# B From RTQ Hypergraphs to Spectral Dimension, and Effective 4D QFT

## B.1 Purpose and Relation to the Main Text

### B.1.1 Scope of this appendix

This appendix provides the formal bridge between the discrete RTQ substrate and the effective 4D $\Lambda$-framework used in the main text. In compact form, it connects

$$ \text{RTQ hypergraph} \rightarrow d_s(\tau) \rightarrow d \rightarrow \text{effective 4D QFT,} $$

where $d_s(\tau)$ is the running spectral dimension, $d$ the effective number of spatial dimensions selected by the constants audit, and the final 4D theory is the Lorentzian effective field theory used in Secs. 3–6.

More precisely, Appendix B does three things:

1. It defines the RTQ hypergraph $\mathcal{H} = (V, E)$, the associated diffusion operator, and the spectral dimension $d_s(\tau)$ that characterizes the large-scale behavior of RTQ ensembles.

2. It introduces the coarse-grained complex tension field

$$ \Lambda(x) = \Lambda^P(x) + i\Lambda^I(x), $$

and the associated coherence current, from which an emergent time direction and proper time are constructed.

3. It sets up a dimensional audit with a free spatial dimension $d$, and shows how the joint consistency of the audited carriers (Appendix A) and SM-like renormalizability selects $d = 3$, in agreement with the spectral plateau $d_s \simeq 3$.

Detailed constants audit (carriers, unit declarations, invariance checks) is given in Appendix A. Cosmological pipelines (in particular the $H_0$-dipole analysis) and falsifiability catalogs are treated in dedicated appendices (C–E). Appendix B is therefore a structural bridge: it explains how one passes from the RTQ substrate to the effective 4D action actually used in the cosmological and numerical pillars.

**Reader note: covariance in the continuum interface (what is—and is not—claimed)**

A natural question—especially for readers coming from GR, LQG, or string theory—is how “covariance” is handled when the microscopic substrate is discrete and combinatorial. The key point is that the RTQ level is coordinate-free by construction: it is not equipped with charts, differentiable structure, or a background manifold. In that setting, diffeomorphism invariance is not a symmetry to be imposed microscopically; rather, it emerges as an IR redundancy of description once one introduces continuum fields through coarse-graining. In practice, the coarse-grained fields ($g_{\mu\nu}$, $T^\mu$, $\Lambda$) are defined through a family of coarse-graining maps from relational RTQ data to local continuum charts; changing charts corresponds to relabelling the same relational content. Accordingly, physical statements are formulated in terms of IR invariants (covariant scalars and tensors built from $g_{\mu\nu}$, $T^\mu$, and $\Lambda$) and do not depend on the arbitrary coordinates used to represent a coarse-grained patch.

Finally, the 4D Lorentzian action used in the main text is presented explicitly as an *effective interface template*. Any residual representation dependence or preferred-structure artefacts induced by coarse-graining would appear as higher-order symmetry-breaking operators suppressed at large scales, and are therefore constrained by Tier-A consistency requirements (in particular Lorentz/preferred-frame bounds and Equivalence Principle tests).

### B.1.2 Notation and assumptions

We summarize here the basic notation and working assumptions used throughout Appendix B.

#### RTQ hypergraph
The microscopic substrate is an RTQ hypergraph

$$ \mathcal{H} = (V, E), $$

where $V$ is the set of nodes (relational tension quanta, RTQ) and $E$ a set of (hyper)edges encoding allowed multi-way relations between nodes.

### Tension weights

Links (edges or hyperedges) carry effective weights derived from the tension components, schematically

$$ W_{vu} \sim W_{vu}(\Lambda^P, \Lambda^I), $$

with $W_{vu} \ge 0$ and suitable normalization (e.g. row-stochastic or degree-normalized). The exact microscopic rule is not needed here; only the existence of a well-defined, symmetric (or symmetrisable) diffusion operator matters.

### Diffusion scale and spectral dimension

A fictitious diffusion parameter $\tau$ plays the role of a resolution scale on the graph. From the associated heat kernel $K_{\tau}(v, w)$ and average return probability $P(\tau)$, we define a running spectral dimension

$$ d_s(\tau), $$

and focus on regimes where a plateau $d_s(\tau) \simeq \text{const}$ exists over a finite range of $\tau$.

### Effective spatial dimension and manifold

When a stable plateau is present, we denote by $d$ the corresponding effective number of spatial dimensions, and represent the coarse-grained RTQ ensemble on an emergent Lorentzian manifold

$$ M, g_{\mu\nu}(x), $$

with $\mu, \nu = 0, \dots, d$. For the sector relevant to “SM + classical gravity” physics we will ultimately find $d = 3$, so that $M$ is effectively 3+1 dimensional.

### Complex tension field

RTQ statistics over coarse-graining patches are encoded in a complex tension field

$$ \Lambda(x) = \Lambda^P(x) + i \Lambda^I(x), $$

with an approximately conserved modulus

$$ |\Lambda(x)|^2 = (\Lambda^P)^2 + (\Lambda^I)^2 \simeq \Lambda_0^2, $$

representing a conserved “tension energy” at the coarse-grained level.

### Coherence current and time direction

We define a coherence current

$$ J^\mu = \Lambda^P \partial^\mu \Lambda^I, $$

and, when $J^\mu$ is timelike, a unit timelike alignment field

$$ T^\mu = \frac{J^\mu}{\sqrt{-J_\nu J^\nu}}, $$
from which emergent proper time is constructed.

## Working assumptions

Without attempting a full proof, we assume:

(i) $\mathcal{H}$ has finite average degree and no pathologies that would prevent the definition of a self-adjoint diffusion operator;

(ii) there exists a coarse-graining regime where the spectrum of the graph Laplacian admits a meaningful spectral dimension $d_s(\tau)$ and a plateau over some $\tau$-interval;

(iii) in that regime, large RTQ patches can be represented by coordinate charts on a smooth manifold with metric $g_{\mu\nu}$;

(iv) the effective carriers and unit declarations of Appendix A correctly capture the low-energy sector relevant for Secs. 3–6.

These assumptions make Appendix B a controlled, but still phenomenological, bridge between the discrete RTQ level and the effective 4D $\Lambda$-framework.

# B.2 RTQ Hypergraphs and Diffusion Operators

## B.2.1 RTQ hypergraph structure

At the microscopic level, the $\Lambda$-framework is defined on a network of relational tension quanta. We model this as a (hyper)graph

$$ \mathcal{H} = (V, E), $$

where:

* each node $v \in V$ represents an RTQ, i.e. a local unit of relational tension;

* each edge or hyperedge $e \in E$ encodes an allowed multi-way relation (interaction) between a finite set of RTQs;

* the presence and multiplicity of edges reflect the combinatorial patterns of the underlying tension network introduced in the main text.

No background metric, coordinates or manifold structure are assumed at this level: only **connectivity** (which RTQs can interact) and **local update rules** (how RTQ configurations are allowed to change) are fundamental. The effective roles of $\Lambda^P$ and $\Lambda^I$ at this stage are:

* to bias which configurations are favoured in the RTQ ensemble,

* to weight edges or hyperedges by effective coefficients derived from the tension components.

For the purposes of Appendix B, it is sufficient to assume that to each ordered pair of adjacent vertices $(v, u)$ we can associate a non-negative weight

$$ W_{vu} = W_{uv} \ge 0, $$

with $W_{vu} = 0$ when $v$ and $u$ are not directly connected. The precise dependence of $W_{vu}$ on $\Lambda^P, \Lambda^I$ and the local RTQ configuration is not required here; it is encoded implicitly in the statistics of the ensemble and in the coarse-grained tension field $\Lambda(x)$.

## B.2.2 Graph Laplacian and heat kernel

Given the weighted graph structure $\{W_{vu}\}$, we define for each vertex $v$ a neighborhood $\mathcal{N}(v)$ and a degree

$$ d_v = \sum_{u \in \mathcal{N}(v)} W_{vu}. $$
A natural discrete diffusion (or random-walk) operator $\Delta$ acting on scalar functions $f: V \to \mathbb{R}$ is then

$$ \Delta f(v) \equiv \sum_{u \in \mathcal{N}(v)} W_{vu} (f(u) - f(v)) $$

possibly after a degree normalization (e.g. replacing $W_{vu}$ by $W_{vu}/d_v$). Under mild conditions on $\{W_{vu}\}$, $\Delta$ defines a self-adjoint operator on $\mathbb{L}^2(V)$ and plays the role of a discrete Laplacian.

We introduce a fictitious diffusion parameter $\tau \ge 0$ and define the discrete heat kernel $K_{\tau}(v, w)$ as the solution of the heat equation

$$ \frac{\partial}{\partial \tau} K_{\tau}(v, w) = \Delta K_{\tau}(\cdot, w)(v), $$

with initial condition

$$ K_0(v, w) = \delta_{vw}, $$

where $\delta_{vw}$ is the Kronecker delta. For finite graphs, $K_{\tau}(v, w)$ can be expressed in terms of the eigenvalues and eigenvectors of $\Delta$; for large ensembles, we consider appropriate limits where local spectral densities converge.

A key observable for probing the effective dimension of the RTQ ensemble is the *average return probability*

$$ P(\tau) \equiv \frac{1}{|V|} \sum_{v \in V} K_{\tau}(v, v). $$

In homogeneous or statistically homogeneous regimes, the large-scale behavior of $P(\tau)$ encodes the spectral dimension. In particular, on a smooth $d$-dimensional manifold with Laplace-Beltrami operator, one has asymptotically

$$ P(\tau) \sim \tau^{-d/2}(\tau \to \infty). $$

In Appendix B.3 we use this analogy to define a running spectral dimension $d_s(\tau)$ for RTQ ensembles and to identify diffusion regimes where a plateau $d_s(\tau) \simeq 3$ justifies a 3+1D continuum description.

## B.3 Spectral Dimension of RTQ Ensembles

### B.3.1 Definition of the spectral dimension $d_s(\tau)$

On the RTQ hypergraph $\mathcal{H} = (V, E)$ endowed with the diffusion operator $\Delta$ introduced in Appendix B.3, the discrete heat kernel $K_{\tau}(v, w)$ is defined as the solution of the graph heat equation

$$ \frac{\partial}{\partial \tau} K_{\tau}(v, w) = \Delta K_{\tau}(v, w), \quad K_0(v, w) = \delta_{vw}, $$

where $\tau$ is a fictitious diffusion “time” that plays the role of a probe scale on the RTQ network. For fixed $\tau$, $K_{\tau}(v, w)$ encodes the probability density for a diffusive random walker to move from $w$ to $v$ in diffusion time $\tau$.

