2},\hfill \end{array}$$

(38)

the hopping term in Eq. (35) is manifestly gauge-invariant: the transformed product is identical to the original one, so the edge unitary is gauge-covariant by construction. A full QCA update step for the lepton sector is then given by

$$U_{\mathrm{lep}}=\prod _x{U}_{x+1/2}^{\mathrm{lep}}{U}_x^{\mathrm{loc}},$$

(39)

where $U_x^{\mathrm{loc}}$ acts only on $L_x$ and preserves the lattice Gauss-law constraint. In the Mathematical Appendix we generalise this construction to the quark sector and to SU(3)×SU(2)×U(1) with the full Standard-Model charge assignments.

6. From Prediction to Inverse Inference: a Closure Test

6.1. Inverse-Inference Principle and Admissible Manifold

The forward statement of the QICT closure is the Golden Relation,

$${\mathcal{R}}_G(m_S,{\chi }_Y,{\kappa }_{\mathrm{eff}},C_{\Lambda })\equiv m_S^2-C_{\Lambda }^2{\kappa }_{\mathrm{eff}}{\chi }_Y=0.$$

(40)

Taken by itself, Eq. (40) is a prediction map: once the microscopic inputs $({\chi }_Y,{\kappa }_{\mathrm{eff}},C_{\Lambda })$ are fixed, it yields a characteristic singlet-scalar mass scale. The inverse problem runs in the opposite direction. External measurements constrain different projections of the same latent closure variables, and an admissible theory point must lie in the intersection

$${\mathcal{M}}_{\mathrm{adm}}={\mathcal{M}}_G\cap {\mathcal{M}}_{\mathrm{DD}}\cap {\mathcal{M}}_{\mathrm{coll}}\cap {\mathcal{M}}_{\mathrm{cosmo}},$$

(41)

where ${\mathcal{M}}_G$ is the Golden-Relation manifold defined by Eq. (40), ${\mathcal{M}}_{\mathrm{DD}}$ is the direct-detection allowed set, ${\mathcal{M}}_{\mathrm{coll}}$ the collider allowed set, and ${\mathcal{M}}_{\mathrm{cosmo}}$ the cosmological allowed set.

This reframing is important conceptually. In the present manuscript, LZ-type direct-detection data and DESI dark-energy data are not treated as isolated checks appended after the prediction. They are treated as coordinates in an inverse map that test whether the particle-physics and cosmological sectors can be generated by the same infrared scale ${\Lambda }_{\mathrm{IR}}=C_{\Lambda }\sqrt{{\chi }_Y}$. The rigorous content of the section is therefore an intersection test: if the data-driven allowed regions fail to intersect Eq. (41), the closure is falsified.

Proposition 2  

(Inverse-closure admissibility criterion). Let $\Theta =(m_S,{\chi }_Y,{\kappa }_{\mathrm{eff}},C_{\Lambda },{\nu }_{\mathrm{DE}},\dots )$ denote the latent variables of the closure. Suppose that the forward constraints defining ${\mathcal{M}}_G$, ${\mathcal{M}}_{\mathrm{DD}}$, ${\mathcal{M}}_{\mathrm{coll}}$, and ${\mathcal{M}}_{\mathrm{cosmo}}$ are all well-defined subsets of the same parameter space. Then:

(i) 

if ${\mathcal{M}}_{\mathrm{adm}}=⌀$, the closure hypothesis is excluded for that choice of model class;

(ii) 

if ${\mathcal{M}}_{\mathrm{adm}}\ne ⌀$, the data determine a non-empty admissible inverse set but not, in general, a unique reconstructed point;

(iii) 

uniqueness requires additional structure, e.g. a likelihood or dynamical map from QICT variables to the dark-energy sector.

Proof. 

Statement (i) is immediate from the definition of ${\mathcal{M}}_{\mathrm{adm}}$ as an intersection of admissible sets. Statement (ii) is equally immediate: any $\Theta \in {\mathcal{M}}_{\mathrm{adm}}$ satisfies all forward constraints simultaneously, so the inverse problem has at least one admissible solution, but set-theoretic non-emptiness alone does not imply uniqueness. Statement (iii) follows because uniqueness is equivalent to the admissible set collapsing to a singleton, which requires additional identifiability information beyond membership in the constraint sets.    □

Status of the claim.

A precise one-parameter curve in the $(w_0,w_a)$ plane requires an explicit map from the QICT infrared sector to a dark-energy equation of state. The present submission does not derive such a map in full generality. Accordingly, we formulate the inverse problem conservatively: present cosmological data define a target domain in $(w_0,w_a)$ space, not yet a uniquely derived trajectory. This keeps the closure claim within what is mathematically justified.

6.2. Hypercharge Susceptibility at a Reference Temperature $T_{★}$

In the symmetric phase, the static susceptibility associated with a conserved U(1) charge Q is defined by

$${\chi }_Q\left(T\right)\equiv {\left.{\displaystyle \frac{\partial n_Q}{\partial {\mu }_Q}}\right|}_T,$$

(42)

where $n_Q$ is the charge density and ${\mu }_Q$ the corresponding chemical potential. In the Standard-Model numerics below we take $T_{★}=260\mathrm{GeV}$ as a representative point above the electroweak crossover, but conceptually $T_{★}$ is treated as an operational plateau scale rather than a fundamental input. In natural units, $\left[n_Q\right]={\mathrm{GeV}}^3$ and $\left[{\mu }_Q\right]=\mathrm{GeV}$, hence $\left[{\chi }_Q\right]={\mathrm{GeV}}^2$. In an ideal-gas approximation it can be written as [17,18,19,20]

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\chi }_Q\left(T\right)}{T^2}}& ={\displaystyle \frac{1}{6}}\sum _{\mathrm{Weyl}\mathrm{fermions}}{d}_f{q}_f^2\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}\sum _{\mathrm{complex}\mathrm{scalars}}{d}_s{q}_s^2.\hfill \end{array}$$

(43)

where $q_{f,s}$ are the Q-charges of fermions and scalars, and $d_{f,s}$ their degeneracies. Applying this to hypercharge depends on the normalisation convention for the Abelian generator coupled to the chemical potential. For the canonical Standard-Model convention $Q=T_3+Y/2$, the ideal-gas result in the electroweak-symmetric phase with three generations and one Higgs doublet is

$${\displaystyle \frac{{\chi }_{Y/2}\left(T\right)}{T^2}}={\displaystyle \frac{11}{6}},$$

(44)

while a rescaling of the charge $Y/2\mapsto s(Y/2)$ rescales $\chi \mapsto s^2\chi$. In this submission we adopt the charge

$$\tilde{Y}\equiv \sqrt{{\displaystyle \frac{3}{5}}}{g}^{\prime }\left(T\right){\displaystyle \frac{Y}{2}},$$

(45)

so that the corresponding susceptibility is ${\chi }_Y\equiv {\chi }_{\tilde{Y}}$. In the ideal-gas limit this gives

$${\displaystyle \frac{{\chi }_Y\left(T\right)}{T^2}}={\left(\sqrt{{\displaystyle \frac{3}{5}}}{g}^{\prime }\left(T\right)\right)}^2{\displaystyle \frac{{\chi }_{Y/2}\left(T\right)}{T^2}}={\displaystyle \frac{11}{10}}{g}^{\prime 2}\left(T\right),$$

(46)

which yields ${\chi }_Y/T^2\simeq 0.14$ for $g^{\prime }\left(T_{★}\right)\simeq 0.36$, consistent with the benchmark below. We summarize this input as

$${\displaystyle \frac{{\chi }_Y\left(T_{★}\right)}{T_{★}^2}}=0.145\pm 0.010,T_{★}=260\mathrm{GeV},$$

(47)

which captures perturbative uncertainties and modest non-perturbative corrections.

6.3. Forward Map: Microscopic QICT Parameters and the Mass Band

On the microscopic side we consider a QCA realization of an effectively one-dimensional hypercharge-carrying channel, with lattice spacing a and hypercharge diffusion constant $D_Y$. Matching the QCA to a thermal plasma suggests benchmark values

$$a=0.197{\mathrm{GeV}}^{-1},D_Y\simeq 0.10{\mathrm{GeV}}^{-1},$$

(48)

with relative uncertainties of order $20\%$.

The QICT analysis then yields an operational time scale ${\tau }_{\mathrm{copy}}\left(Y\right)$. We define the associated information scale by $k_I\equiv {\tau }_{\mathrm{copy}}^{-1}\left(Y\right)$, and identify the infrared matching scale with this information scale up to a dimensionless protocol-dependent factor,

$${\Lambda }_{\mathrm{IR}}\equiv C_{\Lambda }\sqrt{{\chi }_Y}.$$

(49)

In the zero-parameter benchmark we derive $C_{\Lambda }$ directly from the QCA network geometry and transport:

$$C_{\Lambda }\equiv \sqrt{{\displaystyle \frac{dim\left({\mathcal{H}}_{\mathrm{local}}\right)}{D_{\mathrm{eff}}}}f\left(\mathrm{topology}\right)}.$$

(50)

For the gauge-coded 3D QCA used in the closure construction, the relevant microchannel observable is an oriented-link parity bit with effective local dimension $dim\left({\mathcal{H}}_{\mathrm{local}}\right)=2$. Defining $D_{\mathrm{eff}}$ by the long-time slope

$$\mathrm{MSD}\left(t\right)={\langle |\Delta \mathbf{x}\left(t\right)|}^2\rangle =D_{\mathrm{eff}}t(t\to \infty ),$$

(51)

and using the topological factor $f\left(\mathrm{topology}\right)=6/5$, we obtain the benchmark value

$$C_{\Lambda }=1.606\pm 0.044.$$

(52)

We define the scalar dressing parameter ${\kappa }_{\mathrm{eff}}$ microscopically via a ratio of regulated Kubo–Mori second susceptibilities computed within the same QICT dynamics,

$${\kappa }_{\mathrm{eff}}\equiv \underset{T\in {\mathcal{P}}_{\mathrm{symm}}}{lim}{\displaystyle \frac{{\chi }_{\mathrm{micro},S}^{\left(2\right)}\left(T\right)}{{\chi }_{\mathrm{micro},Y}^{\left(2\right)}\left(T\right)}},$$

(53)

and for numerical illustration we propagate the conservative benchmark interval

$${\kappa }_{\mathrm{eff}}=0.1356\pm 0.0714.$$

(54)

Combining the QICT identification

$${\Lambda }_{\mathrm{IR}}=C_{\Lambda }\sqrt{{\chi }_Y}$$

(55)

with the scalar-sector dressing relation

$$m_S^2={\kappa }_{\mathrm{eff}}{\Lambda }_{\mathrm{IR}}^2$$

(56)

yields the Golden Relation

$$m_S=C_{\Lambda }\sqrt{{\kappa }_{\mathrm{eff}}{\chi }_Y},$$

(57)

which is the forward closure map.

Using the derived/benchmark intervals

$$\begin{array}{cc}\hfill C_{\Lambda }& =1.606\pm 0.044,{\kappa }_{\mathrm{eff}}=0.1356\pm 0.0714,\hfill \\ \hfill {\displaystyle \frac{{\chi }_Y}{T_{★}^2}}& =0.145\pm 0.010,T_{★}=260\mathrm{GeV},\hfill \end{array}$$

(58)

and propagating uncertainties in quadrature gives

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\delta m_S}{m_S}}\right)}^2& ={\left({\displaystyle \frac{\delta C_{\Lambda }}{C_{\Lambda }}}\right)}^2+{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\delta {\kappa }_{\mathrm{eff}}}{{\kappa }_{\mathrm{eff}}}}\right)}^2\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\delta ({\chi }_Y/T_{★}^2)}{({\chi }_Y/T_{★}^2)}}\right)}^2.\hfill \end{array}$$

(60)

from which we obtain the benchmark mass scale

$$m_S=58.5\pm 15.6\mathrm{GeV},$$

(61)

with conservative interval

$$m_S\in [43,74]\mathrm{GeV}.$$

(62)

Figure 13. Illustrative one-dimensional probability density for $m_S$ induced by benchmark uncertainties on $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$, and ${\chi }_Y$. The figure represents the forward Golden-Relation inference before external laboratory and cosmological filters are imposed.

Figure 13. Illustrative one-dimensional probability density for $m_S$ induced by benchmark uncertainties on $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$, and ${\chi }_Y$. The figure represents the forward Golden-Relation inference before external laboratory and cosmological filters are imposed.

6.4. Inverse Coordinate from Direct Detection: What LZ Fixes, and What It Does Not

The present inverse step uses direct-detection data as an exclusion coordinate in the Higgs-portal parameter plane. The latest public high-mass WIMP result from LZ reports no excess over background and quotes world-leading spin-independent limits, with the strongest published exclusion at $2.2\times {10}^{-48}{\mathrm{cm}}^2$ near $40\mathrm{GeV}$ [23]. This result does not measure a preferred dark-matter mass, and in particular it does not justify replacing the Golden Relation by an externally imposed point estimate for $m_S$.

The correct inverse-inference statement is therefore the following. In the minimal Higgs-portal singlet-scalar model, direct detection constrains the pair $(m_S,{\lambda }_{HS})$ through the spin-independent cross section derived in Section 7. When combined with the invisible-width constraint, the LZ exclusion surface removes a substantial part of the Golden-Relation band but leaves a surviving corridor near the Higgs-resonance regime, where the relic abundance can still be obtained for sufficiently small portal coupling. In this sense LZ acts as a projector: it does not supply the mass, but it reduces the admissible subset of the forward band.

This distinction is crucial for scientific rigor. Treating LZ as if it singled out a unique mass around $m_h/2$ would overstate the present experimental situation. The publication-safe claim is narrower and stronger: if the Golden Relation is the correct closure rule, then current direct-detection data force the particle-physics inverse solution toward the near-resonant part of the QICT mass band.

6.5. Inverse Coordinate from Cosmology: DESI and the Dark-Energy Sector

The cosmological inverse step is analogous but conceptually different. The latest public DESI DR2 cosmology analysis combines BAO information from three years of data with external datasets and finds that the time-dependent $(w_0,w_a)$ extension provides a better fit than flat $\Lambda$CDM, with a favored region in the quadrant $w_0>-1$ and $w_a<0$; for DESI BAO + CMB the preference reaches $3.1\sigma$, and with supernovae it ranges from $2.8\sigma$ to $4.2\sigma$ depending on the SN sample [36]. This is evidence for a preferred region in dark-energy parameter space, not yet a unique microscopic reconstruction.

Within the QICT framework, the cosmological sector is controlled by the same infrared scale that enters the Golden Relation. The inverse problem is therefore to determine whether the information-fluid / copy-time-noise sector associated with ${\Lambda }_{\mathrm{IR}}$ can be mapped into the DESI-favored region of the $(w_0,w_a)$ plane. At the level achieved in the present manuscript, this is again a consistency-domain statement: the closure is viable only if the same latent scale that fixes $m_S$ also places the effective dark-energy sector in the quadrant favored by DESI.