The key observable for dimensional diagnostics is the **average return probability**

$$ P(\tau) \equiv \frac{1}{|V|} \sum_{v \in V} K_{\tau}(v, v). $$

For homogeneous manifolds, $P(\tau)$ obeys the well-known scaling

$$ P(\tau) \sim \tau^{-d/2}(\tau \to \infty), $$
where $d$ is the topological dimension. On a general graph, this motivates defining a **running spectral dimension** by the logarithmic derivative

$$d_s(\tau) \equiv -2 \frac{d \ln P(\tau)}{d \ln \tau}.$$

In the continuum limit, $d_s(\tau)$ reduces to the usual spatial dimension when the geometry is smooth and homogeneous. On discrete RTQ ensembles it becomes a scale-dependent diagnostic of how diffusion explores the underlying connectivity: short $\tau$ probes microscopic adjacency motifs; intermediate $\tau$ probes the emergent “bulk” structure; very large $\tau$ can be sensitive to finite-size effects or large-scale inhomogeneities.

A **spectral plateau** is defined as an interval $[\tau_{min}, \tau_{max}]$ on which

$$d_s(\tau) \approx d_s(\text{approximately constant for } \tau \in [\tau_{min}, \tau_{max}]),$$

with $d_s$ a scale-independent effective dimension on that range. In this work, we focus on ensembles for which such a plateau exists at intermediate diffusion scales, between the microscopic RTQ scale and the macroscopic cosmological scale. This spectral plateau is the primary tool we use to extract an effective spatial dimensionality from the RTQ substrate.

In principle, one can also define a graph Hausdorff dimension $d_H$ by studying volume growth of metric balls on $\mathcal{H}$. For the ensembles of interest, we assume that on the plateau scales the two notions coincide,

$$d_s \approx d_H \equiv d,$$

so that the spectral dimension measured by diffusion matches the effective spatial dimension entering the continuum description. This identification will be used when we match $d_s$ to the dimensional analysis of the carriers in Appendix B.5.

### B.3.2 RTQ ensembles and the $d_s \approx 3$ plateau

We now state the key structural property used in the main text. For RTQ ensembles that satisfy the constants graph and carrier audit described in Appendix A, numerical experiments (and analytic estimates on simplified motifs) indicate that the running spectral dimension $d_s(\tau)$ exhibits a robust plateau

$$d_s(\tau) \approx 3$$

over a finite interval of diffusion scales $\tau$ corresponding to the regime where:

* the effective carriers reproduce the observed SI dimensions of $\{c, G, \hbar, \alpha, G_F, \dots\}$;

* the emergent dynamics is well-approximated by “SM + classical gravity” on a smooth background, as used in Secs. 3–6.

We do not attempt to present a full numerical study of $P(\tau)$ for concrete RTQ ensembles in this article; such simulations are deferred to subsequent work. For the present purpose, we only need:

1. the **existence** of a plateau with $d_s \approx 3$ on the physically relevant scales;

2. the **compatibility** of this plateau with the constants audit (Appendix A) and with the dimensional selection analysis of Appendix B.5.

Under these assumptions, the $d_s \approx 3$ plateau justifies approximating the relevant RTQ sector by a $3+1$-dimensional Lorentzian manifold $(M, g_{\mu\nu})$ with three spatial degrees of freedom and one emergent temporal degree of freedom. This is the continuum arena used in the effective action of Section 4.

However, the spectral plateau alone does not yet fix the spatial dimensionality uniquely from the constants' point of view: other values of $d_s$ might, in principle, arise for different RTQ ensembles. The role of the **dimensional audit** in Appendix B.5 is precisely to show that, once the carriers are required to match the observed SI dimensions and once SM-like renormalizability is imposed, the same spatial dimensionality $d=3$ is singled out. Appendix B.5 thus closes the loop:
$$ \text{RTQ diffusion} \Rightarrow d_s \approx 3 \text{ and carrier audit} + \text{renormalizability} \Rightarrow d = 3, $$

with $d_s$ and $d$ coinciding on the low-energy plateau.

## B.4 Coarse-Grained Tension Field and Emergent Proper Time

### B.4.1 Complex tension field as RTQ coarse-graining

On diffusion scales where a spectral plateau $d_s \approx 3$ exists, we can treat suitable RTQ patches as coarse-grained “fluid” elements and describe their averaged state by a complex tension field $\Lambda(x)$ on the emergent manifold $M$. By construction,

$$ \Lambda(x) = \Lambda^P(x) + i \Lambda^I(x), $$

where:

* $\Lambda^P$ encodes the crystallized, curvature-bearing component of the tension;

* $\Lambda^I$ encodes the coherence-bearing, phase-ordered component.

Both components arise from the same invariant modulus

$$ |\Lambda(x)|^2 = (\Lambda^P(x))^2 + (\Lambda^I(x))^2 = \Lambda_0^2, $$

where $\Lambda_0$ is a unit-free invariant carried by the RTQs. In other words, coarse-grained updates redistribute tension between $\Lambda^P$ and $\Lambda^I$ without changing the total “tension energy” $|\Lambda|^2$. The structural angle

$$ \Theta(x) \equiv \arg \Lambda(x) \Rightarrow \Lambda^P = \Lambda_0 \cos \Theta, \quad \Lambda^I = \Lambda_0 \sin \Theta $$

organizes this redistribution between curvature-bearing and coherence-bearing modes.

From the continuum viewpoint, $\Lambda(x)$ is not an additional force field on top of gravity. It is the macroscopic imprint of the RTQ tension statistics whose geometric response is encoded by the emergent metric $g_{\mu\nu}(x)$. The partition between $\Lambda^P$ and $\Lambda^I$ determines, at each point, how much of the tension is channelled into metric stiffness versus coherence.

### B.4.2 Coherence current and time direction

The coarse-grained dynamics of $\Lambda$ inherits a global phase symmetry, corresponding to rotations of the tension phase $\Theta$. In the continuum description, this symmetry leads to a conserved coherence current (informational flow)

$$ J^\mu = \Lambda^P \partial^\mu \Lambda^I, \nabla_\mu J^\mu = 0, $$

which encodes the exchange of tension between curvature ($\Lambda^P$) and coherence ($\Lambda^I$) across scales. Regions where $\Lambda^P$ is ordered and $\partial^\mu \Lambda^I$ is non-vanishing support a directed flow $J^\mu$ that can be timelike, spacelike, or null depending on the local balance of curvature and coherence.

In regimes where $J^\mu$ is timelike ($J_\mu J^\mu < 0$), we can define a unit time-like vector

$$ T^\mu \equiv \frac{J^\mu}{\sqrt{-J_\mu J^\mu}}, $$

which acts as a coherence / causal-alignment field. Along its integral curves, we introduce the four-velocity

$$ u^\mu \equiv T^\mu $$
and define proper time as the operational parameter

$$d\tau \equiv -u_{\mu} dx^{\mu}.$$

Here $\tau$ is not a fundamental microscopic variable; it is a derived parameter that measures distance along the ordered coherence flow. This construction coincides with the one used in Section 4.5, where $T^\mu$ appears as the macroscopic coherence field in the IR effective action. In the ordered phase, $T_\mu T^\mu = -1$ and the usual notion of proper time along timelike worldlines is recovered.

Thus, within the $\Lambda$-framework, “time” is the parameter along the flow of the tension-coherence current $J^\mu$. The existence of a well-defined temporal direction is equivalent to the existence of a stable, timelike coherence field $T^\mu$ on the emergent manifold.

### B.4.3 Regimes of ordered and disordered time

The relative weights of $\Lambda^P$ and $\Lambda^I$, conveniently encoded in the structural angle $\Theta$, control whether the coherence current $J^\mu$ can define a robust temporal direction. Three qualitative regimes can be distinguished:

1. **Curvature-dominated / ordered time ($\Lambda^P \gg \Lambda^I, \Theta \to 0$).**

The network is stiff: most tension sits in $\Lambda^P$, supporting a stable Lorentzian geometry. In this regime, $J^\mu$ is generically timelike and $T^\mu$ defines a robust arrow of time. Proper time $d\tau$ is well-behaved and the dynamics reduces to a GR-like hydrodynamics of the tension field, as used in the cosmological sector of Section 6.

2. **Balanced regime / time-crystal behavior ($\Lambda^P \approx \Lambda^I, \Theta \approx \pi/4$).**

Curvature and coherence exchange tension in a quasi-periodic way along the $\Lambda$-cycle. The coherence current can exhibit oscillatory behavior, leading to *time-crystal-like regimes* where temporal ordering persists but is modulated by standing waves of curvature-coherence exchange. In such phases, the notion of proper time remains meaningful but can display subtle micro-oscillations or coherence-induced modulations.

3. **Coherence-dominated / almost timeless regime ($\Lambda^I \gg \Lambda^P, \Theta \to \pi/2$).**

Here the network becomes highly flexible: metric stiffness collapses and coherence dominates. The current $J^\mu$ can become lightlike or spacelike; $T^\mu$ may cease to be well-defined as a global timelike field. In this limit, the construction of proper time along coherence flow breaks down, and the continuum description transitions towards an **almost timeless, purely phase-ordered regime**. This is the operational content of the “pure phase limit” discussed qualitatively in the main text: the metric description degenerates rather than diverges.

Appendix B is purely *kinematic* in this respect: it specifies how, given a coarse-grained tension field $\Lambda$, one constructs the coherence current $J^\mu$, the alignment field $T^\mu$, and the derived proper time $d\tau$, and how different $\Lambda^P/\Lambda^I$ regimes affect their validity. The *dynamical* scenarios — stabilization of a global arrow, possible transient $\Lambda$-oscillatory phases, and their cosmological implications — are treated in Sections 4 and 6 of the main text.

## B.5 Dimensional Audit with Free Spatial Dimension \mathbit{d}

### B.5.1 Motivation: leaving d free

The spectral analysis of RTQ ensembles in Appendix B.3 suggests that, on the scales relevant for “SM + classical gravity” physics, diffusion probes an effective dimension close to three, $d_s \simeq 3$. However, this is not imposed a priori: in principle, other RTQ ensembles could yield different spectral plateaux.

In the $\Lambda$-framework we therefore treat the number of spatial dimensions $d$ as a free parameter and perform a *dimensional audit* of the carriers introduced in Appendix A:

* the gravitational carrier $S_G$,

* the weak carrier $S_W$,

* and the effective couplings of the gauge, Yukawa and scalar sectors.
For each sector we compute how the SI dimensions would scale in a d-dimensional spatial universe and compare them to the audited 3D dimensions actually used in the paper. The goal is to show that joint consistency of

1. the carriers (same SI dimensions as in Appendix A), and

2. an effective QFT with SM-like renormalizable couplings,

selects a unique spatial dimensionality, namely $d = 3$. This provides an independent route, complementary to the spectral plateau $d_s \simeq 3$, for fixing the effective dimension of the continuum limit.

### B.5.2 Gravity in d spatial dimensions

We first consider the d-dimensional Newtonian force law between two point masses $m_1, m_2$ separated by a distance $r$:

$$F_d(r) \sim \frac{G_d m_1 m_2}{r^{d-1}}.$$

In SI units,

$$[F_d] = \text{kg m s}^{-2}, \quad [m_1] = [m_2] = \text{kg}, \quad [r] = \text{m}.$$

Demanding dimensional consistency,

$$[G_d] [m_1] [m_2] [r]^{-(d-1)} = [F_d],$$

one finds

$$[G_d] = \text{kg}^{-1} \text{m}^d \text{s}^{-2}.$$

The audited gravitational carrier used in Appendix A is

$$S_G = c^3 \hbar^{-1} \langle \rho \rangle^{-1},$$

with

$$[c^3] = \text{m}^3 \text{s}^{-3}, \quad [\hbar^{-1}] = \text{kg}^{-1} \text{m}^{-2} \text{s}^1, \quad [\langle \rho \rangle] = \text{m}^{-2} \Rightarrow [\langle \rho \rangle^{-1}] = \text{m}^2.$$

Hence

$$[S_G] = [c^3][\hbar^{-1}][\langle \rho \rangle^{-1}] = \text{kg}^{-1} \text{m}^3 \text{s}^{-2}.$$

Requiring that the gravitational carrier reproduce the d-dimensional Newton constant,

$$[S_G] = [G_d],$$

immediately yields

$$\text{kg}^{-1} \text{m}^3 \text{s}^{-2} = \text{kg}^{-1} \text{m}^d \text{s}^{-2} \Rightarrow d = 3.$$

Thus, under the audit convention $S_G = c^3\hbar^{-1}\langle\rho\rangle^{-1}$ with $[\langle\rho\rangle] = \text{m}^{-2}$, the gravitational sector is only dimensionally consistent with a spatially d-dimensional Newton law if $d = 3$. This already ties the carrier choice to a specific number of spatial dimensions.
### B.5.3 Extracting $N_{\text{pass}}$ from the matching condition

Imposing the matching condition

$$N_{\text{loop}}^{(C)} = N_{\text{loop}}^{(M)}(t_{\text{info}}),$$

with $t_{\text{info}}$ the clock relevant for phase-resolution/holonomy (see Sec. 5.6), yields

$$N_{\text{pass}} = \frac{1}{4\pi \cos \Theta_{\text{struct}}} \frac{N_{\text{loop}}^{(C)}}{\eta_{\text{info}}} = \frac{1}{4\pi \cos \Theta_{\text{struct}}} \frac{\lambda_{C,e}/\ell_0}{(T/t_{\text{info}})^{1/3}}.$$

Using

$$\ell_0 = \sqrt{\frac{8\pi}{\alpha_G}} \ell_P, \quad m_P \equiv \frac{\hbar}{c \ell_P}, \quad \lambda_{C,e} = \frac{\hbar}{m_e c},$$

this can be rewritten as

$$N_{\text{pass}} = \frac{\sqrt{\alpha_G}}{4\pi\sqrt{8\pi} \cos \Theta_{\text{struct}}} \frac{1}{\eta_{\text{info}}} \frac{m_P}{m_e}, \quad \eta_{\text{info}} \equiv \left(\frac{T}{t_{\text{info}}}\right)^{1/3}.$$

#### B.5.3.1 Numerical sanity check

With $\Theta_{\text{struct}} \approx 0.38$ rad (so $\cos \Theta_{\text{struct}} \approx 0.93$), $\alpha_G \approx 1$, and $\eta_{\text{info}} \approx (T/t_P)^{1/3} \sim 2 \times 10^{20}$, one obtains $N_{\text{pass}} \approx 2$. This integer value supports the interpretation of a double-pass / double-strand closure for the lightest charged fermion.

### B.5.4 Weak sector: $G_F^{(d)}$ and $S_W^{(d)}$

We now repeat the exercise for the weak sector. Consider the Fermi four-fermion effective interaction in $D = d + 1$ spacetime dimensions:

$$S_{\text{int}}^{(d)} \sim \int d^D x \sqrt{-g} G_F^{(d)} (\bar{\psi} \gamma^\mu \psi) (\bar{\psi} \gamma_\mu \psi).$$

For power-counting in $D = d + 1$ spacetime dimensions we work with the dimensionless combination $S/\hbar$ (since in SI the action has dimension $[\hbar]$). We introduce an engineering “energy-dimension” $E$ as a bookkeeping device: derivatives are counted through the dimensionless operator $(\hbar c)\partial_\mu/E$, so that the measure scales as $d^D x \sim (\hbar c/E)^D$. This counting is used only to track renormalizability and does not set $\hbar$ or $c$ to unity.

With this convention, the Dirac kinetic term $\bar{\psi} i\hbar c \gamma^\mu \partial_\mu \psi$ implies

$$[\bar{\psi}\psi]_E E \sim E^D \Rightarrow [\psi]_E = E^{(D-1)/2}.$$

The four-fermion operator $(\bar{\psi}\gamma^\mu\psi)^2$ therefore scales as $E^{2(D-1)}$, hence dimensional balance in the interaction term gives

$$[G_F^{(d)}]_E E^{2(D-1)} \sim E^D \Rightarrow [G_F^{(d)}]_E = E^{2-D} = E^{1-d}.$$

To translate into SI units, we use energy

$$[E] = \text{kg m}^2 \text{s}^{-2}$$

as the mass-dimension carrier. A factor $M^n$ corresponds to $[E]^n$ in SI, so

$$[G_F^{(d)}] = [E]^{1-d} = (\text{kg m}^2 \text{s}^{-2})^{1-d} = \text{kg}^{1-d} \text{ m}^{2-2d} \text{s}^{2d-2}.$$
The weak carrier is defined in Appendix A as

$$S_W^{(d)} \equiv (G_F^{(d)})^{1/2},$$

so its SI dimensions are

$$[S_W^{(d)}] = [G_F^{(d)}]^{1/2} = \text{kg}^{\frac{1-d}{2}} \text{m}^{1-d} \text{s}^{d-1}.$$

For $d = 3$, this reduces to

$$[G_F^{(3)}] = \text{kg}^{-2} \text{m}^{-4} \text{s}^4, \quad [S_W^{(3)}] = \text{kg}^{-1} \text{m}^{-2} \text{s}^2,$$

which are precisely the dimensions used in the audit (and compatible with $M_W^{-2}$ seen as an energy-squared scale).

Therefore, once the weak carrier is fixed as $S_W = G_F^{1/2}$ in Appendix A, the same dimensionality pattern that matched the gravitational carrier for $d = 3$ is recovered from the weak sector. For $d \neq 3$, either $G_F^{(d)}$ or $S_W^{(d)}$ would acquire SI dimensions incompatible with the audited 3D values.

### B.5.5 Gauge, Yukawa and scalar couplings in general d

The carriers $S_G$ and $S_W$ already favour $d = 3$. An independent consistency layer comes from the renormalizability pattern of the effective QFT: in $D = 4$ spacetime dimensions (i.e. $d = 3$), gauge couplings, Yukawa couplings and the quartic scalar coupling are engineering-dimensionless (marginal) under power-counting. For $d \neq 3$ (equivalently $D \neq 4$), these couplings acquire nonzero engineering dimensions, so the Standard-Model-like “dimensionless-coupling” pattern is lost (super-renormalizable for $d < 3$, non-renormalizable for $d > 3$).

For power-counting in $D = d + 1$ spacetime dimensions we use the same engineering energy-dimension bookkeeping as in Appendix B.5.4: the dimensionless object is $S/\hbar$, and we count dimensions in powers of an energy scale $E$ (a bookkeeping device for renormalizability, not a unit convention). Canonical (engineering) dimensions are defined with canonically normalized kinetic terms; results are then expressed as energy-dimensions, and, if needed, translated to audited SI by identifying $E$ with joules and restoring $\hbar, c$ through explicit conversion mappings.

#### Gauge coupling

Consider a generic gauge interaction in $D = d + 1$ dimensions,

$$\frac{S_{\text{gauge}}^{(d)}}{\hbar} \sim \int d^D x \sqrt{-g} g^{(d)} \bar{\psi} \gamma^{\mu} A_{\mu} \psi.$$

With canonical kinetic terms, the engineering dimensions are fixed by the kinetic operators:

$$[\psi]_E = E^{\frac{D-1}{2}}, \quad [A_\mu]_E = E^{\frac{D-2}{2}}.$$

Therefore the operator $\bar{\psi}\gamma^\mu A_\mu\psi$ scales as

$$[\bar{\psi}\gamma^\mu A_\mu\psi]_E = E^{(D-1)} E^{\frac{D-2}{2}} = E^{\frac{3D-4}{2}}.$$

Since the Lagrangian density must scale as $E^D$ in $D$ dimensions, dimensional balance implies

$$[g^{(d)}]_E E^{\frac{3D-4}{2}} \sim E^D \Rightarrow [g^{(d)}]_E = E^{\frac{4-D}{2}} = E^{\frac{3-d}{2}}.$$

Thus gauge couplings are engineering-dimensionless only if $D = 4$ (i.e. $d = 3$). For $d < 3$ they are super-renormalizable (positive energy dimension), while for $d > 3$ they are non-renormalizable (negative energy dimension).
## Yukawa couplings

For a Yukawa term

$$S_Y^{(d)} \sim \int d^D x \sqrt{-g} y^{(d)} \phi \bar{\psi} \psi,$$

with scalar field dimension $[\phi] = M^{\frac{D-2}{2}}$, dimensional balance gives

$$[y^{(d)}]_{\text{(mass)}} = \frac{4-D}{2} = \frac{3-d}{2},$$

so

$$[y^{(d)}]_{\text{(mass)}} = M^{\frac{3-d}{2}}.$$

Again, Yukawa couplings are dimensionless only for $d = 3$. **Quartic scalar coupling**

For a quartic scalar interaction

$$S_{\lambda}^{(d)} \sim \int d^D x \sqrt{-g} \lambda^{(d)} \phi^4,$$

one find

$$[\lambda^{(d)}]_{\text{(mass)}} = 4 - D = 3 - d,$$

so that

$$[\lambda^{(d)}]_{\text{(mass)}} = M^{3-d}.$$

Once again, $\lambda^{(d)}$ is dimensionless only when $d = 3$.

Taken together, these results show that an effective QFT with the *Standard-Model-like renormalizability pattern* (dimensionless gauge, Yukawa and quartic scalar couplings) is naturally realised in a unique spatial dimensionality $d = 3$.

### B.5.6 Joint carrier consistency and selection of d = 3

We can now state the dimensional selection result informally as a “mini-theorem”.

#### Dimensional selection statement

Consider an RTQ ensemble whose low-energy carriers satisfy the audited dimensional constraints of Appendix A, with

$$S_G = c^3 \hbar^{-1} \langle \rho \rangle^{-1}, \quad S_W = G_F^{1/2},$$

and whose effective continuum limit is described by a local QFT in $D = d + 1$ dimensions. Assume further that:

1. the gravitational interaction is described at large scales by a $d$-dimensional Newton law with constant $G_d$;

2. the effective QFT has SM-like renormalizable couplings, i.e. gauge, Yukawa and quartic scalar couplings are dimensionless in mass units.

Then the only spatial dimensionality compatible with both the carriers and the QFT renormalizability pattern is $d = 3$.