We emphasize the limit of the present argument. The manuscript does not yet derive a unique analytic trajectory ${\Gamma }_{\mathrm{QICT}}\subset (w_0,w_a)$ on which DESI points must lie. A claim of exact agreement with an observed DESI curve would therefore be premature. The scientifically defensible statement is that DESI supplies the cosmological coordinate of the inverse problem, and that the QICT closure must be tested against that coordinate rather than against $\Lambda$CDM alone.

6.6. Unified Closure Test: One Latent Scale, Two External Coordinates

The mathematical content of the inverse loop can now be stated cleanly. Let $\Theta =({\chi }_Y,{\kappa }_{\mathrm{eff}},C_{\Lambda },{\lambda }_{HS},{\nu }_{\mathrm{DE}})$ denote the latent closure variables, where ${\nu }_{\mathrm{DE}}$ summarizes the effective parameters governing the dark-energy realization of the QICT infrared sector. Then external data impose the system

$$\begin{array}{cc}\hfill {\mathcal{R}}_G(\Theta )& =0,\hfill \\ \hfill {\sigma }_{\mathrm{SI}}(m_S,{\lambda }_{HS})& \le {\sigma }_{\mathrm{LZ}}^{90\%}\left(m_S\right),\hfill \\ \hfill {\mathrm{BR}}_{\mathrm{inv}}(h\to SS)& \le {\mathrm{BR}}_{\mathrm{inv}}^{\max },\hfill \\ \hfill \Psi ({\Lambda }_{\mathrm{IR}};{\nu }_{\mathrm{DE}})& \in {\mathcal{D}}_{\mathrm{DESI}},\hfill \end{array}$$

(63)

where $\Psi$ is the effective map from the infrared QICT sector to dark-energy observables and ${\mathcal{D}}_{\mathrm{DESI}}$ is the region favored by DESI.

Equation (63) formalizes what is meant here by data as coordinates. LZ supplies a particle-physics coordinate in the $(m_S,{\lambda }_{HS})$ projection; DESI supplies a cosmological coordinate in the $(w_0,w_a)$ projection; the Golden Relation locks both to the same latent infrared scale. If future work makes $\Psi$ explicit and one-to-one, the admissible set collapses from a constrained manifold to an actual one-parameter curve. At present, the paper establishes the manifold version of that statement.

The falsifiability criterion is immediate. If the Golden-Relation band is driven by direct-detection plus collider bounds entirely away from the infrared scale needed by the cosmological sector, or conversely if the DESI-favored region cannot be produced by any ${\nu }_{\mathrm{DE}}$ compatible with the particle-physics closure, then the inverse loop fails. The value of the reframed chapter is precisely that it turns what would otherwise be a loose comparison of plots into a sharp intersection test.

6.7. Prior Sensitivity and Identifiability of the Inverse Solution

The numerical interval quoted above is obtained by propagating a specific set of benchmark uncertainties. However, Eq. (57) is a protocol-dependent closure rather than a fit to a complete set of particle-physics and cosmological observations. It depends only on the product $C_{\Lambda }\sqrt{{\kappa }_{\mathrm{eff}}{\chi }_Y}$, so the inferred $m_S$ is not fully identifiable without a convention that specifies how $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$, and ${\chi }_Y$ are to be determined.

To make this dependence explicit, we compare several simple prior choices for the input parameters and sample the induced distribution for $m_S$. The results are summarised in Table 7 and illustrated in Figure 15. Across these alternatives the characteristic scale remains at $\mathcal{O}({10}^1$–${10}^2)$ GeV, while the width of the distribution is dominated by the present uncertainty in ${\kappa }_{\mathrm{eff}}$.

Figure 14. Identifiability diagnostic for the auxiliary MCMC consistency check used in the cosmology module of the benchmark closure package. The figure is included to expose degeneracy directions and prior dependence, not to advertise a definitive global fit.

Figure 14. Identifiability diagnostic for the auxiliary MCMC consistency check used in the cosmology module of the benchmark closure package. The figure is included to expose degeneracy directions and prior dependence, not to advertise a definitive global fit.

Figure 15. Prior-sensitivity illustration for the induced distribution of $m_S$ in Eq. (57). The first curve uses the Gaussian benchmark inputs quoted in the main text; the second applies a uniform stress test around the central values; the third adopts a deliberately broader log-uniform prior for ${\kappa }_{\mathrm{eff}}$. The point is to show stability of the mass scale and sensitivity of the width.

Figure 15. Prior-sensitivity illustration for the induced distribution of $m_S$ in Eq. (57). The first curve uses the Gaussian benchmark inputs quoted in the main text; the second applies a uniform stress test around the central values; the third adopts a deliberately broader log-uniform prior for ${\kappa }_{\mathrm{eff}}$. The point is to show stability of the mass scale and sensitivity of the width.

6.8. Robustness Under Variations of the Matching Temperature

The matching temperature $T_{★}=260\mathrm{GeV}$ is a reference point chosen in the electroweak-symmetric regime, where the relevant Standard Model degrees of freedom are relativistic and the ideal-gas estimate for ${\chi }_Y/T^2$ is expected to be accurate at the $\mathcal{O}(10\%)$ level. Corrections from thermal masses and screening can be incorporated systematically, but for our purposes they are absorbed into the benchmark uncertainty on ${\chi }_Y/T^2$.

To assess how sensitive the Golden Relation is to the precise reference point, we vary the matching temperature in a conservative window around the benchmark,

$$T_{★}\in [240,300]\mathrm{GeV}.$$

(64)

In an ideal-gas approximation the ratio ${\chi }_Y\left(T\right)/T^2$ is temperature independent for relativistic species with fixed charge assignments, so that ${\chi }_Y\left(T_{★}\right)\propto T_{★}^2$ and therefore $m_S\propto T_{★}$ up to the mild residual dependence encoded in the quoted uncertainty of ${\chi }_Y/T^2$. Over the above window, the central value shifts by at most $\pm 15\%$, comparable to the propagated benchmark uncertainty from $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$, and ${\chi }_Y/T^2$. The forward mass band therefore remains in the vicinity of the Higgs-resonance region, while the inverse filters described above continue to act on the same neighborhood of scales.

We therefore conclude that the existence and location of the Golden-Relation band are robust against moderate variations of the matching temperature within the electroweak-symmetric regime. What remains non-trivial is not the existence of a mass scale, but whether the direct-detection and cosmological coordinates continue to intersect the same admissible manifold as the data improve.

7. Phenomenological Consistency Checks

We discuss the minimal $Z_2$ singlet-scalar Higgs-portal model as a consistency check of the Golden-Relation mass band, without relying on any global numerical scan. The portal model depends primarily on the physical singlet mass $m_S$ and the Higgs-portal coupling ${\lambda }_{HS}$ [26,27,28,29,30].

The renormalisable interaction is

$$\begin{array}{ccc}\hfill \mathcal{L}\supset & {\displaystyle \frac{1}{2}}\left({\partial }_{\mu }S\right)\left({\partial }^{\mu }S\right)\hfill & \hfill -{\displaystyle \frac{1}{2}}{m}_{S,0}^2{S}^2-{\displaystyle \frac{{\lambda }_S}{4}}{S}^4\\ \hfill & -{\displaystyle \frac{{\lambda }_{HS}}{2}}{S}^2\left(H^{†}H\right),\hfill \end{array}$$

(65)

with $S\to -S$. After electroweak symmetry breaking, $H=(0,(v+h)/\sqrt{2})$, the portal induces the coupling $hSS$ and fixes the physical singlet mass via

$$m_S^2=m_{S,0}^2+{\displaystyle \frac{1}{2}}{\lambda }_{HS}{v}^2.$$

(66)

7.1. Direct Detection (Spin-Independent)

At tree level the spin-independent nucleon cross section is mediated by Higgs exchange and can be approximated by

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

(67)

where $f_N$ parametrises the scalar nucleon matrix element and ${\mu }_N$ is the DM–nucleon reduced mass. Current leading limits in the tens-of-GeV region come from xenon time-projection chambers (LZ, XENONnT, PandaX) [23,24,25]. In the predicted mass band near the Higgs resonance, consistency typically requires ${\lambda }_{HS}$ to be small enough that ${\sigma }_{\mathrm{SI}}$ stays below these limits.

7.2. Invisible Higgs width

For $m_S<m_h/2$, the Higgs decays invisibly via $h\to SS$ with

$$\Gamma (h\to SS)={\displaystyle \frac{{\lambda }_{HS}^2{v}^2}{32\pi m_h}}\sqrt{1-{\displaystyle \frac{4m_S^2}{m_h^2}}}.$$

(68)

The invisible branching fraction is

$${\mathrm{BR}}_{\mathrm{inv}}={\displaystyle \frac{\Gamma (h\to SS)}{{\Gamma }_{\mathrm{SM}}+\Gamma (h\to SS)}}.$$

(69)

A conservative and widely used bound is ${\mathrm{BR}}_{\mathrm{inv}}<0.107$ (95% CL) from the ATLAS combination of Run 2 searches [22]. Since the Golden-Relation band includes $m_S\lesssim m_h/2$, this constraint provides a clean upper limit on ${\lambda }_{HS}$ that is independent of any relic-density computation.

7.3. Parameter-Free Correlation Between $\Gamma (h\to SS)$ and ${\sigma }_{\mathrm{SI}}$

The Higgs-portal model admits a useful elimination of the portal coupling ${\lambda }_{HS}$ between the invisible width (68) and the spin-independent cross section (67). Eliminating ${\lambda }_{HS}$ yields the parameter-free relation

$$\begin{array}{cc}\hfill \Gamma (h\to SS)=& {\sigma }_{\mathrm{SI}}{\displaystyle \frac{v^2{m}_h^3{m}_S^2}{8f_N^2{\mu }_N^2{m}_N^2}}{\beta }_S,\hfill \\ \hfill {\beta }_S=& \sqrt{1-{\displaystyle \frac{4m_S^2}{m_h^2}}}.\hfill \end{array}$$

(70)

up to the hadronic uncertainty in $f_N$. In particular, an upper bound on ${\mathrm{BR}}_{\mathrm{inv}}$ implies $\Gamma (h\to SS)<{\displaystyle \frac{{\mathrm{BR}}_{\mathrm{inv}}}{1-{\mathrm{BR}}_{\mathrm{inv}}}}{\Gamma }_{\mathrm{SM}}$ and therefore a derived upper bound on ${\sigma }_{\mathrm{SI}}$ as a function of $m_S$. This provides a clean, falsifiable interface between collider constraints and direct-detection limits, independent of any relic-density computation.

For transparency and reproducibility, the submission package includes a small self-contained benchmark script that evaluates Eqs. (68)–(70) across the Golden-Relation mass band and reports the implied constraint region under stated numerical choices for $(f_N,{\Gamma }_{\mathrm{SM}})$.

7.4. Relic Abundance

The observed dark-matter abundance ${\Omega }_c{h}^2\simeq 0.12$ (Planck) [21] can be reproduced in the Higgs-portal model for suitable ${\lambda }_{HS}$, and near the Higgs resonance relatively small ${\lambda }_{HS}$ can suffice. A full relic-density computation (Boltzmann equation with thermal averaging and resonance treatment) is standard but is not required for the present submission package, whose central claim concerns the conditional mass-scale estimate from the micro–macro closure map. We therefore restrict the phenomenology here to the robust, analytic constraints (67) and (68), which already delimit the viable coupling range in the predicted band.

8. Discussion

8.1. Dark Energy as Copy-Time Noise and a DM–DE Relation Controlled by Stiffness

In Planck units defined by the emergent coupling $G_{\mathrm{eff}}$, the dimensionless quantity ${\Lambda }_{\mathrm{eff}}{\ell }_{\mathrm{Pl}}^2$ takes a transparent form. Using ${\ell }_{\mathrm{Pl}}^2=\hslash G_{\mathrm{eff}}/c_{\mathrm{QICT}}^3$ and the stiffness-controlled noise-floor scaling, one finds

$${\Lambda }_{\mathrm{eff}}{\ell }_{\mathrm{Pl}}^2\sim {\left(k_{\mathrm{IR}}{\ell }_{\mathrm{Pl}}\right)}^2,$$

(71)

up to a coefficient fixed by the micro-realisation (no adjustable fit parameter). A derivation and discussion of how Eq. (71) yields the numerical order ${10}^{-122}$ when $k_{\mathrm{IR}}$ is identified with the cosmological coarse-graining scale are given in the Mathematical Appendix, Sec. S10 and Sec. S10’.

A strictly positive infrared noise floor in copy-time fluctuations provides an effective cosmological constant term in the emergent geometry description. Denoting by $\delta \tau$ the long-wavelength variance of copy-time fluctuations, a natural estimate is

$${\Lambda }_{\mathrm{eff}}\sim {\displaystyle \frac{{(\delta \tau /{\tau }_0)}^2}{c_{\mathrm{QICT}}^2{\tau }_0^2}}.$$

(72)

In explicit micro-models, the same stiffness coefficient K that fixes $G_{\mathrm{eff}}$ controls the magnitude of $\delta \tau$ and therefore links DM and DE in a single relation ${\Lambda }_{\mathrm{eff}}={\Lambda }_{\mathrm{eff}}(K,{\tau }_0,a)$. A solvable example deriving this linkage from a microscopic generating functional is given in the Mathematical Appendix, Sec. S8.

8.2. Operational Limiting Speed and Planck Scale from Copy-Time Data (Model Units)

The framework defines a limiting speed and an associated Planck scale without fitting to external measurements, in the following precise sense. Let a denote the microscopic lattice spacing of a specified QCA/QLM realisation and let ${\tau }_0$ be the minimal copy time (the infimum over locations of ${\tau }_{\mathrm{copy}}\left(x\right)$). Because strictly local updates can transport conserved information by at most one edge per copy cycle, the operational causal cone implies

$$c_{\mathrm{QICT}}:={\displaystyle \frac{a}{{\tau }_0}},$$

(73)

A further non-continuum consequence of strictly local updates is the presence of high-momentum dispersion: while the causal cone is fixed by locality, the dispersion relation of microscopic modes deviates from linearity at $ka\sim 1$. In a broad class of translation-invariant nearest-neighbour update rules, one obtains lattice-type dispersions (e.g. $\omega \left(k\right)\propto sin(ka/2)$), implying energy-dependent group velocity that remains bounded by $c_{\mathrm{QICT}}$. A self-contained example and the corresponding group-velocity bound are given in the Mathematical Appendix, Sec. S10 and Sec. S10’.

In addition to the operational definition, a deterministic finite-velocity bound holds for any strictly local bounded update rule: correlations satisfy a Lieb–Robinson-type inequality with velocity $v_{\mathrm{LR}}$ fixed solely by microscopic locality and norm bounds. This yields an observer-independent maximal propagation speed (no observer can witness superluminal signaling beyond the bound). See the Mathematical Appendix, Sec. S7.

as the maximal coarse-grained signal speed in model units. This is a definition forced by locality plus the receiver-optimised operational notion of transfer time, not a calibration to the observed speed of light.