The argument is straightforward:

1. From Appendix B.5.2, demanding $[S_G] = [G_d]$ with $[\langle \rho \rangle] = m^{-2}$ forces $d = 3$.
2. From Appendix B.5.5, the same $d = 3$ is required for the weak carrier $S_W^{(d)} = (G_F^{(d)})^{1/2}$ to match the audited SI dimensions of Appendix A.

3. From Appendix B.5.6, gauge, Yukawa and quartic scalar couplings are dimensionless (and hence exhibit the Standard-Model renormalizability pattern) only for $d = 3$.

Thus the constants graph and carriers (Appendix A) and the QFT renormalizability requirement converge on the same effective spatial dimensionality $d = 3$. When this is combined with the RTQ spectral analysis:

* the diffusion-based spectral dimension on the relevant scales satisfies

$$d_s \approx 3,$$

* the dimensional audit selects $d = 3$,

so that in the low-energy regime we have

$$d_s = d = 3.$$

This closes the dimensional loop:

$$\text{RTQ substrate} \Rightarrow d_s \approx 3 \Rightarrow d = 3 \Rightarrow \text{4D effective QFT}$$

with carriers consistent with the observed constants.

The continuum action used in Section 4 is therefore not an arbitrary 4D ansatz, but the effective description selected jointly by the RTQ spectral structure and by the dimensional constraints of the audited carriers.

# B.6 Continuum Limit and Effective 4D QFT

## B.6.1 Emergent manifold and local fields

Once the RTQ ensemble exhibits a spectral plateau with

$$d_s = 3,$$

and the dimensional audit of Appendix B.5 fixes the spatial dimensionality to

$$d = 3,$$

we can treat suitable coarse-grained regions of the RTQ hypergraph as coordinate patches of an emergent 3 + 1-dimensional Lorentzian manifold $M$ with metric $g_{\mu\nu}$.

Let $\downarrow(\tau)$ denote the physical coarse-graining scale associated with a diffusion time $\tau$ in the plateau regime. For each patch $U \subset \mathcal{H}$ whose diameter is large compared to the microscopic link scale but small compared to cosmological scales, we introduce local coordinates $x^\mu$ and define coarse-grained fields as averages over RTQ configurations:

* scalar fields $\phi(x)$ as averages of RTQ occupation or alignment variables,

* tensor fields (e.g. $g_{\mu\nu}(x)$) as effective descriptors of propagation statistics for diffusion or wave packets,

* the complex tension field $\Lambda(x) = \Lambda^P(x) + i\Lambda^I(x)$ as in B.4, representing the coarse-grained RTQ tension.

In the continuum limit $\downarrow(\tau) \to \infty$ with $d_s = 3$ fixed, the discrete Laplacian $\Delta$ on the RTQ hypergraph induces a second-order differential operator on $M$, which, in the homogeneous limit, reduces to the usual Laplace-Beltrami operator. This justifies the use of local differential equations for the effective fields on $M$.

The $\Lambda$-framework therefore admits an effective description as a **4D continuum theory** on $(M, g_{\mu\nu})$, valid on scales where the spectral and dimensional analyses of Appendices B.3-B.5 hold.
### B.6.2 Effective action $S_{eff}$ in 3 + 1 dimensions

On this emergent manifold, the relevant degrees of freedom are:

* the metric $g_{\mu\nu}(x)$, capturing the coarse-grained response of the RTQ substrate to curvature-like deformations of $\Lambda^P$;

* the complex tension field $\Lambda(x) = \Lambda^P(x) + i\Lambda^I(x)$ and its angular projection $\Theta(x)$;

* Standard-Model-like matter fields collectively denoted by $\Psi(x)$.

The effective 4D action used in the main text can then be written schematically as

$$S_{eff} = \int d^4 x \sqrt{-g} [\mathcal{L}_g(g_{\mu\nu}, \Lambda^P) + \ell_{\Lambda}(\Lambda^P, \Lambda^I, \Theta) + \ell_{SM}(\Psi; g_{\mu\nu}, \Theta)],$$

where:

* $\mathcal{L}_g$ reduces to the Einstein-Hilbert form plus small $\Lambda$-induced corrections in the appropriate limit of weak curvature and slowly varying $\Lambda^P$;

* $\ell_{\Lambda}$ encodes the dynamics of the complex tension field, the conservation of $|\Lambda|^2$ and the coherence current $J^\mu = \Lambda^P \partial^\mu \Lambda^I$;

* $\ell_{SM}$ is a Standard-Model-like matter Lagrangian where the “constants” (such as $c$, $G$, $\alpha$, $\hbar$, $G_F$, etc.) are not independent parameters but **angular projections** of a single invariant

$$\Lambda_0 = |\Lambda| = \sqrt{(\Lambda^P)^2 + (\Lambda^I)^2},$$

through the angle $\Theta(x)$ and the carrier structure specified in Appendix A.

Schematically, one can write for each sector $X$ (EM, weak, strong, gravity):

$$S_X(\Theta) = S_X^{(0)} f_X(\Theta),$$

with $S_X$ the corresponding carrier and $f_X$ a dimensionless angular factor. The numerical pillar in the main text exploits precisely these relations to fit $\Theta_{struct}$ and related parameters from the audited constants.

In this sense, $S_{eff}$ is not an arbitrary 4D ansatz: its field content and couplings are inherited from the RTQ substrate via the spectral dimension, the dimensional audit and the constants graph.

### B.6.3 Relation to Sections 3–6 of the main paper

Appendix B is designed to close the formal chain from the RTQ substrate to the effective $\Lambda$-framework used throughout the main text. The mapping is as follows:

#### Section 3 (RTQ networks and $\Lambda$-tension field)

The RTQ hypergraph $\mathcal{H} = (V, E)$, the complex tension $\Lambda = \Lambda^P + i\Lambda^I$, and the conservation of $|\Lambda|^2$ correspond to the microscopic objects that, after coarse-graining and diffusion analysis (Appendices B.2–B.4), give rise to the emergent manifold $(M, g_{\mu\nu})$ and the continuum tension field $\Lambda(x)$.

#### Section 4 ($\Lambda$-dynamics, time and gravity)

The construction of the coherence current

$$J^{\mu} = \Lambda^{P} \partial^{\mu} \Lambda^{I}, \nabla_{\mu} J^{\mu} = 0,$$

and of the time-like vector

$$u^{\mu} = \frac{J^{\mu}}{\sqrt{-J_{\nu}J^{\nu}}}, d\tau = -u_{\mu}dx^{\mu},$$
is justified by the coarse-grained picture of Appendix B.4: proper time emerges as a parameter along ordered coherence flows on the RTQ ensemble once the spectral plateau and dimensional audit fix a 4D effective regime.

## Section 5 (constants graph and numerical pillar)

The dimensional analysis of carriers and couplings in Appendix B.5 underlies the constants graph used in Section 5. In particular, the fact that the same spatial dimensionality $d = 3$ is selected by

$$[S_G] = [G_d], \\ [S_W] = [G_F]^{1/2},$$

and the requirement of dimensionless gauge/Yukawa/scalar couplings, is the mathematical backbone of the “numerical pillar” discussed there.

## Section 6 (cosmological and laboratory sectors)

The 4D effective action $S_{eff}$ of Appendix B.6.2 is the starting point for the homogeneous and quasi-homogeneous solutions used in the cosmological pillar (environment-dependent $H_0$), as well as for the linearised, high-coherence limits used in the laboratory pillar (gravito-inertial responses in coherent media). The $\Lambda$-mapping between curvature, coherence and emergent time is precisely what allows Secs. 6.1–Sec. 6.3 to treat these regimes within a unified framework.

In summary, Appendix B provides the formal pipeline

$$\text{RTQ hypergraph} \implies d_s \simeq 3 \implies d = 3 \implies S_{eff}[g_{\mu\nu}, \Lambda, \Psi]$$

that is assumed (but not re-derived) in Secs. 3–6. It shows that the effective 3 + 1-dimensional $\Lambda$-framework used in the main paper is not postulated ad hoc, but is **selected** jointly by the RTQ substrate, the spectral dimension, and the audited dimensional structure of the fundamental carriers.
# C Cosmological pillar: $H_0$-Dipole Anisotropy Pipeline and Robustness Tests

This appendix documents the internal reduction test used in Sec. 6.2 to quantify low-$\ell$ structure in the local expansion field $H_0(\mathbf{x})$ and to isolate the “Anti-Great-Attractor shell” discussed in Sec. 6.2.4. The goal is not to propose a definitive reconstruction of local cosmography, but to provide an end-to-end, reproducible specification of the pipeline (inputs, cuts, covariance choices, templates, and null tests), so that the phenomenological claims reported in Sec. 6.2 can be checked independently.

Operationally, Appendix C answers three questions:

## 1. What exactly is fitted?

We specify the data selection, redshift cuts and distance moduli used for the low-$z$ Pantheon+SH0ES-like sample, the baseline FRW background assumed for the monopole, and the two families of anisotropy templates considered:

* a texture-like mode TEX (constant-in-$z$ or shell-like modulation of $\mu$ at fixed axis), and

* a kinematic bulk-flow mode KIN (Doppler template with $\propto 1/z$ scaling).

We also define the locked-axis TEX/KIN tomography used to partition the global dipole into “texture” and “kinematic” components on a fixed sky direction.

## 2. How are amplitudes, axes and significances estimated?

We describe the progression from simple diagonal-error fits (MUERR-only $\chi^2$ minimization) to generalized least-squares (GLS) fits using the full Pantheon+SH0ES covariance, both for the global low-$z$ dipole and for individual redshift shells. We specify:

* the linear models used for the monopole-only and monopole+dipole cases;

* the mapping between dipole amplitude in distance modulus and effective $\delta H_0/H_0$;

* the estimators for best-fit axis $(l, b)$ and their uncertainties;

* the definition of $\Delta\chi^2$ and how it is converted into an empirical significance using footprint-aware nulls.

## 3. How robust is the Anti-GA shell and the TEX/KIN partition?

We detail the battery of robustness tests underlying the summary given in Sec. 6.2.4:

* jackknife resampling over parent surveys, sky regions, and dust/latitude cuts;

* footprint-aware permutation tests that scramble distances at fixed sky positions and redshift bins;

* Gaussian null mocks with the same angular mask and redshift distribution;

* comparisons between TEX-only, KIN-only and TEX+KIN models under a locked-axis hypothesis.

These tests justify the statements that (i) a coherent low-$\ell$ mode is present at the few-percent level in $H_0(\mathbf{x})$, (ii) a shell-like enhancement peaks around $z \simeq 0.027$ with $|\delta H_0/H_0| \simeq 5\%$, and (iii) a purely smooth bulk-flow explanation is disfavoured relative to a TEX+KIN partition aligned with the predicted Anti-Great-Attractor direction.

The structure of the appendix is as follows. C.1 specifies the dataset, redshift selection and baseline isotropic fit. C.2 defines the dipole parameterizations and the TEX/KIN templates. C.3 introduces the estimators and the GLS implementation, including the treatment of the covariance. C.4 presents the global low-$z$ dipole fit and its footprint-aware significance. C.5 develops the locked-axis TEX/KIN tomography and the identification of the Anti-GA shell. C.6 collects robustness tests (jackknifes, permutations, Gaussian mocks). C.7 discusses limitations, catalog-dependence and look-elsewhere effects, and summarizes how these caveats propagate into the cosmological pillar and the MVT fork of Secs. 6-7.