To connect the copy-time geometry to a dynamical metric theory, the leading infrared action takes the diffeomorphism-invariant form $\int d^{4}x\sqrt{-g}(M_{*}^2/2)R+\dots$. In an explicit micro-model, $M_{*}^2$ is not an adjustable parameter: it is a response coefficient (an “information stiffness”) relating variations of ${\tau }_{\mathrm{copy}}$ to conserved-sector stress. In a QCA/QLM where this stiffness K can be computed microscopically, one obtains $M_{*}^2\propto K/{\tau }_0^2$ and hence an emergent gravitational coupling $G_{\mathrm{eff}}\sim 1/M_{*}^2$. This yields a Planck scale in model units,

$$m_{\mathrm{Pl},\mathrm{QICT}}:=\sqrt{{\displaystyle \frac{\hslash c_{\mathrm{QICT}}}{G_{\mathrm{eff}}}}},$$

(74)

again fixed by micro-dynamics once K is computed. A concrete toy computation of K and $G_{\mathrm{eff}}$ for an explicit discrete update geometry is provided in the Mathematical Appendix, Sec. S6. Importantly, Eqs. (73)–(74) are obtained without inserting experimental values as inputs; mapping from model units to SI units is a separate step (choice of a and ${\tau }_0$), not used to define the theory’s internal predictions.

8.3. Uniqueness of Copy Time as an Infrared Geometric Datum (Within Stated Axioms)

A recurring question is whether the “copy time” field is merely one convenient parametrisation of transport, or whether it is singled out by basic consistency requirements. Here we state the precise sense in which it is unique in this submission.

Axioms.

We assume (i) strictly local updates on a bounded-degree graph with microscopic length scale a, (ii) unitary evolution, (iii) existence of at least one non-trivial conserved density channel, and (iv) a receiver–optimised operational definition of transfer time as used throughout this paper. Under these axioms, any coarse-grained notion of a causal cone for information must be constructed from a positive local “slowness” functional that is additive under concatenation of paths and monotone under local slowing of the update rule.

Proposition (operational uniqueness).

Among all local functionals satisfying additivity and monotonicity, the copy-time field ${\tau }_{\mathrm{copy}}\left(x\right)$ is unique within this axiomatic class up to a global multiplicative constant and a change of units. In particular, the induced optical metric $g_{ij}^{\mathrm{info}}\left(x\right)={\tau }_{\mathrm{copy}}{\left(x\right)}^{-2}{\delta }_{ij}$ is the unique quadratic form whose geodesic distance reproduces leading-order receiver–optimised transfer times in the hydrodynamic scaling limit. A detailed proof is provided in the Mathematical Appendix, Sec. S5.

8.4. Gravity from Transport Consistency: Why the Einstein–Hilbert Term Is the Unique Two-Derivative Option

Once an effective geometry $g_{\mu \nu }^{\mathrm{info}}$ is built from ${\tau }_{\mathrm{copy}}$, infrared consistency requires that the coarse-grained theory (i) preserve unitarity of the conserved sector and (ii) preserve the operational causal cone set by the variational speed limit. Locality and reparametrisation invariance of the long-wavelength description imply a derivative expansion for $S_{\mathrm{info}}\left[g\right]$. At two derivatives, within the class of local, parity-even, diffeomorphism-invariant pure-metric actions with no additional low-derivative fields, the Einstein–Hilbert operator is the only scalar density compatible with these requirements: any alternative leading term either violates the Noether identity associated with coordinate redundancy (hence breaks covariant conservation of the effective stress tensor) or introduces extra low-derivative degrees of freedom that generically permit super-transport in the effective theory.

About the gravitational coupling.

The framework identifies $G_{\mathrm{eff}}$ as a response coefficient (an “information stiffness”) computable from the microscopic update rule in explicit QCA/QLM realisations. Changing it independently would break the matching between local update rates, the copy-time causal cone, and covariant conservation.

8.5. Isotropy from Discrete Networks and Sharp Anisotropy Diagnostics

A discrete network need not imply observable anisotropy. In this construction, isotropy is an infrared property of the universality class: when the conserved sector flows to an isotropic hydrodynamic fixed point, anisotropies appear only as irrelevant operators. Operationally, the leading anisotropy enters as direction-dependent corrections to copy-time geodesics, ${\tau }_{\mathrm{copy}}\left(\widehat{n}\right)={\tau }_0[1+{\epsilon }_2\left(\widehat{n}\right){\left(ka\right)}^2+\mathcal{O}\left({\left(ka\right)}^4\right)]$, with ${\epsilon }_2$ determined by microscopic lattice symmetries. This yields an observable: a frequency-dependent birefringence/dispersion of the effective propagation cone at order ${\left(ka\right)}^2$. In an explicit model, ${\epsilon }_2$ is computable; the absence of measurable anisotropy constrains a (or forces ${\epsilon }_2\to 0$ at the fixed point).

8.6. An eXclusive Footprint: Copy-Time Induced Phase Structure in Strong-Field Collision Waveforms

A distinctive target is a log-periodic modulation in the phase of gravitational-wave signals from strong-field collisions, arising from discrete-scale structure in copy-time coarse-graining. The prediction is a small oscillatory correction to the Fourier-domain phase,

$$\delta \Psi \left(f\right)=A_{\mathrm{CT}}cos\left({\omega }_{\mathrm{CT}}ln(f/f_0)+{\varphi }_{\mathrm{CT}}\right),$$

(75)

A conservative scaling estimate follows from the fact that lattice-induced anisotropy enters the optical geometry as an irrelevant operator of order ${\left(ka\right)}^2$. In the waveform problem, $k\sim 2\pi f/c_{\mathrm{QICT}}$, so one expects

$$A_{\mathrm{CT}}\lesssim \mathcal{O}\left(1\right)\times {\left({\displaystyle \frac{2\pi fa}{c_{\mathrm{QICT}}}}\right)}^2,$$

(76)

A simple detectability criterion follows by comparing the induced phase modulation to typical per-event phase uncertainties. For ground-based detectors (LIGO/Virgo/KAGRA), the most sensitive band is $f\sim {10}^2\mathrm{Hz}$, giving $A_{\mathrm{CT}}\lesssim \mathcal{O}\left(1\right){(2\pi \times {10}^2{\tau }_0)}^2$. For space-based detectors (LISA), $f\sim {10}^{-2}\mathrm{Hz}$, giving $A_{\mathrm{CT}}\lesssim \mathcal{O}\left(1\right){(2\pi \times {10}^{-2}{\tau }_0)}^2$. Thus a non-observation at fixed ${\omega }_{\mathrm{CT}}$ constrains the microscopic copy-time scale ${\tau }_0$ (in model units) rather than re-fitting a continuous frequency.

Figure 16. Forecast. Conservative bound on ${\tau }_0$ vs SNR from non-observation of the log-periodic GW phase modulation (generated by code/predictions/forecast_gw.py).

Figure 16. Forecast. Conservative bound on ${\tau }_0$ vs SNR from non-observation of the log-periodic GW phase modulation (generated by code/predictions/forecast_gw.py).

Non-degeneracy. Standard post-Newtonian phase corrections are linear combinations of power laws $f^p$ (and at most polynomial powers of $lnf$). By contrast, the copy-time imprint is oscillatory in $lnf$. Over a finite detector band, the inner product between $cos({\omega }_{\mathrm{CT}}lnf)$ and any finite PN basis function is parametrically suppressed unless ${\omega }_{\mathrm{CT}}\to 0$; hence the effect cannot be absorbed by re-fitting PN coefficients when ${\omega }_{\mathrm{CT}}$ is fixed by the coarse-graining factor b.

up to model-dependent coefficients fixed by the same micro-parameters that determine ${\epsilon }_2$ in Sec. Section 8.5. Because ${\omega }_{\mathrm{CT}}$ is fixed (not freely fit), the modulation is not degenerate with standard post-Newtonian phasing terms, which are power-law in f rather than log-periodic.

where ${\omega }_{\mathrm{CT}}=2\pi /lnb$ is fixed by the coarse-graining factor b of the underlying update geometry (binary $b=2$ is the natural case). Because ${\omega }_{\mathrm{CT}}$ is not a free continuous parameter, this constitutes an “exclusive footprint”: either a fixed-frequency log-periodic pattern is present across events, or this class of copy-time geometries is ruled out.

We briefly summarise robustness, assumptions, and scope, to keep the main narrative self-contained and easy to follow.

Assumptions and regime of validity.

The technical appendix (Copy-time bound / Point (6)) establishes the copy–susceptibility exponent in a variational form that does not assume diffusion or ergodicity: for any local, causal, unitary QCA one has bounds ${\tau }_{\mathrm{copy}}=\Theta \left({\left({\chi }_B^{\left(2\right)}\right)}^{1/2}\right)$ when ${\chi }_B^{\left(2\right)}$ is defined as a receiver-optimised Liouvillian-squared susceptibility. In the present closure benchmark we further use a diffusive universality class to calibrate the overall normalisation relating ${\tau }_{\mathrm{copy}}$ and ${\chi }_Y$, and we match to electroweak-symmetric plasma susceptibilities at a reference temperature $T_{★}$ (Sec. Section 6). The scalar dressing parameter ${\kappa }_{\mathrm{eff}}$ is defined microscopically from QCA susceptibilities with a deterministic regulator selection (the technical appendix, Point (6)), and is propagated as a conservative benchmark interval.

Dimensional analysis and matching constant.

The Golden Relation is dimensionally consistent once the thermodynamic (static) hypercharge susceptibility ${\chi }_Y$ is specified (here $\left[{\chi }_Y\right]={\mathrm{GeV}}^2$ in natural units), so that the QICT matching constant $C_{\Lambda }$ is dimensionless and encodes protocol- and protocol/geometry-dependent factors (threshold, region sizes, separation, and diffusion parameters). The microscopic QICT object is the Liouvillian-squared susceptibility ${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}$, which reduces to ${\chi }_Y$ in the diffusive regime with an explicit transport prefactor (see Appendix E).

Scope.

For the electroweak-symmetric reference point we use a standard benchmark interval for ${\chi }_Y/T^2$, including the leading perturbative corrections beyond the ideal-gas limit (Sec. Section 6). The technical appendix (Copy-time bound / Point (6)) provides an explicit, fully interacting thermal-QCA computation protocol for a hypercharge-like susceptibility as a proof of principle that the susceptibility can be computed without an ideal-gas approximation within a local Floquet-QCA. The numerical band should be interpreted as a closure prediction at this explicit level of approximation and convention fixing.

9. Falsifiability and Experimental Signatures

One-line criterion: the minimal-closure chain is falsified if future Higgs-portal searches and direct-detection limits exclude the entire resonance-centred band implied by Eq. (57) under the stated microscopic conventions.

The QICT closure chain is intended to be experimentally and numerically falsifiable. We summarise four concrete tests and the corresponding failure modes.

9.1. Prediction 1: Resonance-Centred Mass Band

Given microscopically defined closure inputs $(C_{\Lambda },{\kappa }_{\mathrm{eff}},{\chi }_Y)$ (as specified by the protocol and convention fixing in the technical appendix), the Golden Relation (57) implies a resonance-centred mass band for a $Z_2$ singlet scalar. The framework is falsified (as a closure for the minimal $Z_2$ portal) if future Higgs-portal searches exclude the entire band under the stated microscopic conventions.

9.2. Prediction 2: ${\Gamma }_{\mathrm{inv}}$–${\sigma }_{\mathrm{SI}}$ Correlation

Equation (70) provides a parameter-free correlation between an invisible Higgs width and the spin-independent cross section, once $m_S$ is fixed. Joint collider and direct-detection constraints can therefore test the closure band without invoking relic-density calculations. A statistically significant violation of this correlation in Higgs-portal interpretations would falsify the minimal closure.

9.3. Independent Test: Thermal Relic Abundance Consistency

The mass-band closure can be challenged by a constraint that is largely orthogonal to the band itself: the requirement that the same minimal portal model reproduce the observed dark-matter abundance under standard thermal freeze-out. In the $Z_2$ singlet-scalar portal, fixing $m_S$ determines the annihilation kinematics, while the relic abundance primarily constrains the coupling ${\lambda }_{HS}$. Thus, for each $m_S$ one can infer the coupling ${\lambda }_{HS}^{\mathrm{relic}}\left(m_S\right)$ required to match ${\Omega }_{\mathrm{DM}}{h}^2$ in a minimal thermal history. This provides an independent, coupling-level diagnostic that does not rely on the QICT calibration itself.

Figure 17 displays the inferred ${\lambda }_{HS}^{\mathrm{relic}}\left(m_S\right)$ across the resonance-centred band together with conservative experimental upper envelopes on ${\lambda }_{HS}$ derived from invisible Higgs-decay kinematics (for $m_S<m_h/2$) and spin-independent scattering. The key point is not the detailed numerical value of the envelopes, which depend on the chosen inputs and nuclear matrix elements, but the existence of a narrow window in which the minimal portal can simultaneously satisfy the relic target and remain experimentally viable. If future data exclude this window over the entire resonance-centred band (under the stated minimal assumptions), the minimal closure is falsified.

The figure and the associated summary table are generated by the relic-density benchmark module shipped with the package.

Table 8. Illustrative orthogonal diagnostic in the minimal $Z_2$ Higgs-portal scalar: the coupling inferred from a canonical thermal target compared to conservative experimental upper envelopes. The envelopes are shown for orientation and depend on the chosen inputs (e.g. $f_N$ and the adopted cross-section bound).

${\mathit{m}}_{\mathit{S}}$ (GeV)	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{relic}}$	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{inv}}$	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{SI}}$
43.0	0.172	0.0114	1.07e-18
58.5	0.0406	0.0163	1.07e-18
74.0	0.129	–	1.06e-18

Table 8. Illustrative orthogonal diagnostic in the minimal $Z_2$ Higgs-portal scalar: the coupling inferred from a canonical thermal target compared to conservative experimental upper envelopes. The envelopes are shown for orientation and depend on the chosen inputs (e.g. $f_N$ and the adopted cross-section bound).

${\mathit{m}}_{\mathit{S}}$ (GeV)	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{relic}}$	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{inv}}$	${\mathit{\lambda }}_{\mathit{H}\mathit{S}}^{\mathbf{SI}}$
43.0	0.172	0.0114	1.07e-18
58.5	0.0406	0.0163	1.07e-18
74.0	0.129	–	1.06e-18

9.4. Prediction 3: Discrete Scale Invariance Imprint

A microscopic QCA with discrete coarse-graining can induce a log-periodic modulation of primordial perturbations,

$${\mathcal{P}}_{\mathcal{R}}\left(k\right)={\mathcal{P}}_{\Lambda \mathrm{CDM}}\left(k\right)\left[1+A_{\mathrm{DSI}}cos\left({\omega }_{\mathrm{DSI}}ln{\displaystyle \frac{k}{k_{★}}}+{\phi }_0\right)\right],$$

(77)

with a frequency set by the QCA coarse-graining factor b via ${\omega }_{\mathrm{DSI}}=2\pi /lnb$. In the simplest binary refinement of a local qubit QCA (a natural choice for explicit coarse-grainings), one has $b=2$ and hence a fixed frequency ${\omega }_{\mathrm{DSI}}=2\pi /ln2$. In QICT, the amplitude $A_{\mathrm{DSI}}$ is controlled by the fraction of the energy density carried by the information sector at equality; thus a non-observation of such modulations at the fixed frequency directly constrains that fraction within this class of QCA coarse-grainings. Figure 18 illustrates the modulation at percent level.

9.5. Prediction 4: Deterministic Lorentz Emergence in the Continuum Limit

For the lattice QCA dynamics assumed here, Lorentz invariance must emerge in the continuum limit with controlled anisotropy corrections. The technical appendix (Point (2)) provides an explicit group-theoretic argument that, under stated symmetry and locality hypotheses, the leading continuum kinetic operator is rotationally invariant and Lorentz-symmetric up to $\mathcal{O}\left({\left(qa\right)}^2\right)$ corrections. Large-scale numerical simulations of the QCA provide an independent falsification channel by measuring velocity anisotropy as a function of lattice size.

10. Constraints

We provide concrete numerical bounds implied by current data together with a minimal illustrative likelihood pipeline (shipped in the code package) that outputs numbers.

10.1. Gravitational-Wave Constraints

Injection–recovery cross-check. We implement a minimal injection–recovery mismatch test on a leading-order inspiral waveform with analytic aLIGO PSD weighting, maximised over time and phase shifts. The shipped benchmark script outputs conservative upper bounds on ${\tau }_0$ by requiring the mismatch to remain below $1/\left(2{\mathrm{SNR}}^2\right)$.

Figure 19. Constraints from public GW events. Conservative upper bounds on ${\tau }_0$ using GWTC-1 network SNR values (GW150914, GW151226) and GW170817. The plot is generated by the benchmark Fisher/PSD module shipped with the package.

Figure 19. Constraints from public GW events. Conservative upper bounds on ${\tau }_0$ using GWTC-1 network SNR values (GW150914, GW151226) and GW170817. The plot is generated by the benchmark Fisher/PSD module shipped with the package.

Figure 20. Injection–recovery mismatch. Noise-weighted mismatch between an injected waveform with copy-time phase modulation and the best-matched unmodulated inspiral template (maximised over time and phase). The dashed line is the conservative threshold $1/\left(2{\mathrm{SNR}}^2\right)$ for a GW150914-like band.

Figure 20. Injection–recovery mismatch. Noise-weighted mismatch between an injected waveform with copy-time phase modulation and the best-matched unmodulated inspiral template (maximised over time and phase). The dashed line is the conservative threshold $1/\left(2{\mathrm{SNR}}^2\right)$ for a GW150914-like band.

Figure 21. Injection–recovery (3.5PN phase). Same mismatch test as Figure 20, but using a nonspinning TaylorF2 phase through 3.5PN. The plot is generated by the corresponding benchmark PN module in the submission package.

Figure 21. Injection–recovery (3.5PN phase). Same mismatch test as Figure 20, but using a nonspinning TaylorF2 phase through 3.5PN. The plot is generated by the corresponding benchmark PN module in the submission package.

Requiring a log-periodic phase modulation $\delta \Psi \left(f\right)\sim A_{\mathrm{CT}}cos({\omega }_{\mathrm{CT}}lnf)$ to satisfy $|\delta \Psi |\lesssim 1/\mathrm{SNR}$ yields a conservative bound. Using $A_{\mathrm{CT}}\propto {\left(2\pi f{\tau }_0\right)}^2$ gives

$${\tau }_0\lesssim {\displaystyle \frac{1}{2\pi f}}\sqrt{{\displaystyle \frac{1}{\mathrm{SNR}}}}.$$

(78)

The shipped benchmark module evaluates a full Fisher integral on the phase derivative with an analytic aLIGO PSD proxy; a simple illustrative likelihood script evaluates Eq. (78) and prints numerical bounds for representative events.

10.2. CMB/LSS scaling constraints

Irrelevant lattice operators imply an amplitude envelope $\lesssim {\left(ka\right)}^2$ for log-periodic oscillations in $lnk$. If observations constrain the oscillatory amplitude to $A_{\mathrm{obs}}$ at some $k_{★}$, then $a\lesssim \sqrt{A_{\mathrm{obs}}}/k_{★}$. The same script evaluates this scaling and returns a bound on a in chosen units.

11. Conclusions

The central contribution of this paper is now deliberately narrow and explicit: a theorem-level operator-theoretic framework for receiver-optimised copy time, together with a variational lower bound that survives both continuous-time and discrete-time local dynamics after projection away from the conserved zero mode. The rest of the manuscript is organised around the consequences, limits, and possible uses of that framework.

Theorem-level core.

The finite-volume QICT construction provides: (i) a precise operational definition of copy time for a conserved charge; (ii) a projected Kubo–Mori susceptibility built from the inverse squared projected generator; (iii) a variational receiver-side speed limit; and (iv) the anomaly-theoretic uniqueness of hypercharge within the admissible Abelian span for one Standard-Model-like generation. These statements are the parts of the paper that should be judged as formal results.

Conditional consequences.

The practical scaling law ${\tau }_{\mathrm{copy}}\propto {\left({\chi }^{\left(2\right)}\right)}^{-1/2}$ requires an additional infrared assumption: the relevant channel must be governed by a single diffusive slow mode. That hypothesis is benchmarked, stress-tested, and used when the closure programme is specialised to diffusive transport, but it is not claimed as a consequence of locality alone. Likewise, the uniqueness of the minimal light-sector gauge algebra requires an explicit model-selection principle and remains a conditional uniqueness statement rather than a first-principles classification theorem.

Model-dependent closure.

The optical-metric reinterpretation, the worked gauge-coded QCA construction in 3+1 dimensions, the microscopic benchmark for ${\kappa }_{\mathrm{eff}}$, and the Golden-Relation mass band are all downstream closure modules. Their scientific role is to show that the theorem-level core can be propagated into a coherent benchmark phenomenology with a transparent uncertainty budget. They should not be read as enlarging the formal core of the paper.

What would strengthen the programme further.

The clearest route to a stronger future paper is also the most conservative one: derive stronger infrared spectral information directly from local microscopic dynamics, or else isolate a broader class of models in which the diffusive reduction can itself be promoted. Until then, the present manuscript is intentionally structured so that the strongest formal claims do not depend on the success of the closure modules.

Final scientific claim.

The paper establishes a nontrivial information-theoretic constraint on conserved-charge transport and demonstrates, in a logically controlled way, how that constraint can be interfaced with gauge selection and infrared matching. The theorem-level core is independent of the later closure choices; the later closure choices are included because they are explicit, testable, and falsifiable, not because they are formally forced by locality alone.

Appendix A.1. Definition

Let $\mathcal{U}$ be a local, causal, unitary QCA on a cubic lattice of linear size L with local Hilbert space ${\mathcal{H}}_{\mathrm{local}}$, and let ${\rho }_T$ denote the thermal state at temperature T for the (effective) QCA Hamiltonian used to define equilibrium. Denote by $Y_x$ the local hypercharge density and by ${\Theta }_x$ the local scalar mass operator (the microscopic operator whose long-wavelength component sources the singlet-scalar mass term in the infrared matching).

We define the (dimensionless) static susceptibilities per unit volume,

$$\begin{array}{cc}\hfill {\chi }_Y\left(T\right)& \equiv {\displaystyle \frac{1}{V}}{〈{\left(\sum _{x\in \Lambda }{Y}_x\right)}^2〉}_T,\hfill \\ \hfill {\chi }_{\Theta }\left(T\right)& \equiv {\displaystyle \frac{1}{V}}{〈{\left(\sum _{x\in \Lambda }{\Theta }_x\right)}^2〉}_T.\hfill \end{array}$$

(A1)

with $V=L^3$ and $\Lambda \equiv {\{1,\dots ,L\}}^3$ (or the corresponding periodic torus), and ${\langle ·\rangle }_T\equiv \mathrm{Tr}\left({\rho }_T·\right)$.

The dimensionless dressing parameter entering the Golden Relation is then defined by

$${\kappa }_{\mathrm{eff}}\equiv {\mathcal{N}}_{\Theta /Y}{\displaystyle \frac{{\chi }_{\Theta }\left(T_{★}\right)}{{\chi }_Y\left(T_{★}\right)}}.$$

(A2)

Here $T_{★}$ is the electroweak-symmetric matching temperature defined operationally in the technical appendix (Point (6)) (plateau criterion in the gauge-coded QCA), and ${\mathcal{N}}_{\Theta /Y}$ is a fixed convention factor that converts the microscopic generator normalisations to the standard continuum conventions used for Y and for the singlet-scalar mass operator. Crucially, ${\mathcal{N}}_{\Theta /Y}$ is not a fit parameter: it is fixed once and for all by trace conventions (e.g. the usual GUT-normalisation factor for hypercharge).

In the present implementation we use

$${\mathcal{N}}_{\Theta /Y}={\displaystyle \frac{6}{5}},$$

(A3)

which corresponds to the standard rescaling between the microscopic $U\left(1\right)$ generator normalisation used in the QCA update rule and the continuum $g_1$ convention.1

Appendix A.2. Numerical Extraction from the 3D Gauge-Coded QCA Dataset

Using the benchmark 3D gauge-coded QCA thermal ensemble discussed in the technical appendix, at the plateau temperature $T_{★}=0.5$ (lattice units) one finds

$$\begin{array}{cc}\hfill {\chi }_Y\left(T_{★}\right)& =251.131,\hfill \\ \hfill {\chi }_{\Theta }\left(T_{★}\right)& =-28.374.\hfill \end{array}$$

(A4)

The minus sign reflects the microscopic definition of $\Theta$ in the dataset; the ratio in Eq. (A2) uses its absolute value. Therefore

$$\begin{array}{cc}\hfill {\displaystyle \frac{|{\chi }_{\Theta }\left(T_{★}\right)|}{{\chi }_Y\left(T_{★}\right)}}& =0.11298,\hfill \\ \hfill {\kappa }_{\mathrm{eff}}& ={\displaystyle \frac{6}{5}}\times 0.11298=0.13558.\hfill \end{array}$$

(A5)

This value is the one used in the updated closure chain; no asymptotic-safety input is required.

For reproducibility, the computation is implemented in the benchmark extraction script included in the package, which reads the JSON file and prints ${\kappa }_{\mathrm{eff}}$ together with a bootstrap error estimate when multiple ensembles are provided.

Appendix A.3. Minimality and Robustness

The construction (A2) makes explicit what is (and is not) assumed:

• One assumes the existence of a well-defined electroweak-symmetric thermal regime in which both Y and $\Theta$ are conserved or approximately conserved on the timescales relevant for susceptibility measurement (validated numerically in the technical appendix (Point (6))).
• One fixes generator normalisations by a standard convention factor ${\mathcal{N}}_{\Theta /Y}$, which is not tunable.

Given these two ingredients, ${\kappa }_{\mathrm{eff}}$ is a derived, dimensionless number. The remaining uncertainty is purely statistical/systematic (finite size, thermalisation, finite sampling) and can be reduced by larger-volume runs.

Appendix B.1. Axioms on the Microscopic Model and Emergent Gauge Theory

We consider a microscopic gauge-coded QCA in $(3+1)$ effective dimensions, with strictly local update rules and a finite-dimensional on-site Hilbert space. The emergent long-wavelength physics is assumed to be described by a relativistic quantum field theory with gravity, gauge fields, and chiral fermions.

Assumption A5 

(QCA locality and relativistic continuum limit). The microscopic dynamics is given by a strictly local, causal QCA on a regular lattice. Its long-wavelength, low-energy limit admits an effective description by a local, unitary, Lorentz-invariant quantum field theory in $(3+1)$ dimensions, coupled to gravity.

Assumption A6 

(Compact, connected gauge group). The gauge sector of the emergent QFT is described by a compact, connected Lie group G with Lie algebra $\mathfrak{g}=\mathrm{Lie}\left(G\right)$. The corresponding gauge fields are massless at the QICT matching scale and couple minimally to chiral fermions and scalars.

Assumption A7 

(Chiral fermions and complex representations). The matter sector contains a finite set of Weyl fermions transforming in (possibly reducible) complex representations of G, such that:

(a) 

the theory is genuinely chiral (no pairing into vectorlike multiplets that render all gauge interactions parity-invariant);

(b) 

in the light sector at and below the QICT matching scale $T_{★}$ introduced in Sec. Section 6, the representation content coincides exactly with one Standard-Model-like generation of left-handed quarks and leptons, plus, optionally, right-handed neutrinos and a real gauge-singlet scalar S;

(c) 

there are no additional light chiral fermions charged under the non-abelian factors of G beyond this Standard-Model-like content.

Assumption A8 

(Anomaly cancellation). All local and global gauge anomalies, as well as mixed gauge–gravitational anomalies, cancel exactly for the given set of fermion representations. In particular, the cubic gauge anomaly and the mixed gauge–gravitational anomaly vanish for each simple factor of G and for every gauged abelian subgroup.

Assumption A9 

(Asymptotic safety and finite number of relevant directions). The combined gravity+gauge+matter system admits a UV completion by an asymptotically safe non-Gaussian fixed point in the space of dimensionless couplings. The linearised flow around this fixed point has afinitenumber of IR-relevant directions, compatible with the observed number of free parameters at low energy, including the three gauge couplings, the Yukawa couplings of the light fermions, the Higgs self-coupling, the singlet-scalar self-coupling and portal coupling, and the singlet mass parameter. In particular, additional gauge factors or large fermion representations that would require extra independent relevant directions beyond these are excluded.

Assumption A10 

(Minimality at fixed low-energy content). At fixed low-energy field content (namely, one chiral generation of light fermions with observed quantum numbers, one light Higgs doublet, and a real singlet scalar S, plus optionally gauge-singlet right-handed neutrinos), the gauge group G is chosen to minimise

(i) 

the total dimension of G,

(ii) 

the total dimension of the fermion representation space, and

(iii) 

the number of independent gauge couplings,

subject to Assumptions A5–A9 and to the requirement that QICT can be implemented on at least one non-trivial conserved $U\left(1\right)$ charge with an information susceptibility that matches the hypercharge susceptibility of a thermal plasma at the QICT matching scale.

The last requirement ensures that the distinguished $U\left(1\right)$ charge used in the QICT analysis has a well-defined embedding in the gauge sector of the emergent theory.

Status of the minimality axioms.

Assumption A10 is not presented as a theorem of representation theory. Its role is to convert a broad admissible class into a sharply testable selection principle: among gauge algebras compatible with the light chiral spectrum, anomaly cancellation, and the local QCA closure, one retains the smallest algebraic realisation that does not introduce additional low-energy gauge bosons, extra light chiral multiplets, or extra independent gauge couplings. The resulting uniqueness claim is therefore a conditional uniqueness statement inside a physically motivated axiom class, rather than an unconditional derivation of the Standard-Model gauge group from locality alone.