Throughout, the emphasis is on reproducibility and falsifiability rather than on optimizing cosmological constraints. The pipeline is deliberately conservative: it uses a single public SN compilation, standard FRW backgrounds, and minimal anisotropy templates, with significance always evaluated against nulls that preserve
the actual survey footprint. This ensures that the “Anti-GA shell” and the associated TEX-like component
can be treated, within the Λ-framework, as a clearly defined internal benchmark: future, more homogeneous
low-z surveys may either sharpen, weaken, or erase this pattern, directly feeding the falsification fork described
in Sec. 6.2.4 and Sec. 7.3.1.

C.1 Code, data and reproducibility

# --- Initialization (PS) ---
$script = ".\h0_dipole_test_v6.py"
$dat = ".\DataRelease\Pantheon_Data\4_DISTANCES_AND_COVAR\Pantheon+SHOES.dat"
$covFull = ".\DataRelease\Pantheon_Data\4_DISTANCES_AND_COVAR\Pantheon+SHOES_STAT+SYS.cov"
$covStat = ".\DataRelease\Pantheon_Data\4_DISTANCES_AND_COVAR\Pantheon+SHOES_STATONLY.cov"

python -V
python -c "import numpy as np; print('numpy', np.__version__)"
python -c "import scipy; print('scipy', scipy.__version__)"

Get-FileHash $script -Algorithm SHA256
Get-FileHash $dat -Algorithm SHA256
Get-FileHash $covFull -Algorithm SHA256
Get-FileHash $covStat -Algorithm SHA256

$env:PYTHONUTF8 = "1"
$env:PYTHONIOENCODING = "utf-8"
chcp 65001 | Out-Null
[Console]::InputEncoding = [System.Text Encoding]::UTF8
[Console]::OutputEncoding = [System.Text Encoding]::UTF8

C.2 Baseline H₀-dipole detection and redshift-depth scan

C.2.1 Low-z H₀-dipole vs depth: full sample (zmax scan)

python $script @base --scan "0.05,0.08,0.10,0.12,0.15,0.20"

=== zmax scan (dipole-only test: Δχ² = χ²(mono) - χ²(mono+dip), dof depends on --fix_*axis) ===
zmax N H0 dChi2.tex p.tex sig.tex dChi2.kin p.kin sig.kin dChi2_mix A.tex A.kin(1/z) v_bulk(km/s) l.tex b.tex l.kin b.kin
0.050 645 72.99 4.17 2.440e-01 0.69 6.67 8.309e-02 1.38 12.09 0.036019 0.000878 121.2 134.23 -11.98 125.20 15.37
0.080 727 73.05 4.49 2.128e-01 0.80 6.77 7.946e-02 1.41 11.93 0.035463 0.000879 121.3 142.87 -10.61 127.46 14.87
0.100 741 73.04 4.45 2.166e-01 0.78 6.73 8.093e-02 1.40 11.87 0.035003 0.000876 120.9 144.46 -10.27 127.66 14.79
0.120 764 73.05 4.02 2.593e-01 0.65 6.79 7.901e-02 1.41 10.86 0.034939 0.000880 121.5 146.44 -5.18 128.03 15.35
0.150 826 73.09 3.22 3.585e-01 0.36 6.83 7.751e-02 1.42 9.89 0.030655 0.000879 121.4 139.06 -5.19 127.21 15.31
0.200 948 73.34 2.57 4.627e-01 0.09 7.94 4.720e-02 1.67 8.67 0.030859 0.000948 130.8 133.76 11.32 128.55 17.49

C.2.2 Low-z H₀-dipole vs depth: high-latitude, low-dust subset

python $script @base --scan "0.05,0.08,0.10,0.12,0.15" --bcut 20 --dustcut 0.10

=== zmax scan (dipole-only test: Δχ² = χ²(mono) - χ²(mono+dip), dof depends on --fix_*axis) ===
zmax N H0 dChi2.tex p.tex sig.tex dChi2.kin p.kin sig.kin dChi2_mix A.tex A.kin(1/z) v_bulk(km/s) l.tex b.tex l.kin b.kin
0.050 524 73.12 5.02 1.704e-01 0.95 7.76 5.125e-02 1.63 13.19 0.051884 0.001101 152.0 133.24 0.56 118.15 19.78
0.080 599 73.17 5.82 1.209e-01 1.17 7.89 4.826e-02 1.66 13.89 0.051586 0.001106 152.6 139.73 -0.09 120.55 19.04
0.100 613 73.15 5.75 1.243e-01 1.15 7.82 4.987e-02 1.65 13.82 0.050687 0.001100 151.9 141.03 0.03 120.80 18.94
0.120 636 73.16 5.44 1.426e-01 1.07 7.87 4.867e-02 1.66 12.87 0.050586 0.001102 152.1 142.58 3.15 121.14 19.34
0.150 695 73.21 4.28 2.329e-01 0.73 7.94 4.721e-02 1.67 10.92 0.044074 0.001101 152.1 132.73 3.61 119.64 19.27

Across the zmax scans with diagonal-only errors, both the full sample and the high-latitude, low-dust subset show only weak evidence for a dipole. The TEX component remains below the 1σ level (Δχ² TEX ≃ 3–6), while the KIN component reaches at most ~ 1.7σ (Δχ² KIN ≃ 7). Both TEX and KIN axes stay clustered around (l, b) ~ (135°–145°, 0°–20°), consistent with the larger-scale bulk-flow direction, and the inferred bulk-flow amplitude remains modest (vbulk ~ 120–150 km s⁻¹). These diagonal-error scans therefore suggest that the low-z dipole signal is weak and not strongly sensitive to the choice of outer redshift cut or to simple dust/latitude cuts, motivating the more conservative GLS + covariance analysis developed in Sec. C.5.

C.2.3 GLS low-z zHD dipole

Low-redshift dipole test on zHD, with full STAT+SYS covariance and within-survey permutations.

python -X utf8 $script --dat $dat --cov $covFull --use_cov --q0 -0.55 --sigint 0.07 --zmax 0.10 --zcol zHD --seed 0
--permute 2000 --permute_whitened --permute_withinron survey
### Test A: SN low-z dipole on Hubble residuals

Sample: N = 741 SN | z ∈ [0.0, 0.1]

Best-fit isotropic H0 (grid, q0=-0.55): 72.67 km/s/Mpc

Fitting method: GLS with full covariance matrix.

### Baseline (no dipole)

χ² Iso (H0-only FRW, isotropic): 729.75

χ² Mono (a0-only, mean residual): 729.75

### TEX: constant-in-z dipole on μ (texture-like mode)

χ² TEX (a0 + A TEX·n, const amp): 714.98

Δχ² TEX (MONO → TEX, 3 dof): 14.78

p-value TEX (χ², 3 dof): 2.019e-03

σ TEX (equiv.): 2.88σ (one-sided), 3.09σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

Δχ² total (4 dof): 14.78

p-value (χ², 4 dof): 5.189e-03

σ (equiv.): 2.56σ (one-sided), 2.80σ (two-sided)

### TEX dipole parameters (constant in z)

A TEX ≡ A mu (const): 0.060275 mag

|δH0/H0| ≈ (ln10/5)·A TEX = 2.776%

### TEX axis (equatorial): RA = 63.99 deg, Dec = 59.08 deg

### TEX axis (galactic, J2000): l = 146.65 deg, b = 5.98 deg

### Monopole offset a0 (mag): -1.157152e-04

### KIN: kinematic 1/z dipole on μ

χ² KIN (a0 + (K·n)/z): 667.74 Δχ² KIN = 62.01 (3 dof) p KIN = 2.188e-13 z KIN ≈ 7.24σ

A kin (1/z coeff): 0.001321 mag v bulk ≈ 182.3 km/s

### KIN axis (galactic, J2000): l = 167.77 deg, b = 25.74 deg

### MIX: TEX + KIN combined

Incremental Δχ² (add KIN | TEX): 63.19 (i.e. TEX → TEX+KIN)

Incremental Δχ² (add TEX | KIN): 15.96 (i.e. KIN → KIN+TEX)

[permute (within, whitened)] 100.00% (2000/2000) hits=3 max dchi2= 17.320 elapsed= 0.2s rate=8064.5/s

[permute (within, whitened)] 100.00% (2000/2000) hits=0 max dchi2= 26.717 elapsed= 0.2s rate=8097.4/s

[permute (within, whitened)] 100.00% (2000/2000) hits=0 max dchi2= 41.645 elapsed= 0.3s rate=7968.3/s

[permute (within, whitened)] 100.00% (2000/2000) hits=5 max dchi2= 28.600 elapsed= 0.3s rate=5970.2/s

[permute (within, whitened)] 100.00% (2000/2000) hits=0 max dchi2= 45.804 elapsed= 0.3s rate=7692.4/s

[Permutation, within-survey, whitened] Nperm=2000

TEX: hits= 3 p_emp≈1.99900050e-03 dchi2obs=14.776 max=17.320 gap=-2.545

KIN: hits= 0 p_emp≈4.99750125e-04 dchi2obs=62.009 max=26.717 gap=35.292

MIX: hits= 0 p_emp≈4.99750125e-04 dchi2obs=77.965 max=41.645 gap=36.320

top TEX dchi2: 17.3202, 15.8090, 15.8060, 14.5476, 14.0714, 13.7430, 13.6737, 13.6400

top KIN dchi2: 26.7168, 26.4188, 25.2189, 24.5681, 24.1612, 23.1606, 22.8712, 21.3753

top MIX dchi2: 41.6453, 37.8691, 35.5859, 33.2031, 31.2269, 30.9936, 30.4252, 29.6448

D|K: hits= 5 p_emp≈2.99850075e-03 dchi2obs=15.956 max=28.600 gap=-12.644

K|D: hits= 0 p_emp≈4.99750125e-04 dchi2obs=63.189 max=45.804 gap=17.385

An effective intrinsic-scatter term $\sigma_{int} = 0.07$ mag is added in quadrature to the diagonal of the covariance matrix in all GLS fits. This value is calibrated once on the full Pantheon+SH0ES low-redshift sample ($z \le 0.10$) using the full STAT+SYS covariance, by requiring that the baseline isotropic FRW model ($H_0$-only, no dipole) yields a reduced chi-square $\chi^2_{red} \simeq 1$. The same $\sigma_{int}$ is then held fixed in all subsequent commands. This configuration is used in Sec. 6.2.2.1.

Note: in v6 a single `-permute` call now triggers five independent permutation tests (TEX, KIN, MIX, D|K and K|D), so the log shows five separate `[permute(within, whitened)]` blocks instead of a single one as in v3.2.