Promotion test for the gauge-selection claim.

The strongest possible upgrade would be to derive Assumption A10 from the preceding axioms themselves. In the present framework this promotion fails for a precise reason: anomaly cancellation and chirality restrict the admissible light-sector representations, but they do not exclude larger simple embeddings, semi-simple enlargements with spectator factors, or UV completions whose low-energy sector reduces to the observed multiplets only after additional threshold structure. Locality of the QCA and exact conservation of the distinguished $U\left(1\right)$ likewise do not order these admissible gauge algebras by minimal dimension. The uniqueness statement is therefore left in its correct logical class: a conditional theorem inside the explicitly declared minimality axiom class, not a first-principles classification theorem.

Appendix B.2. Structural Constraints from Chirality and Anomalies

We now analyse the constraints imposed by Assumptions A6–A8 on the possible gauge algebras $\mathfrak{g}$ and their representations.

Let G decompose into simple and abelian factors,

$$G\simeq G_{\mathrm{s}.\mathrm{s}.}\times U{\left(1\right)}^k,G_{\mathrm{s}.\mathrm{s}.}=G_1\times \dots \times G_n,$$

(A7)

with simple compact Lie groups $G_i$ and integer $k\ge 0$. The Lie algebra then decomposes as

$$\mathfrak{g}\simeq \underset{i=1}{\overset{n}{⨁}}{\mathfrak{g}}_i\oplus \mathfrak{u}{\left(1\right)}^k.$$

(A8)

Proposition A1 

(Necessity of at least two non-abelian factors). Under Assumptions A7 and A8, with a low-energy spectrum containing colour and weak interactions of the observed type, the semi-simple part $G_{\mathrm{s}.\mathrm{s}.}$ must contain at least two non-abelian factors, one of which is isomorphic to $\mathrm{SU}\left(3\right)$ and one of which is locally isomorphic to $\mathrm{SU}\left(2\right)$.

Proof. (i) Colour confinement and the existence of hadrons with three-valued colour charge in the observed spectrum require a non-abelian gauge group with a complex fundamental representation of dimension 3. Among simple compact Lie groups, the only ones with a three-dimensional complex fundamental representation are $\mathrm{SU}\left(3\right)$ and groups containing it as a subgroup. By Assumption A10, we exclude larger simple groups when a smaller one suffices to realise the same low-energy representation content. Thus one factor must be isomorphic to $\mathrm{SU}\left(3\right)$.

(ii) The observed weak interactions involve left-handed doublets and right-handed singlets, with parity violation and massive charged gauge bosons. The minimal simple group with a non-trivial two-dimensional representation that can implement such a structure is $\mathrm{SU}\left(2\right)$. Other candidates (e.g. $\mathrm{SO}\left(3\right)\simeq \mathrm{SU}\left(2\right)/{\mathbb{Z}}_2$) are locally isomorphic to $\mathrm{SU}\left(2\right)$ at the algebra level. Again by minimality, we take a factor locally isomorphic to $\mathrm{SU}\left(2\right)$.

(iii) If there were only a single non-abelian factor (e.g. a grand unified $\mathrm{SU}\left(5\right)$ or $\mathrm{SO}\left(10\right)$), the low-energy decomposition would necessarily embed colour and weak interactions into a single simple algebra. This is phenomenologically possible but would typically introduce additional gauge bosons and representations beyond those observed. By Assumption A10 we then prefer the product of two smaller simple groups over a single larger group, provided both constructions yield the same low-energy content. Combining (i)–(iii) yields the stated result. □

Proposition A2 

(Existence of at least one abelian factor). Under Assumptions A7 and A8, the gauge group G must contain at least one $U\left(1\right)$ factor whose charge assignments are non-trivial on both quark and lepton multiplets.

Proof. 

The observed electric charges of quarks and leptons are fractional and not all identical in magnitude. In a purely semi-simple gauge group, electric charge would arise as a linear combination of Cartan generators; however, reproducing the observed pattern of fractional charges with a single simple group generally forces a unification scheme in which quarks and leptons sit in common multiplets (e.g. $\mathbf{5}\oplus \overline{\mathbf{10}}$ of $\mathrm{SU}\left(5\right)$). This introduces additional gauge bosons mediating transitions between quarks and leptons, which are severely constrained by proton decay and lepton-flavour violation. To avoid such extra light gauge bosons while preserving chiral gauge interactions and the observed charge pattern, we require at least one abelian factor $U\left(1\right)$ acting diagonally on the fermion multiplets. This $U\left(1\right)$ must be non-trivial on both quark and lepton sectors in order to reproduce the phenomenology of neutral currents. The anomaly constraints then restrict its charge assignments; in particular, purely baryonic or purely leptonic $U\left(1\right)$ charges are anomalous, whereas a hypercharge-like combination can be anomaly-free. □

Combining Propositions A1 and A2, we obtain the following structural statement.

Corollary A1. 

Under Assumptions A6–A8 and the requirement of reproducing the qualitative structure of QCD and weak interactions, the gauge algebra $\mathfrak{g}$ has a subalgebra isomorphic to

$$\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right),$$

(A9)

acting non-trivially on the light chiral fermions. Any additional simple or abelian factors either decouple from the light sector or are broken at scales above the QICT matching scale.

Proof. See the Mathematical Appendix, Sec. S2.

At this stage we have not excluded the possibility that $\mathfrak{g}$ is strictly larger than $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$, e.g. a grand-unified simple algebra containing this subalgebra. This is addressed below.

Appendix B.3. Hypercharge from Anomaly Cancellation and QICT

Within the subspace spanned by baryon number B, lepton number L and an abelian generator Y, the analysis in the main text shows that hypercharge Y is the unique non-trivial anomaly-free combination that couples to both quark and lepton sectors, for a single Standard-Model-like generation. We now encode this in a theorem that also incorporates the QICT requirement.

Theorem A1 

(Uniqueness of hypercharge as QICT-compatible $U\left(1\right)$). Let G be a gauge group satisfying Assumptions A6–A8, with fermion content matching one chiral Standard-Model-like generation without right-handed neutrinos at scales around a matching temperature $T_{★}$. Consider the three-dimensional space of global charges spanned by $(B,L,Y)$, where Y is a generic abelian generator acting on both quark and lepton sectors.

Then:

(i) 

The subspace of charge combinations whose associated gauged $U\left(1\right)$ is anomaly-free and couples to both quarks and leptons is one-dimensional and spanned by hypercharge $Y_{\mathrm{SM}}$.

(ii) 

Among all such anomaly-free abelian generators, the information-theoretic susceptibility at temperature $T_{★}$, computed from the Kubo–Mori metric in an ideal-gas approximation, has an extremum (in fact, a local maximum or minimum depending on conventions) along the hypercharge direction.

(iii) 

The QICT requirements on the distinguished charge used in the Golden Relation (existence of a diffusive channel, finite and positive susceptibility, and compatibility with the microscopic QCA encoding) single out precisely this hypercharge direction as the unique viable $U\left(1\right)$ candidate.

Proof. See the Mathematical Appendix, Sec. S3.

Proof. (i) The anomaly polynomial for a general linear combination $Q(\beta ,\gamma ,\delta )=\beta B+\gamma L+\delta Y$ can be written as a cubic form in $(\beta ,\gamma ,\delta )$, with coefficients determined by the traces of charge products over Weyl fermions. For the Standard-Model chiral content, the conditions that all gauge anomalies and mixed gauge–gravitational anomalies vanish define a system of homogeneous linear equations in $(\beta ,\gamma ,\delta )$, whose solution space is one-dimensional and spanned by the hypercharge assignment $Y_{\mathrm{SM}}$. This is a standard textbook result; we reproduce the explicit sums in the Mathematical Appendix.

(ii) The static susceptibility matrix in the $(B,L,Y)$ space is given by

$${\Xi }_{ab}\left(T\right)={\displaystyle \frac{{\partial }^2\Omega }{\partial {\mu }_a\partial {\mu }_b}}{|}_{\mu =0},a,b\in \{B,L,Y\},$$

(A10)

where $\Omega$ is the thermodynamic potential. In the ideal-gas approximation, $\Xi \left(T\right)$ is positive-definite and symmetric. Restricting to the anomaly-free subspace (one-dimensional in this case) and considering the quadratic form $\mathcal{S}\left[\overrightarrow{q}\right]={\overrightarrow{q}}^T\Xi \overrightarrow{q}$ on unit-norm charge vectors $\overrightarrow{q}$, the extremum condition reduces to an eigenvalue problem. Since the anomaly-free subspace is one-dimensional, hypercharge is automatically an eigen-vector and therefore an extremum direction of $\mathcal{S}$.

(iii) The QICT analysis requires a conserved charge with a diffusive channel, finite and positive information susceptibility, and an operationally defined copy time. Charges that are anomalous at the quantum level cannot satisfy these requirements consistently, because they fail to be exactly conserved at all scales. Purely baryonic or purely leptonic $U\left(1\right)$ charges are anomalous; their susceptibilities and transport properties are contaminated by the anomaly. The only remaining candidate in the $(B,L,Y)$ space that is both anomaly-free and couples to quarks and leptons is $Y_{\mathrm{SM}}$. Hence the QICT conditions single out hypercharge as the unique viable abelian generator. □

The Theorem shows that, given the Standard-Model fermion content and our microscopic QCA/QICT assumptions, the distinguished QICT charge used in the Golden Relation must be hypercharge.

Appendix B.4. Excluding Larger Simple Unification Groups

We now address the possibility that the full gauge group G is a larger simple group containing $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$ as a subgroup, such as $\mathrm{SU}\left(5\right)$ or $\mathrm{SO}\left(10\right)$. In such scenarios the low-energy gauge group arises from spontaneous symmetry breaking, and the observed hypercharge is embedded as a Cartan generator of the unified group.

From the perspective of the QICT–QCA–FRG framework, we require that:

• the QCA admit a local encoding of the full gauge group and its representations with a finite on-site Hilbert space;
• the FRG flow for the full gravity+gauge+matter system admit an asymptotically safe fixed point with a finite number of relevant directions; and
• the additional heavy gauge bosons and matter fields required by unification do not introduce extra light degrees of freedom or instabilities incompatible with the observed low-energy spectrum.

These constraints are difficult to analyse in complete generality, but we can formulate a physically motivated axiom capturing their effect.

Assumption A11 

(Asymptotic-safety minimality of the gauge algebra). Among all gauge algebras $\tilde{\mathfrak{g}}$ that

(a) 

contain $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ as a subalgebra acting in the same way on the light chiral fermions,

(b) 

admit an asymptotically safe fixed point with a finite number of relevant directions compatible with low-energy data, and

(c) 

can be implemented as a local gauge-coded QCA with finite on-site Hilbert space,

the actual gauge algebra realised in nature isminimalwith respect to inclusion: there is no strictly larger algebra $\tilde{\mathfrak{g}}⊋\mathfrak{g}$ satisfying (a)–(c).

This is an asymptotic-safety analogue of the minimality principle: among all QCA/QFT realisations consistent with observations and asymptotic safety, the one realised in nature uses the smallest gauge algebra compatible with the data.

Proposition A3 

(Exclusion of simple grand-unified algebras). Under Assumptions A9 and A11, any simple Lie algebra $\tilde{\mathfrak{g}}$ that strictly contains $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ and acts non-trivially on the light chiral fermions is excluded as the full gauge algebra at the QICT matching scale.

Proof. 

Let $\tilde{\mathfrak{g}}$ be a simple Lie algebra such as $\mathfrak{su}\left(5\right)$ or $\mathfrak{so}\left(10\right)$, with a decomposition under its $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ subalgebra that reproduces the observed light representations, plus additional heavy fields. In such a theory the FRG flow must be considered in the larger theory space of couplings associated with $\tilde{\mathfrak{g}}$ and the extra matter fields.

If $\tilde{\mathfrak{g}}$ admits an asymptotically safe fixed point with finitely many relevant directions, then by Assumption A11 the realised gauge algebra must be the minimal one satisfying the conditions (a)–(c). But the subalgebra $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ also admits an asymptotically safe fixed point with the same light matter content and fewer gauge degrees of freedom, and can be implemented as a simpler local QCA. Therefore $\tilde{\mathfrak{g}}$ cannot be minimal, and is excluded.

Conversely, if $\tilde{\mathfrak{g}}$ does not admit such an asymptotically safe fixed point, it is excluded directly by Assumption A9. In both cases, simple grand-unified algebras strictly larger than $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ are ruled out as candidates for the full gauge algebra at the QICT matching scale. □

Appendix B.5. Conditional Uniqueness Theorem

We can now assemble the previous statements into a single conditional uniqueness result.

Theorem A2 

(Conditional uniqueness of the Standard-Model gauge group). Assume:

(i) 

the microscopic dynamics is given by a gauge-coded QCA satisfying Assumption A5;

(ii) 

the emergent low-energy theory has a compact, connected gauge group G satisfying Assumptions A6–A8;

(iii) 

the combined gravity+gauge+matter system is asymptotically safe with a finite number of relevant directions, as in Assumption A9;

(iv) 

the low-energy chiral fermion content matches one Standard-Model-like generation with a single light Higgs doublet and a real singlet scalar S;

(v) 

QICT can be implemented on at least one non-trivial conserved $U\left(1\right)$ charge whose information susceptibility matches the thermal hypercharge susceptibility at a matching temperature $T_{★}$, as in Theorem A1;

(vi) 

the minimality principles of Assumptions A10 and A11 hold.

Then the gauge algebra $\mathfrak{g}=\mathrm{Lie}\left(G\right)$ acting on the light chiral fermions at the QICT matching scale is, up to finite abelian quotients and possible fully-decoupled spectator factors,

$$\mathfrak{g}\simeq \mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right),$$

(A11)

with the $\mathfrak{u}\left(1\right)$ factor identified with hypercharge $Y_{\mathrm{SM}}$.

Proof. See the Mathematical Appendix, Sec. S3.

Proof. 

By Proposition A1, the semi-simple part of $\mathfrak{g}$ must contain $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)$ acting non-trivially on the light fermions. By Proposition A2 and Theorem A1, there must be at least one abelian factor whose generator is hypercharge $Y_{\mathrm{SM}}$, on which QICT is implemented. Corollary A1 then implies that $\mathfrak{g}$ contains a subalgebra isomorphic to $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ acting exactly as in the Standard Model on the light sector.

Any strictly larger gauge algebra with this property is excluded by Proposition A3 and Assumption A11, which encode the asymptotic-safety and QCA minimality requirements. Therefore, up to finite quotients and spectator factors that decouple from the light sector, the full gauge algebra must coincide with $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$, with the abelian generator identified with hypercharge. This completes the proof. □

Appendix B.6. Status and Limitations of the Gauge-Selection Result

Theorem A2 is, in a precise sense, as strong a statement as the present QICT–QCA–FRG framework can support without going beyond what is known or reasonably conjectured:

• The logical implication is clear: if Assumptions A5–A11 hold, then the gauge algebra at the QICT matching scale is essentially that of the Standard Model.
• The physical content of the assumptions is non-trivial: they encode locality and causality at the QCA level, the presence of a relativistic continuum limit, anomaly cancellation and asymptotic safety in the FRG sense, and a minimality principle informed by both the QCA representation and the FRG flow.
• What is not proven is that any microscopic QCA satisfying Assumption A5 must realise precisely this gauge group; nor is it proven that asymptotic safety holds only for the Standard-Model gauge algebra and not for any larger unification group. These are encoded as axioms rather than derived facts.

In other words, the present framework does not yet solve the full “gauge-group selection problem” in an absolute sense. It does, however, provide a mathematically controlled conditional derivation:

Given locality, chiral matter, anomalies, QICT, and asymptotic safety,

and given a minimality principle at the level of the gauge algebra,

the unique consistent choice is $\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$ for the light sector.

This is the precise sense in which the QICT–QCA–FRG framework can currently be said to “derive” the Standard-Model gauge group. It turns an empirical input into the unique solution of a well-posed structural problem under explicit, physically motivated, and falsifiable assumptions.

Appendix C.1. Microscopic–Macroscopic Link and Strong Assumptions

The connection between the microscopic QCA-based description and the macroscopic continuum observables used in the phenomenological analysis rests on a set of strong assumptions:

• Emergent diffusive hydrodynamics. The QICT scaling theorem is formulated under explicit assumptions of emergent diffusive hydrodynamics for the distinguished conserved charge (dynamic exponent $z=2$, absence of ballistic contributions in the relevant channel, controlled finite-size effects, etc.). These properties are verified rigorously only in restricted classes of models (e.g. specific Lindblad generators) and numerically in stabiliser-code examples, but are not derived from the most general gauge-coded QCA dynamics considered in this work.
• Single matching scale and thermal equilibrium. The identification of the QICT scale with a thermal hypercharge susceptibility at a benchmark temperature $T_{★}=260\mathrm{GeV}$ assumes that the relevant degrees of freedom can be described by an approximately equilibrated plasma with ideal-gas susceptibilities, and that higher-order interactions and non-perturbative effects do not qualitatively modify the matching. This is a physically motivated but non-trivial hypothesis.
• Parametric robustness vs. quantitative accuracy. While the qualitative structure of the Golden Relation is expected to be robust under moderate variations of microscopic and matching-scale assumptions, the quantitative mass band for the singlet scalar inherits all uncertainties and potential biases associated with these choices. In particular, the adopted priors on $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$ and ${\chi }_Y^{\left(2\right)}$ are not uniquely determined by first principles.

Taken together, these points imply that the microscopic–macroscopic link constructed here should be viewed as a concrete scenario rather than a model-independent consequence of QICT.

Appendix C.2. Conditional Nature of the Gauge-Selection Theorem

The partial “derivation” of the Standard-Model gauge group presented in Appendix B is explicitly conditional on a set of axioms and minimality assumptions:

• The existence of a relativistic continuum limit of the gauge-coded QCA, with a compact, connected gauge group G acting on genuinely chiral fermions in complex representations.
• Exact cancellation of all local and mixed gauge–gravitational anomalies for the given fermion content.
• The existence of an asymptotically safe non-Gaussian fixed point for the combined gravity+gauge+matter system with a finite number of IR-relevant directions.
• Minimality assumptions on the gauge algebra and matter content at fixed low-energy spectrum, used to exclude larger simple unification groups in favour of $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$.
• The additional requirement that the distinguished $U\left(1\right)$ charge on which QICT is implemented coincides with the unique anomaly-free direction that couples to both quark and lepton sectors, identified with hypercharge.

None of these axioms is derived in this paper; they are motivated by current knowledge of chiral gauge theories, anomaly cancellation and asymptotic safety, but remain assumptions. Theorem A2 should therefore be interpreted strictly as a conditional statement: given QCA locality, chiral matter, anomaly cancellation, asymptotic safety and the adopted minimality principles, the gauge algebra is forced to be $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$. It is not a classification of all possible microscopic dynamics or continuum limits.

Appendix C.3. Theoretical Status and Lack of Immediate Experimental Validation

Although parts of the construction interface with phenomenology (e.g. the singlet-scalar mass band and direct-detection cross sections), the overall framework remains theoretical at this stage:

• The QICT scaling relation, the existence of a gauge-coded QCA realising a full Standard-Model-like generation, and the asymptotically safe FRG fixed point for gravity+SM+singlet are all subject to ongoing theoretical scrutiny. Their mutual consistency is plausible but not proven from a more fundamental microscopic theory.
• The numerical values adopted for $C_{\Lambda }$, ${\kappa }_{\mathrm{eff}}$ and ${\chi }_Y^{\left(2\right)}$ rely on specific truncations, approximations and matching prescriptions. Further improvements in FRG technology, lattice simulations or non-equilibrium QCA analyses may shift these values or even challenge some of the underlying assumptions.
• The most concrete phenomenological predictions (such as a resonance-centred mass window for the singlet scalar around the Higgs resonance and an associated range of direct-detection cross sections) are, by construction, scenario-dependent. They become meaningful only if one accepts the full chain of assumptions and identifications implemented in this work.

In summary, the microscopic–macroscopic link developed here relies on strong hypotheses (emergent diffusive hydrodynamics and matching at a single temperature $T_{★}=260\mathrm{GeV}$), and the “derivation” of the Standard-Model gauge group in Appendix B is conditional on a specific set of ad hoc axioms about chirality, anomalies, asymptotic safety and minimality. In the absence of immediate experimental validation of the QICT scaling or of the Golden-Relation mass window, the entire framework should therefore be regarded as a speculative but internally consistent theoretical proposal, rather than as an established or uniquely compelling description of nature.

Appendix D.1. Lorentzian Hydrodynamic Limit for Interacting Gauge-Coded QCA

The QICT analysis in the main text is formulated for channels whose long-wavelength dynamics is diffusive and whose low-energy dispersion relations are relativistic, $\omega \simeq c\left|k\right|$, up to controlled corrections. For free or weakly interacting QCA with suitable lattice symmetries, this can be established explicitly. For the fully interacting, gauge-coded QCA relevant to the Standard-Model-like sector, this was treated only at the level of assumptions.

In this subsection we define a concrete class of interacting, gauge-coded QCA for which: (i) a Lorentzian dispersion relation can be derived at low energy in perturbation theory, and (ii) isotropy of the emergent signal velocity can be quantified and tested numerically.

Appendix D.1.1. Class of Interacting QCA and Assumptions

We consider a family of translation-invariant, gauge-coded QCA on a cubic lattice ${\mathbb{Z}}^3$, with local Hilbert space ${\mathcal{H}}_x\cong {\mathbb{C}}^d$ per site and gauge links on edges, and a one-step update unitary U of the form

$$U=exp\left(-i(H_0+\lambda V)\right),$$

(A12)

where:

• $H_0$ is a strictly local Hamiltonian generating a free, relativistic QCA with dispersion ${\omega }_0\left(\mathbf{k}\right)=c\left|\mathbf{k}\right|+{\mathcal{O}\left(\right|\mathbf{k}|}^3)$ near $\mathbf{k}=0$ and a finite Lieb–Robinson velocity $v_{\mathrm{LR}}$.
• V is a local, gauge-invariant interaction term encoding the minimal couplings (gauge and Yukawa) required to reproduce a Standard-Model-like spectrum in the continuum.
• $\lambda \in \mathbb{R}$ is a dimensionless interaction parameter, assumed small (weakly interacting regime): $\left|\lambda \right|\ll 1$.
• The microscopic update is strictly local and causal, and respects the discrete symmetry group of the cubic lattice (rotations by $\pi /2$ around lattice axes and reflections).

We assume that the one-particle sector of $H_0$ can be diagonalised by a Bloch–Floquet transform, with bands labelled by an index a and momenta $\mathbf{k}$ in the Brillouin zone $\mathcal{B}$, such that

$$H_0\mathbf{k},a={\omega }_0^{\left(a\right)}\left(\mathbf{k}\right)\mathbf{k},a,$$

(A13)

and that the band hosting the light excitations of interest is non-degenerate near $\mathbf{k}=0$.

Appendix D.1.2. Perturbative Emergent Lorentz Invariance

We first state a perturbative result showing that Lorentzian dispersion is stable under weak, local, gauge-invariant interactions.

Proposition A4  

(Perturbative Lorentzian dispersion). Let U be a QCA update of the form (A12), with $H_0$ and V as above, and let ${\omega }_{\lambda }^{\left(a\right)}\left(\mathbf{k}\right)$ denote the interacting dispersion relation for band a. Assume:

(A1) 

The free dispersion near $\mathbf{k}=0$ is ${\omega }_0^{\left(a\right)}\left(\mathbf{k}\right)=c\left|\mathbf{k}\right|+{\mathcal{O}\left(\right|\mathbf{k}|}^3)$, with $c>0$.

(A2) 

The interaction V is local, gauge-invariant, and analytic in momentum space; its action on one-particle states is relatively bounded with respect to $H_0$.

(A3) 

There is a gap ${\Delta }_0>0$ separating the light band a from other bands in a neighbourhood of $\mathbf{k}=0$.

Then, for $\left|\lambda \right|$ sufficiently small, there exists a neighbourhood $\mathcal{U}$ of $\mathbf{k}=0$ such that

$${\omega }_{\lambda }^{\left(a\right)}\left(\mathbf{k}\right)=c_{\mathrm{eff}}\left(\lambda \right)\left|\mathbf{k}\right|+\mathcal{O}\left({\left|\mathbf{k}\right|}^3\right),\mathbf{k}\in \mathcal{U},$$

(A14)

with $c_{\mathrm{eff}}\left(\lambda \right)=c+\mathcal{O}\left(\lambda \right)$. Moreover, the $\mathcal{O}\left(\right|\mathbf{k}{|}^3)$ term is analytic in λ and $\left|\mathbf{k}\right|$.

Proof 

(Sketch of proof). The proof is standard degenerate perturbation theory for analytic families of operators. The assumed spectral gap (i.e., an isolated low-energy sector separated from the rest of the spectrum) allows us to define a Bloch Hamiltonian $H(\mathbf{k},\lambda )$ acting on a finite-dimensional internal space, analytic in $(\mathbf{k},\lambda )$ near $(0,0)$, with an isolated non-degenerate eigenvalue corresponding to band a. Kato’s theory of analytic perturbations ensures that the eigenvalue ${\omega }_{\lambda }^{\left(a\right)}\left(\mathbf{k}\right)$ is analytic in $(\mathbf{k},\lambda )$ in a neighbourhood of $(0,0)$. Rotational invariance of $H_0$ at leading order, combined with the discrete symmetry group of the lattice and the locality of V, implies that the only rotationally invariant scalar linear in $\left|\mathbf{k}\right|$ is $\left|\mathbf{k}\right|$ itself, with a coefficient renormalised by interactions. Terms quadratic in $\mathbf{k}$ are forbidden by parity; the first allowed non-linear corrections are cubic in $\left|\mathbf{k}\right|$, which yields the stated expansion. □

This proposition shows that, within a well-defined perturbative regime, the low-energy dispersion remains relativistic up to controllable corrections. Extending this result beyond perturbation theory and including strong coupling remains open.

Conjecture 1  

(Non-perturbative Lorentzian hydrodynamic limit). For gauge-coded QCA that is local and translation invariant, admits such a spectral separation, and admits a diffusive hydrodynamic limit for conserved charges, the long-wavelength, low-frequency modes of the associated continuity equations propagate on an emergent Lorentzian background with effective metric $g_{\mu \nu }^{\mathrm{eff}}$ and characteristic velocity $c_{\mathrm{eff}}$, in the sense that the retarded Green’s functions of charge and energy densities solve, at leading order,

$$\left({\square }_{g_{\mathrm{eff}}}+\dots \right)G_{\mathrm{ret}}\left(x\right)={\delta }^{\left(4\right)}\left(x\right),$$

(A15)

with Lorentz-violating corrections suppressed by powers of the lattice spacing a and the interaction strength λ.

A rigorous derivation of Conjecture 1 for non-trivial interacting examples remains a central open problem.

Appendix D.1.3. Numerical Test of Isotropy in Higher Dimensions

Beyond the formal analysis, the isotropy of information propagation can be tested numerically.

Definition of the anisotropy indicator.

For a given QCA update U, we define the maximal group velocity in the direction $\widehat{\mathbf{n}}$ as

$$c\left(\widehat{\mathbf{n}}\right)=\underset{a,\mathbf{k}\Vert \widehat{\mathbf{n}}}{max}\left|{\nabla }_{\mathbf{k}}{\omega }_{\lambda }^{\left(a\right)}\left(\mathbf{k}\right)\right|,$$

(A16)

and the anisotropy indicator as

$$\Delta c/c={\displaystyle \frac{{max}_{\widehat{\mathbf{n}}}c\left(\widehat{\mathbf{n}}\right)-{min}_{\widehat{\mathbf{n}}}c\left(\widehat{\mathbf{n}}\right)}{\frac{1}{4\pi }\int c\left(\widehat{\mathbf{n}}\right)d{\Omega }_{\widehat{\mathbf{n}}}}}.$$

(A17)

Numerical protocol.

For a given interacting gauge-coded QCA:

(N1)

Diagonalise the one-step update in momentum space on a discrete grid in $\mathbf{k}$ for 2D or 3D lattices of increasing size, extracting ${\omega }_{\lambda }^{\left(a\right)}\left(\mathbf{k}\right)$.

(N2)

Estimate $c\left(\widehat{\mathbf{n}}\right)$ along a dense set of directions $\widehat{\mathbf{n}}$ and compute $\Delta c/c$ as a function of the lattice spacing a and the interaction strength $\lambda$.

(N3)

Extrapolate to the continuum limit $a\to 0$ (or large system sizes) and weak-coupling limit to test whether $\Delta c/c\to 0$, and quantify the rate of convergence.

Conjecture 2  

(Isotropy bound). For gauge-coded QCA in the class defined above, there exist constants $C_1,C_2>0$ such that, for a sufficiently small and $\left|\lambda \right|$ sufficiently small,

$$\Delta c/c\le C_1{(a\Lambda )}^2+C_2{\lambda }^2,$$

(A18)

where Λ is a microscopic cutoff (e.g. inverse lattice spacing or maximal physical momentum). In particular, for realistic values of $(a,\lambda )$ compatible with the QICT matching scale, one expects $\Delta c/c\lesssim {10}^{-4}$.

A numerical verification of Conjecture 2 in 2D and 3D for concrete gauge-coded QCA families would provide a direct test of the credibility of the emergent Lorentzian metric in this framework.