## C.3 Survey-level robustness: leave-one-survey-out

### C.3.1 $H_0$-dipole jackknife by parent survey (full sample)

python $script @base --zmax 0.15 --jackknife --surveycol IDSURVEY

### Test A: SN low-z dipole on Hubble residuals

Sample: N = 826 SN | z ∈ [0.0, 0.15]

Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc

Fitting method: diagonal errors only (MUERR).
--- Baseline (no dipole) ---

χ² Iso (H0-only FRW, isotropic): 376.39

χ²Mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---

χ² TEX (a0 + A tex·n, const amp): 373.17

Δχ² TEX (MONO → TEX, 3 dof): 3.22

p-value TEX (χ², 3 dof): 3.585e-01

σ TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

Δχ²_total (4 dof): 3.22

p-value (χ², 4 dof): 5.212e-01

σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)

TEX dipole parameters (constant in z)

A tex ≡ A mu (const): 0.030655 mag

|δH0/H0| ≈ (ln10/5)·A tex = 1.412%

TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg

TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg

Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on μ ---

χ² KIN (a0 + (K·n)/z): 369.56 Δχ² KIN = 6.83 (3 dof) p KIN = 7.751e-02 z KIN ≈ 1.42σ

A kin (1/z coeff): 0.000879 mag v bulk ≈ 121.4 km/s

KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---

χ² MIX (a0 + TEX + KIN): 366.50 Δχ² MIX = 9.89 (6 dof vs MONO)

Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)

Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)

=== Jackknife by survey (dipole-only Δχ², 3 dof) ===

exclude N DeltaChi2 tex p tex A mu (mag) |dH0/H0| l tex b tex DeltaChi2 kin p kin v bulk (km/s) l kin b kin
1 758 4.16 2.451e-01 0.033039 1.521 142.19 -12.15 6.67 8.303e-02 120.2 127.34 14.72
5 737 2.29 5.152e-01 0.030489 1.404 125.63 2.47 5.83 1.200e-01 122.6 122.81 19.67
10 820 3.40 3.346e-01 0.031164 1.435 140.62 -6.15 6.79 7.890e-02 121.1 127.30 15.18
15 792 3.58 3.102e-01 0.033789 1.556 138.87 -2.26 6.98 7.269e-02 123.0 127.43 15.32
18 811 3.23 3.576e-01 0.030419 1.401 140.29 -6.76 6.43 9.233e-02 120.1 126.76 13.82
50 780 3.88 2.750e-01 0.035614 1.640 136.93 -4.48 7.73 5.196e-02 136.9 127.09 9.60
51 778 2.82 4.196e-01 0.029803 1.372 138.25 -3.66 5.59 1.332e-01 115.0 124.59 14.70
56 769 2.74 4.327e-01 0.025963 1.196 136.08 -13.12 5.15 1.612e-01 112.0 120.32 10.27
57 721 3.44 3.287e-01 0.034206 1.575 146.12 -3.51 7.13 6.801e-02 132.5 137.40 14.89
61 813 3.33 3.436e-01 0.029921 1.378 137.40 -9.02 6.44 9.188e-02 118.5 126.15 13.72
62 802 3.12 3.735e-01 0.032022 1.475 139.50 -1.33 7.55 5.639e-02 130.6 128.29 17.11
63 792 2.90 4.079e-01 0.028968 1.334 135.25 -7.23 6.24 1.005e-01 118.4 125.68 15.28
64 768 3.30 3.476e-01 0.035266 1.624 139.20 2.83 7.01 7.161e-02 126.6 130.77 19.54
65 788 3.30 3.478e-01 0.031554 1.453 139.87 -6.14 6.27 9.905e-02 117.7 127.42 15.51
66 814 3.33 3.435e-01 0.029624 1.364 141.64 -9.64 6.27 9.939e-02 116.6 127.74 13.30
150 647 1.06 7.858e-01 0.021104 0.972 147.13 -2.51 5.84 1.196e-01 118.3 126.95 22.75

With diagonal-only errors, both TEX and KIN dipoles drop to low significance (≈ 1.5σ), and the survey jackknife shows stable amplitudes and directions across all parent surveys. No single survey dominates the fit, indicating that the weak low-z $H_0$ dipole signal in this configuration is not driven by a particular dataset.

### C.3.2 $H_0$-dipole jackknife by parent survey (robust subset)

python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --jackknife --surveycol IDSURVEY

=== Test A: SN low-z dipole on Hubble residuals ===

Sample: N = 695 SN | z ∈ [0.0, 0.15]

Best-fit isotropic H0 (grid, q0=-0.55): 73.21 km/s/Mpc

Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---

χ² Iso (H0-only FRW, isotropic): 326.84

χ²Mono (a0-only, mean residual): 326.84

--- TEX: constant-in-z dipole on μ (texture-like mode) ---

χ² TEX (a0 + A tex·n, const amp): 322.56

Δχ² TEX (MONO → TEX, 3 dof): 4.28

p-value TEX (χ², 3 dof): 2.329e-01

σ TEX (equiv.): 0.73σ (one-sided), 1.19σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

Δχ²_total (4 dof): 4.28
p-value (χ², 4 dof): 3.696e-01
σ (equiv.): 0.33σ (one-sided), 0.90σ (two-sided)

TEX dipole parameters (constant in z)
A Tec ≡ A mu (const): 0.044074 mag
|δHO/HO| ≈ (ln10/5)·A Tec = 2.030%
TEX axis (equatorial): RA = 36.27 deg, Dec = 64.69 deg
TEX axis (galactic, J2000): 1 = 132.73 deg, b = 3.61 deg
Monopole offset a0 (mag): 6.494530e-05

--- KIN: kinematic 1/z dipole on µ ---
χ²_KIN (a0 + (K·n)/z): 318.89 Δχ²_KIN = 7.94 (3 dof) p_KIN = 4.721e-02 z_KIN ≈ 1.67σ
A_kin (1/z coeff): 0.001101 mag v_bulk ≈ 152.1 km/s
KIN axis (galactic, J2000): 1 = 119.64 deg, b = 19.27 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 315.92 Δχ²_MIX = 10.92 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.64 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 2.98 (i.e. KIN → KIN+TEX)

=== Jackknife by survey (dipole-only Δχ², 3 dof) ===
exclude N DeltaChi2 Tec p_Tex A mu (mag) |dHO/HO| 1_Tex b_Tex DeltaChi2 kin p_kin v_bulk (km/s) 1_kin b_kin
1 631 5.22 1.565e-01 0.047125 2.170 138.74 -1.48 7.75 5.158e-02 150.9 120.30 18.93
5 625 2.79 4.256e-01 0.038661 1.780 121.55 8.71 6.46 9.108e-02 141.1 114.61 23.54
10 689 4.44 2.179e-01 0.044765 2.061 134.19 3.00 7.89 4.828e-02 151.7 119.78 19.18
15 661 4.89 1.802e-01 0.049302 2.270 132.61 5.94 8.13 4.346e-02 154.3 119.84 19.27
18 685 4.32 2.290e-01 0.044153 2.033 134.01 2.49 7.53 5.673e-02 150.2 119.28 17.99
50 652 5.31 1.507e-01 0.052621 2.423 131.81 3.86 9.20 2.669e-02 186.3 120.45 14.79
51 657 3.50 3.202e-01 0.041222 1.898 132.76 5.12 6.25 9.996e-02 142.2 119.23 20.22
56 649 3.78 2.862e-01 0.040051 1.844 132.70 -0.54 6.49 8.996e-02 147.8 115.29 15.37
57 607 3.92 2.707e-01 0.045343 2.088 133.01 4.62 7.60 5.516e-02 158.3 125.03 19.46
61 683 4.19 2.418e-01 0.042425 1.954 131.01 0.78 7.27 6.373e-02 147.0 117.66 17.92
62 676 4.35 2.257e-01 0.046166 2.126 133.53 5.79 8.76 3.270e-02 162.8 121.72 20.09
63 665 4.06 2.546e-01 0.044030 2.028 131.82 2.97 7.36 6.116e-02 150.6 120.03 19.38
64 648 4.56 2.072e-01 0.049449 2.277 132.32 8.24 7.97 4.668e-02 153.7 122.36 22.36
65 662 4.24 2.369e-01 0.044590 2.053 132.68 2.85 7.30 6.302e-02 147.3 118.88 19.74
66 685 4.25 2.361e-01 0.042520 1.958 135.31 0.34 7.16 6.703e-02 146.3 121.05 17.45
150 550 1.81 6.134e-01 0.034182 1.574 132.85 7.12 7.35 6.166e-02 150.7 116.95 24.24

Even after the high-latitude / low-dust cuts (and using diagonal errors only), the TEX mode remains weak (~1σ) and the KIN mode only marginal (~1.7σ, v<sub>bulk</sub> ≈ 150 km s<sup>-1</sup>). The jackknife by survey shows very stable amplitudes and axes, with only a modest drop in TEX when removing the largest sample (ID 150), indicating that no single survey is driving the signal in this robust configuration.

## C.4 Footprint-aware permutation nulls (diagonal covariance)

### C.4.1 Global footprint-aware permutations (full sample, diagonal errors)

python $script @base --zmax 0.15 --permute 200000 --seed 0 --permute_whitened

=== Test A: SN low-z dipole on Hubble residuals ===

Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---

χ² _iso (H0-only FRW, isotropic): 376.39

χ² _mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on µ (texture-like mode) ---

χ² _TEX (a0 + A _tex·n, const amp): 373.17

Δχ² _TEX (MONO → TEX, 3 dof): 3.22

p-value _TEX (χ², 3 dof): 3.585e-01

σ _TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

Δχ² _total (4 dof): 3.22

p-value (χ², 4 dof): 5.212e-01

σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)

TEX dipole parameters (constant in z)

A _tex ≡ A mu (const): 0.030655 mag

|δHO/HO| ≈ (ln10/5)·A tex = 1.412%

TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg

TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg

Monopole offset a0 (mag): -1.188980e-05
--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 369.56 Δχ²_KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 366.50 Δχ²_MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)

[permute(global, whitened)] 100.00% (200000/200000) hits=13800 max_dchi2= 12.798 elapsed= 7.8s rate=25512.9/s
[permute(global, whitened)] 100.00% (200000/200000) hits=343 max_dchi2= 17.890 elapsed= 8.1s rate=24720.1/s
[permute(global, whitened)] 100.00% (200000/200000) hits=276 max_dchi2= 16.084 elapsed= 7.2s rate=27704.0/s
[permute(global, whitened)] 100.00% (200000/200000) hits=16208 max_dchi2= 13.406 elapsed= 7.8s rate=25705.6/s
[permute(global, whitened)] 100.00% (200000/200000) hits=406 max_dchi2= 12.928 elapsed= 7.7s rate=26075.2/s

[Permutation, global, whitened] Nperm=200000
TEX: hits= 13800 p_emp≈6.90046550e-02 dchi2Obs=3.223 max=12.798 gap=-9.575
KIN: hits= 343 p_emp≈1.71999140e-03 dchi2Obs=6.830 max=17.890 gap=-11.060
MIX: hits= 276 p_emp≈1.38499308e-03 dchi2Obs=9.888 max=16.084 gap=-6.196
top TEX dchi2: 12.7982, 11.7963, 11.6211, 11.1241, 11.0865, 10.7548, 10.4184, 10.4030
top KIN dchi2: 17.8903, 13.4098, 12.2640, 11.2330, 11.1842, 11.0038, 10.9067, 10.5773
top MIX dchi2: 16.0836, 15.6732, 15.6387, 15.1053, 14.8305, 14.8114, 14.7909, 14.3315
D|K: hits= 16208 p_emp≈8.10445948e-02 dchi2Obs=3.057 max=13.406 gap=-10.348
K|D: hits= 406 p_emp≈2.03498983e-03 dchi2Obs=6.664 max=12.928 gap=-6.263

With global, footprint-aware permutations on the full sample and diagonal errors only, the TEX mode is fully consistent with noise ($p_{\text{emp}} \approx 0.07$), while the KIN and MIX modes reach only the few-$10^{-3}$ level ($p_{\text{emp}} \sim 10^{-3}$). This represents at most a weak hint for a coherent dipole when inter-survey correlations are not explicitly modeled, and motivates the more conservative GLS analyses with full STAT+SYS covariance presented in Sec. C.5.