Appendix D.2. Gauge-Group Selection from QICT Functionals and Stabiliser Algebra

The main text and Appendix B showed that, under explicit axioms (chiral matter, anomaly cancellation, asymptotic safety, minimality), the gauge algebra acting on the light sector is forced to be $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$. Here we sketch how this “minimality” can be tied more closely to QICT and to the stabiliser structure of gauge-coded QCA.

Appendix D.2.1. A QICT-Based Functional of the Gauge Group

We define a functional $F\left[G\right]$ that assigns to each candidate gauge group G a real number quantifying the “QICT efficiency” and microscopic complexity of its gauge-coded QCA realisation.

Let $\mathcal{C}\left(G\right)$ be the class of gauge-coded QCA whose emergent gauge group is G and whose matter content matches a fixed chiral spectrum (e.g. one SM-like generation). For each $U\in \mathcal{C}\left(G\right)$ we define:

• ${\tau }_{\mathrm{copy}}\left[U\right]$: a suitably normalised average information copy time for a set of distinguished conserved charges (including the hypercharge-like one used in QICT), e.g. averaged over directions and channels.
• ${\mathcal{K}}_{\mathrm{loc}}\left[U\right]$: a measure of local complexity, such as the minimal circuit depth per time step required to implement U with local unitaries acting on a fixed radius, or the minimal number of non-commuting local stabiliser generators per site.
• $\mathcal{A}\left[G\right]$: an anomaly-penalty functional, which is zero if all gauge and mixed anomalies cancel and positive otherwise; for example, $\mathcal{A}\left[G\right]$ could be the sum of squares of anomaly coefficients.

We then define

$$F\left[G\right]=\underset{U\in \mathcal{C}\left(G\right)}{sup}\left[\alpha {\tau }_{\mathrm{copy}}\left[U\right]-\beta {\mathcal{K}}_{\mathrm{loc}}\left[U\right]-\gamma \mathcal{A}\left[G\right]\right],$$

(A19)

with positive weights $(\alpha ,\beta ,\gamma )$ encoding the relative importance of efficient information propagation, microscopic simplicity, and anomaly freedom.

Proposition A5  

(Basic properties of F[G). ] Let G be a compact, connected Lie group for which the class $\mathcal{C}\left(G\right)$ of gauge–coded QCA, with the prescribed chiral matter content, is non-empty. Assume moreover that, for all $U\in \mathcal{C}\left(G\right)$, both ${\tau }_{\mathrm{copy}}\left[U\right]$ and ${\mathcal{K}}_{\mathrm{loc}}\left[U\right]$ are finite. Then:

(i) 

$F\left[G\right]$ is finite for every such G.

(ii) 

If G admits no anomaly-free embedding with the given chiral content, then $F\left[G\right]<0$ for any choice of $\gamma >0$ in Eq. (A19).

(iii) 

If G admits at least one anomaly-free embedding, there exists $U\in \mathcal{C}\left(G\right)$ with $\mathcal{A}\left[G\right]=0$, so that $F\left[G\right]$ is bounded from below by a strictly positive function of ${\tau }_{\mathrm{copy}}\left[U\right]$ and ${\mathcal{K}}_{\mathrm{loc}}\left[U\right]$.

The precise computation of $F\left[G\right]$ is highly non-trivial. However, it provides a concrete mathematical object that ties together QICT (i.e. ${\tau }_{\mathrm{copy}}$), microscopic QCA complexity, and anomaly constraints.

Conjecture 3  

(QICT optimality of the SM gauge group). For fixed light chiral spectrum matching one SM-like generation and for any positive weights $(\alpha ,\beta ,\gamma )$ in Eq. (A19), the functional $F\left[G\right]$ defined above is maximised (or at least admits a strict local maximum) for

$$G\simeq \mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right),$$

(A20)

with the $\mathrm{U}\left(1\right)$ factor identified with hypercharge $Y_{\mathrm{SM}}$.

A proof of Conjecture 3 would upgrade the “minimality” argument of Appendix B into a QICT-based optimality principle.

Appendix D.2.2. Stabiliser Algebra and Non-Abelian Structure

Gauge-coded QCA are naturally formulated in terms of local stabiliser operators (e.g. products of Pauli matrices) enforcing local constraints (Gauss laws, code conditions). These stabilisers generate an operator algebra whose commutation relations reflect the underlying gauge structure.

Let $\left\{S_{\alpha }\right\}$ be a set of local, Hermitian stabiliser generators acting on a finite neighbourhood of each lattice site, such that:

(S1)

The stabilisers close under commutation: $[S_{\alpha },S_{\beta }]=i{f}_{\alpha \beta }^{\gamma }{S}_{\gamma }$, with real structure constants $f_{\alpha \beta }^{\gamma }$.

(S2)

The representation of the algebra generated by $\left\{S_{\alpha }\right\}$ on the local code space is irreducible.

(S3)

The stabilisers implement local gauge transformations on the matter and link degrees of freedom of the QCA.

Proposition A6  

(Lie-algebra structure of stabilisers). Under assumptions (S1)–(S3), the real span of $\left\{S_{\alpha }\right\}$ with the commutator as Lie bracket is a compact, semisimple Lie algebra $\mathfrak{h}$, and the local code space furnishes a unitary representation of $\mathfrak{h}$.

Proof 

(Sketch of proof). (S1) implies that the $S_{\alpha }$ generate a finite-dimensional real Lie algebra. Hermiticity and unitarity of the representation ensure that the corresponding group is compact. The absence of abelian factors acting trivially on the code space (because stabilisers are non-trivial constraints) implies that the algebra is semisimple. The representation on the local code space is unitary by construction. □

In principle, many compact semisimple Lie algebras are possible. However, additional constraints from QCA locality, code distance, and the requirement of matching the chiral SM-like matter content are expected to restrict $\mathfrak{h}$ to a small subset.

(N) series).Conjecture 4 (Stabiliser efficiency and SU Among all compact semisimple Lie algebras $\mathfrak{h}$ that can be realised as stabiliser algebras satisfying (S1)–(S3) on a fixed local Hilbert space dimension d, the classical series $\mathfrak{su}\left(N\right)$ maximise a suitable “efficiency ratio”

$$\mathcal{E}\left[\mathfrak{h}\right]={\displaystyle \frac{dim\left(\mathrm{fundamental}\mathrm{rep}\right)}{dim\left(\mathfrak{h}\right)}},$$

(A21)

subject to the requirement that the emergent gauge theory admits chiral fermions with SM-like quantum numbers and anomaly cancellation. In particular, for the colour and weak sectors, the choices $\mathfrak{su}\left(3\right)$ and $\mathfrak{su}\left(2\right)$ are singled out by this criterion within the space of stabiliser-compatible algebras.

A rigorous classification of stabiliser algebras satisfying (S1)–(S3), together with anomaly and matter-content constraints, would go a long way towards turning Conjecture 4 into a theorem.

Appendix D.3. Cosmological Sector: Boltzmann Implementation and Data Confrontation

The Golden Relation connects the singlet-scalar mass $m_S$ to QICT and FRG parameters, and the singlet-scalar dark matter model is already confronted with direct-detection and collider bounds. A natural next step is to embed the QICT sector into cosmology and confront it with CMB and large-scale-structure data via a Boltzmann code.

We outline here a concrete cosmological extension in which:

• the singlet scalar S is treated as a standard cold dark matter (CDM) component with mass fixed (or sharply constrained) by the Golden Relation;
• an additional “information fluid” with energy density ${\rho }_{\mathrm{info}}$ and pressure $p_{\mathrm{info}}$ is added to the energy budget, representing the QICT contribution to the effective stress-energy tensor;
• both background and perturbation equations are modified accordingly, and the model is implemented in a Boltzmann code such as CLASS or CAMB.

Appendix D.3.1. Background Evolution with an Information Fluid

We work in a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric with scale factor $a\left(t\right)$ and Hubble rate $H=\dot{a}/a$. The Friedmann equations are modified to include ${\rho }_{\mathrm{info}}$:

$$\begin{array}{cc}\hfill H^2& ={\displaystyle \frac{8\pi G}{3}}\left({\rho }_{\mathrm{r}}+{\rho }_{\mathrm{b}}+{\rho }_{\mathrm{cdm}}+{\rho }_{\Lambda }+{\rho }_{\mathrm{info}}\right),\hfill \end{array}$$

(A22)

$$\begin{array}{cc}\hfill \dot{H}& =-4\pi G\left({\rho }_{\mathrm{tot}}+p_{\mathrm{tot}}\right),\hfill \end{array}$$

(A23)

where ${\rho }_{\mathrm{cdm}}$ includes the singlet scalar S, ${\rho }_{\Lambda }$ is a (possibly residual) cosmological constant, and ${\rho }_{\mathrm{info}}$ is the QICT-induced component.

We postulate an effective equation of state

$$w_{\mathrm{info}}\left(a\right)\equiv {\displaystyle \frac{p_{\mathrm{info}}}{{\rho }_{\mathrm{info}}}}\simeq -1+\delta w\left(a\right),$$

(A24)

with $\left|\delta w\right(a\left)\right|\ll 1$ over the redshift range constrained by CMB and large-scale-structure data. The continuity equation for ${\rho }_{\mathrm{info}}$ reads

$${\dot{\rho }}_{\mathrm{info}}+3H\left(1+w_{\mathrm{info}}\left(a\right)\right){\rho }_{\mathrm{info}}=0.$$

(A25)

A QICT-motivated parametrisation could be

$$w_{\mathrm{info}}\left(a\right)=-1+ϵ{\left({\displaystyle \frac{a}{a_{*}}}\right)}^n,$$

(A26)

with small $ϵ$ and integer n, where $a_{*}$ is the scale factor corresponding to the QICT matching temperature $T_{*}$. This is only an illustrative example; more refined parametrisations could be derived from the microscopic dynamics of ${\tau }_{\mathrm{copy}}$ in an expanding background.

Appendix D.3.2. Linear Perturbations and Boltzmann Hierarchy

In Newtonian gauge, the scalar-perturbed FLRW metric reads

$$d{s}^2=-(1+2\Psi )d{t}^2+a^2\left(t\right)(1-2\Phi )d{\mathbf{x}}^2.$$

(A27)

For each fluid species i (radiation, baryons, CDM, etc.), the density contrast ${\delta }_i$ and velocity divergence ${\theta }_i$ satisfy the usual linearised conservation equations. The information fluid contributes additional perturbations $({\delta }_{\mathrm{info}},{\theta }_{\mathrm{info}})$ satisfying

$$\begin{array}{cc}\hfill {\dot{\delta }}_{\mathrm{info}}& =-(1+w_{\mathrm{info}})\left({\theta }_{\mathrm{info}}-3\dot{\Phi }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -3H\left(\delta w_{\mathrm{info}}{\delta }_{\mathrm{info}}+(1+w_{\mathrm{info}}){\delta }_{\mathrm{info}}\right),\hfill \end{array}$$

(A28)

$$\begin{array}{cc}\hfill {\dot{\theta }}_{\mathrm{info}}& =-H(1-3w_{\mathrm{info}}){\theta }_{\mathrm{info}}+{\displaystyle \frac{c_{\mathrm{s},\mathrm{info}}^2}{1+w_{\mathrm{info}}}}{k}^2{\delta }_{\mathrm{info}}+k^2\Psi ,\hfill \end{array}$$

(A29)

where $c_{\mathrm{s},\mathrm{info}}^2$ is the effective sound speed of the information fluid in its rest frame. For a nearly cosmological-constant component, one expects $c_{\mathrm{s},\mathrm{info}}^2\simeq 1$.

The singlet scalar S is treated as a standard CDM-like component with negligible pressure and sound speed, with perturbations ${\delta }_S$ and ${\theta }_S$ obeying the usual CDM perturbation equations.

To implement this in a Boltzmann code such as CLASS or CAMB, one adds the information fluid as an additional species with background evolution governed by $w_{\mathrm{info}}\left(a\right)$ and linear perturbations governed by the above equations. The total gravitational potentials $\Phi$ and $\Psi$ are then obtained from the Einstein equations with the modified total stress-energy tensor, and the CMB and matter power spectra are computed in the standard way.

Appendix D.3.3. MCMC Analysis and Observational Constraints

A full confrontation of the QICT cosmological sector with data requires a Markov-Chain Monte Carlo (MCMC) exploration of the parameter space, including:

• Standard cosmological parameters: $({\Omega }_b{h}^2,{\Omega }_{\mathrm{cdm}}{h}^2,H_0,n_s,A_s,{\tau }_{\mathrm{reio}})$.
• Singlet scalar parameters: $m_S$ (constrained or fixed by the Golden Relation) and possible residual freedom in the Higgs-portal coupling ${\lambda }_{HS}$, subject to consistency with relic density and collider constraints.
• QICT/information-fluid parameters: initial energy density ${\Omega }_{\mathrm{info}}$, equation-of-state parameters (e.g. $ϵ,n$ in the illustrative parametrisation), and sound speed $c_{\mathrm{s},\mathrm{info}}^2$.

An MCMC analysis could use Planck 2018 CMB likelihoods and large-scale-structure data (e.g. SDSS, DESI), together with local $H_0$ measurements if desired. The key questions are:

(Q1)

Is there a region of parameter space in which the QICT cosmological sector is consistent with current data at the same level as $\Lambda$CDM?

(Q2)

Does the inclusion of the information fluid alleviate any known tensions (e.g. $H_0$ or $S_8$) without spoiling the fit to CMB and LSS?

(Q3)

To what extent do cosmological data constrain the QICT parameters $({\Omega }_{\mathrm{info}},w_{\mathrm{info}}\left(a\right),c_{\mathrm{s},\mathrm{info}}^2)$ and the singlet scalar mass $m_S$ beyond the direct-detection and collider bounds?

A positive answer to (Q1) and (Q2), together with non-trivial constraints from (Q3), would elevate the QICT–QCA–FRG framework from a purely theoretical construction to a quantitatively tested cosmological model. A negative result (e.g. strong exclusion of any non-negligible ${\Omega }_{\mathrm{info}}$ or tight bounds forcing $w_{\mathrm{info}}\to -1$ and $m_S$ far from the Golden-Relation band) would falsify significant parts of the current implementation, thereby providing a clear empirical verdict on this aspect of the framework.

Appendix D.4. Status Summary of Level-4 Extensions

For clarity, we summarise the status of the Level-4 components:

• Lorentzian hydrodynamic limit: Proposition A4 gives a perturbative derivation of relativistic dispersion for a non-trivial class of interacting, gauge-coded QCA. Conjectures 1 and 2 define precise non-perturbative and numerical targets.
• Gauge-group selection: The functional $F\left[G\right]$ in Eq. (A19) ties together QICT, microscopic QCA complexity and anomaly cancellation. Conjectures 3 and 4 formulate the idea that the Standard-Model gauge group is singled out by a QICT-based optimality principle and by stabiliser-algebra efficiency, turning the heuristic “minimality” into a precise optimisation problem.
• Cosmological sector: The inclusion of an information fluid with nearly $w\simeq -1$, together with the singlet scalar dark matter candidate, defines a concrete extension of $\Lambda$CDM that can be implemented in a Boltzmann code and tested against Planck and LSS data through MCMC. This yields a clear path to falsifying or supporting the QICT framework at the cosmological level.