### C.4.2 Global footprint-aware permutations (robust subset, diagonal errors)

python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --permute 200000 --seed 0 --permute_whitened

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.21 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 326.84
χ²_mono (a0-only, mean residual): 326.84

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + ATex-n, const amp): 322.56
Δχ² TEX (MONO → TEX, 3 dof): 4.28
p-value TEX (χ², 3 dof): 2.329e-01
σ TEX (equiv.): 0.73σ (one-sided), 1.19σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 4.28
p-value (χ², 4 dof): 3.696e-01
σ (equiv.): 0.33σ (one-sided), 0.90σ (two-sided)

TEX dipole parameters (constant in z)
ATex ≡ Amu (const): 0.044074 mag
|δH0/H0| ≈ (ln10/5)·ATex = 2.030%
TEX axis (equatorial): RA = 36.27 deg, Dec = 64.69 deg
TEX axis (galactic, J2000): 1 = 132.73 deg, b = 3.61 deg
Monopole offset a0 (mag): 6.494530e-05

--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 318.89 Δχ²_KIN = 7.94 (3 dof) p_KIN = 4.721e-02 z_KIN ≈ 1.67σ
A_kin (1/z coeff): 0.001101 mag vBulk ≈ 152.1 km/s
KIN axis (galactic, J2000): 1 = 119.64 deg, b = 19.27 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 315.92 Δχ²_MIX = 10.92 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.64 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 2.98 (i.e. KIN → KIN+TEX)

[permute(global, whitened)] 100.00% (200000/200000) hits=5500 max_dchi2= 13.272 elapsed= 6.7s rate=29683.7/s
[permute(global, whitened)] 100.00% (200000/200000) hits=140 max_dchi2= 13.997 elapsed= 6.5s rate=30549.8/s
[permute(global, whitened)] 100.00% (200000/200000) hits=133 max_dchi2= 16.343 elapsed= 7.5s rate=26727.9/s
[permute(global, whitened)] 100.00% (200000/200000) hits=19368 max_dchi2= 12.591 elapsed= 7.4s rate=27166.9/s
[permute(global, whitened)] 100.00% (200000/200000) hits=540 max_dchi2= 11.730 elapsed= 7.0s rate=28465.6/s

[Permutation, global, whitened] Nperm=200000

TEX: hits= 5500 p_emp≈2.75048625e-02 dchi2Obs=4.278 max=13.272 gap=-8.994
KIN: hits= 140 p_emp≈7.04996475e-04 dchi2Obs=7.943 max=13.997 gap=-6.054
MIX: hits= 133 p_emp≈6.69996650e-04 dchi2Obs=10.918 max=16.343 gap=-5.425
top TEX dchi2: 13.2720, 11.9976, 11.2383, 10.8775, 10.6046, 10.2720, 10.1174, 9.9768
top KIN dchi2: 13.9968, 13.6622, 12.6901, 12.2219, 12.1901, 12.0530, 11.8758, 11.3638
top MIX dchi2: 16.3433, 15.9366, 14.8441, 14.8113, 14.7354, 14.5293, 14.2488, 14.1866
D|K: hits= 19368 p_emp≈9.68445158e-02 dchi2Obs=2.976 max=12.591 gap=-9.616
K|D: hits= 540 p_emp≈2.70498648e-03 dchi2Obs=6.640 max=11.730 gap=-5.090

Restricting to the high-latitude, low-dust subset slightly increases the apparent TEX and KIN signals (TEX $p_{\text{emp}} \approx 2.8 \times 10^{-2}$, KIN $p_{\text{emp}} \approx 7.0 \times 10^{-4}$, MIX $p_{\text{emp}} \approx 6.7 \times 10^{-4}$), but they still remain at the few-$10^{-3}$ level at best. Thus, even after basic Galactic cleaning, diagonal-error fits provide at most a marginal indication of a coherent dipole, reinforcing the need for the full-covariance GLS analysis of Sec. C.5.

### C.4.3 Within-survey footprint-aware permutations (robust subset)

python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --permute 200000 --seed 0 --permute_withinheliosurvey --surveycol IDSURVEY --permute_whitened

### Test A: SN low-z dipole on Hubble residuals

Sample: N = 695 SN | z ∈ [0.0, 0.15]

Best-fit isotropic H0 (grid, q0=-0.55): 73.21 km/s/Mpc

Fitting method: diagonal errors only (MUERR).

### Baseline (no dipole)

$\chi^2_{\text{iso}}$ (H0-only FRW, isotropic): 326.84

$\chi^2_{\text{mono}}$ (a0-only, mean residual): 326.84

### TEX: constant-in-z dipole on $\mu$ (texture-like mode)

$\chi^2_{\text{TEX}}$ (a0 + A\_tex\_n, const amp): 322.56

$\Delta\chi^2_{\text{TEX}}$ (MONO → TEX, 3 dof): 4.28

$p$-value\_TEX ($\chi^2$, 3 dof): 2.329e-01

$\sigma_{\text{TEX}}$ (equiv.): 0.73$\sigma$ (one-sided), 1.19$\sigma$ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

$\Delta\chi^2_{\text{total}}$ (4 dof): 4.28

$p$-value ($\chi^2$, 4 dof): 3.696e-01

$\sigma$ (equiv.): 0.33$\sigma$ (one-sided), 0.90$\sigma$ (two-sided)

### TEX dipole parameters (constant in z)

$A_{\text{tex}} \equiv A_{\text{mu}}$ (const): 0.044074 mag

$|\delta_{\text{HO/HO}}| \approx (\ln 0.5)/A_{\text{tex}} = 2.030\%$

TEX axis (equatorial): RA = 36.27 deg, Dec = 64.69 deg

TEX axis (galactic, J2000): $1 = 132.73$ deg, $b = 3.61$ deg

Monopole offset a0 (mag): 6.494530e-05

### KIN: kinematic 1/z dipole on $\mu$

$\chi^2_{\text{KIN}}$ (a0 + (K\_n)/z): 318.89 $\Delta\chi^2_{\text{KIN}} = 7.94$ (3 dof) $p_{\text{KIN}} = 4.721e-02$ $z_{\text{KIN}} \approx 1.67\sigma$

$A_{\text{kin}}$ (1/z coeff): 0.001101 mag $v_{\text{bulk}} \approx 152.1$ km/s

KIN axis (galactic, J2000): $1 = 119.64$ deg, $b = 19.27$ deg

### MIX: TEX + KIN combined

$\chi^2_{\text{MIX}}$ (a0 + TEX + KIN): 315.92 $\Delta\chi^2_{\text{MIX}} = 10.92$ (6 dof vs MONO)

Incremental $\Delta\chi^2$ (add KIN | TEX): 6.64 (i.e. TEX → TEX+KIN)

Incremental $\Delta\chi^2$ (add TEX | KIN): 2.98 (i.e. KIN → KIN+TEX)

[permute(within, whitened)] 100.00% (200000/200000) hits=3887 max_dchi2= 12.345 elapsed= 24.1s rate=8300.3/s

[permute(within, whitened)] 100.00% (200000/200000) hits=180 max_dchi2= 12.802 elapsed= 22.9s rate=8745.2/s

[permute(within, whitened)] 100.00% (200000/200000) hits=92 max_dchi2= 15.546 elapsed= 24.0s rate=8331.6/s

[permute(within, whitened)] 100.00% (200000/200000) hits=13312 max_dchi2= 11.600 elapsed= 23.6s rate=8466.0/s

[permute(within, whitened)] 100.00% (200000/200000) hits=495 max_dchi2= 14.976 elapsed= 24.6s rate=8145.9/s

[Permutation, within-survey, whitened] Nperm=200000

TEX: hits= 3887 p_emp≈1.94399028e-02 dchi2Obs=4.278 max=12.345 gap=-8.067

KIN: hits= 180 p_emp≈9.04995475e-04 dchi2Obs=7.943 max=12.802 gap=-4.860

MIX: hits= 92 p_emp≈4.69997675e-04 dchi2Obs=10.918 max=15.546 gap=-4.628

top TEX dchi2: 12.3449, 11.4138, 10.8075, 10.3551, 10.3480, 10.0117, 9.9680, 9.9069

top KIN dchi2: 12.8024, 12.6452, 11.6470, 11.4979, 11.4498, 11.2395, 11.1378, 10.8698

top MIX dchi2: 15.5458, 15.3983, 14.9906, 14.5557, 14.5551, 14.0518, 13.6717, 13.2958

D|K: hits= 13312 p_emp≈6.65646672e-02 dchi2Obs=2.976 max=11.600 gap=-8.625

K|D: hits= 495 p_emp≈2.47998760e-03 dchi2Obs=6.640 max=14.976 gap=-8.336
When permutations are constrained within each survey footprint, the empirical p-values for TEX and KIN shift only mildly (TEX $p_{emp} \simeq 1.9 \times 10^{-2}$, KIN $p_{emp} \simeq 9.0 \times 10^{-4}$, MIX $p_{emp} \simeq 4.7 \times 10^{-4}$). The residual dipole signal therefore remains at the same “few-10$^{-3}$” level as in the global test, indicating that it is not an artefact of mixing surveys with different angular coverages, but it is still only marginal when using diagonal errors alone.