In all three directions, the problems are now formulated in a way that is both structurally constrained by the existing QICT–QCA–FRG framework and operationally falsifiable, in the sense that progress can be made by a combination of rigorous analysis, controlled numerics, and confrontation with experimental and observational data. For completeness, and to make the closure chain explicit in one place, Appendix E rewrites the Golden Relation as a conditional closure formula starting from the microscopic QICT definitions together with the diffusive thermal matching ansatz.

Appendix E.1. Definitions, Regime, and Notational Separation

This appendix makes the Golden Relation fully explicit from the microscopic definitions, while keeping the notational separation between (i) the thermodynamic hypercharge susceptibility and (ii) the QICT (Liouvillian-squared) susceptibility.

Thermodynamic susceptibility.

We denote by

$${\chi }_Y\equiv {\left.{\displaystyle \frac{\partial n_Y}{\partial {\mu }_Y}}\right|}_T$$

(A30)

the standard static (equilibrium) hypercharge susceptibility. In natural units, $\left[{\chi }_Y\right]={\mathrm{GeV}}^2$, so $\sqrt{{\chi }_Y}$ has units of GeV.

QICT susceptibility.

We denote by ${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}$ the QICT susceptibility defined through the Kubo–Mori metric and the squared inverse Liouvillian as in Sec. II. Under the additional assumption that the relevant thermal channel is governed by a diffusive reduction, ${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}$ can be matched to a hydrodynamic expression proportional to ${\chi }_Y$, so that the Golden Relation may be written directly in terms of ${\chi }_Y$ evaluated at the benchmark temperature $T_{★}$.

Information scale and matching.

We define the information scale

$$k_I\equiv {\tau }_{\mathrm{copy}}^{-1}\left(Y\right),$$

(A31)

and identify the infrared matching scale for the scalar mass with this information scale, $m_S=\sqrt{{\kappa }_{\mathrm{eff}}}{k}_I$, where ${\kappa }_{\mathrm{eff}}$ is the dimensionless FRG mass parameter defined in Sec. IV.

Chaotic mixing scale.

We define the Lyapunov exponent ${\lambda }_L$ operationally from the exponential growth rate of an OTOC in the pre-saturation regime (when present), and introduce the dimensionless ratio

$$u\equiv {\displaystyle \frac{{\lambda }_L}{2\pi T}},$$

(A32)

so that ${\lambda }_L=u2\pi T$ is an identity by definition of u (no additional postulate is required). When the MSS bound applies [31], one has $u\le 1$, but the derivation below does not require saturating any bound; it only uses ${\lambda }_L>0$ to define a finite microscopic mixing time ${\tau }_{\mathrm{mix}}\equiv {\lambda }_L^{-1}$.

With these definitions, we obtain the following conditional closure steps: (i) the exponent $\alpha =\frac{1}{2}$ once the diffusive reduction is assumed, (ii) an explicit expression for the normalisation $C_{\Lambda }$ in terms of u and transport data, (iii) a two-loop stability criterion for the predicted band, and (iv) a cosmological closure check through freeze-out/freeze-in.

Appendix E.2. From the Liouvillian definition to the diffusive exponent α=1 2

We recall the QICT definition of the Liouvillian-squared susceptibility for hypercharge:

$${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}\equiv {〈\dot{Y},{\mathcal{L}}^{-2}\dot{Y}〉}_{\mathrm{KM}},$$

(A33)

where $\mathcal{L}$ is the Liouvillian superoperator generating time evolution, and ${\langle ·,·\rangle }_{\mathrm{KM}}$ is the Kubo–Mori inner product. Using the spectral representation (details as in Sec. Section 2 and Supplementary), one may rewrite

$${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}={\int }_0^{\infty }dtt{\langle \dot{Y}\left(t\right)\dot{Y}\left(0\right)\rangle }_{\mathrm{KM}}.$$

(A34)

In the diffusive hydrodynamic regime, the slow mode is the conserved density. For a single diffusive mode at wave number k, the relaxation rate is ${\Gamma }_k=D_Y{k}^2$. The relevant correlator decays as $\langle {\dot{Y}}_k\left(t\right){\dot{Y}}_k\left(0\right)\rangle \propto e^{-{\Gamma }_{k}t}$. Inserting into (A34) yields (up to a universal numerical factor fixed by normalisation)

$${\chi }_{Y,\mathrm{micro};k}^{\left(2\right)}\propto {\displaystyle \frac{{\chi }_{Y,k}}{{\Gamma }_k^2}}={\displaystyle \frac{{\chi }_{Y,k}}{D_Y^2{k}^4}},$$

(A35)

where ${\chi }_{Y,k}$ is the static susceptibility of the k-mode. A local operational copying protocol at resolution length ℓ probes modes $k\sim \pi /\ell$; taking $\ell \simeq a$ gives the microscopic estimate

$${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}\propto {\displaystyle \frac{{\chi }_Y}{D_Y^2}}{a}^4.$$

(A36)

By the variational copy–susceptibility bounds established in the technical appendix (Copy-time bound / Point (6)),

$${\tau }_{\mathrm{copy}}\left(Y\right)\propto {\left({\chi }_{Y,\mathrm{micro}}^{\left(2\right)}\right)}^{-1/2}.$$

(A37)

Using (A36) immediately gives

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{copy}}\left(Y\right)& \propto {\displaystyle \frac{D_Y}{a^2\sqrt{{\chi }_Y}}},\hfill \\ \hfill {\tau }_{\mathrm{copy}}\left(Y\right)& \propto {\left({\chi }_Y\right)}^{-1/2}.\hfill \end{array}$$

(A38)

Thus the exponent is fixed to $\alpha =\frac{1}{2}$ in any regime where the dominant slow mode is diffusive and where the Liouvillian-squared susceptibility reduces to (A35). More generally, the $-1/2$ exponent is enforced by the variational formulation of the technical appendix (Copy-time bound / Point (6)) once the susceptibility is defined operationally as a receiver-optimised Liouvillian-squared object; the diffusive analysis here is used to connect the microscopic quantity ${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}$ to the thermodynamic ${\chi }_Y$ employed in the electroweak matching.

Appendix E.3. Deriving C Λ from quantum chaos (Lyapunov-controlled mixing)

The remaining normalisation constant is fixed by the crossover between (i) microscopic chaotic mixing (as diagnosed by OTOCs/Lyapunov growth) and (ii) hydrodynamic diffusion. The integral in (A34) is dominated by times up to the mixing time ${\tau }_{\mathrm{mix}}$ beyond which the slow diffusive description applies. A minimal controlled interpolation is to introduce a short-time cutoff at ${\tau }_{\mathrm{mix}}$, yielding the estimate

$$\begin{array}{cc}\hfill {\chi }_{Y,\mathrm{micro}}^{\left(2\right)}\simeq & {\int }_0^{{\tau }_{\mathrm{mix}}}dtt{\langle \dot{Y}\left(t\right)\dot{Y}\left(0\right)\rangle }_{\mathrm{KM}}\hfill \\ \hfill & +{\int }_{{\tau }_{\mathrm{mix}}}^{\infty }dtt{\langle \dot{Y}\left(t\right)\dot{Y}\left(0\right)\rangle }_{\mathrm{hyd}}.\hfill \end{array}$$

(A39)

The first term is controlled by microscopic mixing; the second by diffusion and produces the scaling (A36). The microscopic piece fixes the dimensionless prefactor.

Under local chaotic mixing, microscopic mixing implies that the charge-current autocorrelation decays on ${\tau }_{\mathrm{mix}}\sim {\lambda }_L^{-1}$. Writing the short-time correlator as ${\langle \dot{Y}\left(t\right)\dot{Y}\left(0\right)\rangle }_{\mathrm{KM}}\approx {\langle {\dot{Y}}^2\rangle }_{\mathrm{KM}}{e}^{-t/{\tau }_{\mathrm{mix}}}$ for $t\lesssim {\tau }_{\mathrm{mix}}$, the first term in (A39) gives

$${\chi }_{Y,\mathrm{micro}}^{\left(2\right)}\propto {\langle {\dot{Y}}^2\rangle }_{\mathrm{KM}}{\tau }_{\mathrm{mix}}^2.$$

(A40)

Fluctuation–dissipation in a thermal state relates ${\langle {\dot{Y}}^2\rangle }_{\mathrm{KM}}$ to ${\chi }_Y$ and transport data; matching the micro and hydro regimes yields a unique dimensionless prefactor that depends only on the ratio of the mixing time to the thermal time.

Using the definition $u\equiv {\lambda }_L/\left(2\pi T\right)$ we identify

$${\tau }_{\mathrm{mix}}={\displaystyle \frac{1}{{\lambda }_L}}={\displaystyle \frac{1}{u2\pi T}}.$$

(A41)

Combining (A38) with the identification $m_S=\sqrt{{\kappa }_{\mathrm{eff}}}{\tau }_{\mathrm{copy}}^{-1}$ we obtain

$$\begin{array}{cc}\hfill m_S& =\sqrt{{\kappa }_{\mathrm{eff}}}{\tau }_{\mathrm{copy}}^{-1}\hfill \\ \hfill & =\left({\mathcal{C}}_{\mathrm{KM}}u2\pi {\displaystyle \frac{a}{D_Y}}\right)\sqrt{{\kappa }_{\mathrm{eff}}{\chi }_Y}.\hfill \end{array}$$

(A42)

where ${\mathcal{C}}_{\mathrm{KM}}$ is a pure number fixed by the Kubo–Mori normalisation convention used in Sec. Section 2 (and therefore not a fit parameter once that convention is fixed). Equation (A42) is the Golden Relation with a derived constant,

$$C_{\Lambda }={\mathcal{C}}_{\mathrm{KM}}u2\pi {\displaystyle \frac{a}{D_Y}},$$

(A43)

This replaces an unconstrained geometric postulate by a chaos-controlled benchmark formula: $C_{\Lambda }$ is fixed by the microscopic Lyapunov exponent through u, and by transport through $a/D_Y$, with the remaining factor ${\mathcal{C}}_{\mathrm{KM}}$ determined by the information-metric convention.

Numerical estimate.

For the benchmark values used in the main text, $a=0.197{\mathrm{GeV}}^{-1}$ and $D_Y=0.10{\mathrm{GeV}}^{-1}$, so $a/D_Y\simeq 1.97$. Taking ${\mathcal{C}}_{\mathrm{KM}}\simeq 1$ and a weak-coupling chaos ratio $u\simeq 0.13$ (well below the maximal bound $u\le 1$), Eq. (A43) gives

$$C_{\Lambda }\simeq \left(1\right)\times \left(0.13\right)\times \left(2\pi \right)\times \left(1.97\right)\simeq 1.6,$$

(A44)

consistent with the benchmark $C_{\Lambda }=1.6\pm 0.2$ adopted in Sec. Section 6. The quoted uncertainty is intended to cover moderate variations in u and in the transport ratio $a/D_Y$.

Dimensional check.

In natural units, $\left[a\right]=\left[D_Y\right]={\mathrm{GeV}}^{-1}$ so $a/D_Y$ is dimensionless; u and ${\mathcal{C}}_{\mathrm{KM}}$ are dimensionless; hence $C_{\Lambda }$ is dimensionless, and (A42) has $\left[m_S\right]=\mathrm{GeV}$ because $\sqrt{{\chi }_Y}$ has units of GeV.

Appendix E.4. Two-Loop Radiative Stability of the Predicted Band

We summarise the two-loop stability requirement in the perturbative portal regime where two-loop running applies in the $Z_2$ singlet-scalar Higgs-portal model (Sec. Section 7). Writing the pole mass as

$$\begin{array}{cc}\hfill m_{S,\mathrm{pole}}^2=& m_S^2\left(\mu \right)+{\Pi }_S^{\left(1\right)}(p^2=m_S^2;\mu )\hfill \\ \hfill & +{\Pi }_S^{\left(2\right)}(p^2=m_S^2;\mu )+\dots .\hfill \end{array}$$

(A45)

radiative stability of the Golden-Relation band requires that higher-order corrections remain subdominant compared to the quoted uncertainty:

$$\left|{\Pi }_S^{\left(2\right)}\right|\lesssim \delta m_S^2,\mathrm{and}\left|{\Pi }_S^{\left(1\right)}\right|\mathrm{is}\mathrm{absorbed}\mathrm{in}\mathrm{the}\mathrm{matching}\mathrm{scheme}.$$

(A46)

At the parametric level, the dominant portal contribution scales as ${\Pi }_S^{\left(1\right)}\sim ({\lambda }_{HS}/16{\pi }^2)m_h^2$ and ${\Pi }_S^{\left(2\right)}\sim ({\lambda }_{HS}^2/{\left(16{\pi }^2\right)}^2)m_h^2$ up to logarithms and thresholds. Therefore, for portal couplings in the phenomenologically viable regime near the Higgs resonance, the two-loop correction is naturally suppressed by an additional ${\left(16{\pi }^2\right)}^{-1}$ factor. In practice, the stability check consists of running $({\lambda }_{HS},{\lambda }_S)$ with the two-loop RGEs and verifying that the induced shift in the pole mass remains within the Golden-Relation uncertainty band.

Appendix E.5. Cosmological Closure: Freeze-Out / Freeze-In and Planck Abundance

To close the bridge to cosmology, the predicted parameter region must reproduce the observed relic abundance. For freeze-out, the comoving abundance $Y=n_S/s$ satisfies

$${\displaystyle \frac{dY}{dx}}=-{\displaystyle \frac{s\langle \sigma v\rangle }{Hx}}\left(Y^2-Y_{\mathrm{eq}}^2\right),x\equiv {\displaystyle \frac{m_S}{T}},$$

(A47)

with $\langle \sigma v\rangle$ determined by the Higgs-portal interaction (Sec. Section 7). The relic density follows from

$${\Omega }_S{h}^2={\displaystyle \frac{m_S{s}_0{Y}_{\infty }}{{\rho }_c/h^2}}.$$

(A48)

The Golden Relation fixes $m_S$ in terms of $({\kappa }_{\mathrm{eff}},{\chi }_Y)$; cosmological closure is achieved by showing that the corresponding portal coupling window yields ${\Omega }_S{h}^2$ equal to the Planck value within uncertainties.

For freeze-in, the abundance is instead sourced by the production rate ${\Gamma }_{\mathrm{prod}}$ (portal-mediated scatterings/decays),

$${\displaystyle \frac{dY}{dx}}={\displaystyle \frac{1}{sHx}}{\Gamma }_{\mathrm{prod}}\left(T\right),$$

(A49)

and closure requires that the same $m_S$ band admits a portal coupling range producing the observed ${\Omega }_{\mathrm{DM}}{h}^2$ without violating laboratory bounds (direct detection and Higgs invisible), as implemented in Sec. Section 7.