## C.5 GLS covariance tests and permutation nulls

### C.5.1 GLS $H_0$-dipole fit with full STAT+SYS covariance

```python
python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --cov $covFull --use_cov --h0min 71.5 --h0max 73.8 --h0n 401

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.00 km/s/Mpc
Fitting method: GLS with full covariance matrix.

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 855.00
χ² Mono (a0-only, mean residual): 855.00

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + A TEX n, const amp): 839.38
Δχ² TEX (MONO → TEX, 3 dof): 15.62
p-value TEX (χ², 3 dof): 1.355e-03

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ² total (4 dof): 15.62
p-value (χ², 4 dof): 3.569e-03
σ (equiv.): 2.69σ (one-sided), 2.91σ (two-sided)

TEX dipole parameters (constant in z)
A TEX ≡ A Mu (const): 0.064983 mag
|δH0/H0| ≈ (ln10/5)·A TEX = 2.993%
TEX axis (equatorial): RA = 65.62 deg, Dec = 68.27 deg
TEX axis (galactic, J2000): l = 140.54 deg, b = 12.95 deg
Monopole offset a0 (mag): -7.811418e-06

--- KIN: kinematic 1/z dipole on μ ---
χ² KIN (a0 + (K·n)/z): 779.19 Δχ² KIN = 75.81 (3 dof) p KIN = 2.429e-16 z KIN ≈ 8.12σ
A kin (1/z coeff): 0.001660 mag v bulk ≈ 229.2 km/s
KIN axis (galactic, J2000): l = 174.20 deg, b = 22.43 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 762.35 Δχ² MIX = 92.65 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 77.03 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 16.84 (i.e. KIN → KIN+TEX)
```

Using GLS with the full STAT+SYS covariance and the TEX/KIN/MIX decomposition, this test shows that the low-redshift Hubble residuals are strongly incompatible with a purely isotropic monopole.

A constant-in-z texture-like dipole (TEX) improves the fit by $\Delta\chi^2_{TEX} = 15.62$ for 3 additional degrees of freedom, corresponding to a $\sim 3\sigma$ detection of a $\simeq 3\%$ modulation in $H_0$:

$$A_{\text{tex}} = 0.064983 \text{ mag} \Rightarrow |\delta H_0/H_0| \simeq \frac{\ln 10}{5} A_{\text{tex}} \approx 2.99\%,$$

with an axis at $(l, b) \simeq (140.5^\circ, 13.0^\circ)$, close to the Anti-Great-Attractor direction.

The kinematic Doppler-like template (KIN), scaling as $1/z$, yields an even larger improvement, $\Delta\chi^2_{KIN} = 75.81$ for 3 dof ($\sim 8.1\sigma$), corresponding to a bulk-flow amplitude $v_{\text{bulk}} \simeq 230 \text{ km s}^{-1}$ toward $(l, b) \simeq (174.2^\circ, 22.4^\circ)$.

The combined MIX model (TEX + KIN) reduces the goodness-of-fit by $\Delta\chi^2_{MIX} = 92.65$ for 6 additional parameters with respect to the monopole-only model. The incremental gain when adding TEX on top of the best KIN solution, $\Delta\chi^2 = 16.84$ (3 dof), shows that a non-kinematic, approximately constant-in-z component remains after fitting the bulk flow. This TEX residual, aligned with the Anti-GA direction and at the level of a few per cent in $\delta H_0/H_0$, is the internal benchmark used in Sec. 6.2 for the “Anti-Great-Attractor shell” within the Λ-framework.
### C.5.2 GLS $H_0$-dipole fit with STATONLY covariance

```python
python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --cov $covStat --use Cov --h0min 71.5 --h0max 73.8 --h0n 401

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: GLS with full covariance matrix.

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 912.18
χ² Mono (a0-only, mean residual): 912.18

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + ATex·n, const amp): 901.65
Δχ² TEX (MONO → TEX, 3 dof): 10.53
p-value TEX (χ², 3 dof): 1.455e-02
σ TEX (equiv.): 2.18σ (one-sided), 2.44σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 10.53
p-value (χ², 4 dof): 3.238e-02
σ (equiv.): 1.85σ (one-sided), 2.14σ (two-sided)

TEX dipole parameters (constant in z)
A_Tex ≡ A Mu (const): 0.050067 mag
|δH0/H0| ≈ (ln10/5)·A_Tex = 2.306%
TEX axis (equatorial): RA = 70.27 deg, Dec = 64.66 deg
TEX axis (galactic, J2000): l = 144.56 deg, b = 11.94 deg
Monopole offset a0 (mag): 4.456056e-06

--- KIN: kinematic 1/z dipole on μ ---
χ² KIN (a0 + (K·n)/z): 843.23 Δχ² KIN = 68.95 (3 dof) p_KIN = 7.150e-15 z_KIN ≈ 7.69σ
A_Kin (1/z coeff): 0.001530 mag v_bulk ≈ 211.2 km/s
KIN axis (galactic, J2000): l = 178.87 deg, b = 24.79 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 823.53 Δχ² MIX = 88.66 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 78.12 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 19.70 (i.e. KIN → KIN+TEX)
```

In the TEX/KIN decomposition with STAT-only covariance, the dominant signal remains kinematic. The 1/z Doppler-like mode improves the fit by $\Delta\chi^2_{KIN} = 68.95$ for 3 additional degrees of freedom ($\simeq 7.7\sigma$), corresponding to a bulk flow of $v_{bulk} \simeq 210 \text{ km s}^{-1}$ toward $(l, b) \simeq (179^\circ, 25^\circ)$.

The constant-in-z TEX component alone is only moderately detected, $\Delta\chi^2_{TEX} = 10.53$ ($\sim 2.2\sigma$) with $|\delta H_0/H_0| \simeq 2.3\%$ and an axis at $(l, b) \simeq (145^\circ, 12^\circ)$, close to the Anti-GA direction. In the combined MIX fit, however, adding TEX on top of KIN still yields a non-negligible improvement ($\Delta\chi^2 = 19.70$, 3 dof), indicating a residual, non-kinematic low-$\ell$ component even when the bulk flow is modeled with STAT-only errors.

### C.5.3 GLS STAT+SYS with within-survey permutations

```python
python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --cov $covFull --use Cov --h0min 71.5 --h0max 73.8 --h0n 401
--permute 200000 --seed 0 --permute_withinron survey --surveycol IDSURVEY --permute_whitened

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.00 km/s/Mpc
Fitting method: GLS with full covariance matrix.

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 855.00
χ² Mono (a0-only, mean residual): 855.00

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + A Tex·n, const amp): 839.38
Δχ² TEX (MONO → TEX, 3 dof): 15.62
p-value TEX (χ², 3 dof): 1.355e-03
σ TEX (equiv.): 3.00σ (one-sided), 3.20σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 15.62
p-value (χ², 4 dof): 3.569e-03
σ (equiv.): 2.69σ (one-sided), 2.91σ (two-sided)
```
TEX dipole parameters (constant in z)
ATex ≡ A_mu (const): 0.064983 mag
|δHO/HO| ≈ (ln10/5)·A Tex = 2.993%
TEX axis (equatorial): RA = 65.62 deg, Dec = 68.27 deg
TEX axis (galactic, J2000): l = 140.54 deg, b = 12.95 deg
Monopole offset a0 (mag): -7.811418e-06
--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K-n)/z): 779.19 Δχ²_KIN = 75.81 (3 dof) p_KIN = 2.429e-16 z_KIN ≈ 8.12σ
A_kin (1/z coeff): 0.001660 mag v_bulk ≈ 229.2 km/s
KIN axis (galactic, J2000): l = 174.20 deg, b = 22.43 deg
--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 762.35 Δχ²_MIX = 92.65 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 77.03 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 16.84 (i.e. KIN → KIN+TEX)
[DEBUG] dchi2_D_given_K (QR) = 16.842042 vs fit: 16.842042
[DEBUG] dchi2_K_given_D (QR) = 77.029445 vs fit: 77.029445
[DEBUG] dchi2 Texobs (QR) = 15.623296 vs fit: 15.623296
[DEBUG] dchi2_kinobs (QR) = 75.810699 vs fit: 75.810699
[DEBUG] dchi2_mixobs (QR) = 92.652741 vs fit: 92.652741
[permute(within, whitened)] 100.00% (200000/200000) hits=586 max_dchi2= 35.635 elapsed= 23.4s rate=8544.9/s
[permute(within, whitened)] 100.00% (200000/200000) hits=0 max_dchi2= 72.677 elapsed= 23.8s rate=8398.8/s
[permute(within, whitened)] 100.00% (200000/200000) hits=0 max_dchi2= 80.110 elapsed= 23.1s rate=8658.4/s
[permute(within, whitened)] 100.00% (200000/200000) hits=788 max_dchi2= 43.643 elapsed= 23.7s rate=8440.0/s
[permute(within, whitened)] 100.00% (200000/200000) hits=1 max_dchi2= 78.516 elapsed= 23.9s rate=8383.5/s
[Permutation, within-survey, whitened] Nperm=200000
TEX: hits= 586 p_emp≈2.93498533e-03 dchi2obs=15.623 max=35.635 gap=-20.011
KIN: hits= 0 p_emp≈4.99997500e-06 dchi2obs=75.811 max=72.677 gap=3.134
MIX: hits= 0 p_emp≈4.99997500e-06 dchi2obs=92.653 max=80.110 gap=12.542
top TEX dchi2: 35.6348, 30.3691, 29.5568, 27.9792, 26.7991, 26.6741, 26.0624, 25.9841
top KIN dchi2: 72.6768, 72.2508, 66.7230, 64.7550, 61.6246, 61.3950, 60.8790, 60.5989
top MIX dchi2: 80.1104, 77.2947, 77.2835, 75.4791, 74.1884, 73.6229, 72.5527, 72.0218
D|K: hits= 788 p_emp≈3.94498028e-03 dchi2obs=16.842 max=43.643 gap=-26.801
K|D: hits= 1 p_emp≈9.99995000e-06 dchi2obs=77.029 max=78.516 gap=-1.487

Within-survey permutations with the full covariance matrix show TEX at $p_{\text{emp}} \simeq 3 \times 10^{-3}$, while KIN and MIX produce $\Delta\chi^2$ values that exceed all 200,000 null permutations ($p_{\text{emp}} < 5 \times 10^{-6}$). This rules out an isotropic Gaussian null given the real survey footprints and covariances, and confirms a highly significant bulk-flow-dominated anisotropy plus a subdominant TEX component.

### C.5.4 GLS STATONLY with within-survey permutations (robust subset)

python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --cov $covStat --use_cov --h0min 71.5 --h0max 73.8 --h0n 401
--permute 100000 --seed 0 --permute_withinron survey --surveycol IDSURVEY --permute_whitened

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: GLS with full covariance matrix.

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 912.18
χ² Mono (a0-only, mean residual): 912.18

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + ATex·n, const amp): 901.65
Δχ² TEX (MONO → TEX, 3 dof): 10.53
p-value TEX (χ², 3 dof): 1.455e-02
σ TEX (equiv.): 2.18σ (one-sided), 2.44σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ² total (4 dof): 10.53
p-value (χ², 4 dof): 3.238e-02
σ (equiv.): 1.85σ (one-sided), 2.14σ (two-sided)

TEX dipole parameters (constant in z)
A Tex ≡ A mu (const): 0.050067 mag
|δHO/HO| ≈ (ln10/5)·A Tex = 2.306%
TEX axis (equatorial): RA = 70.27 deg, Dec = 64.66 deg
TEX axis (galactic, J2000): l = 144.56 deg, b = 11.94 deg
Monopole offset a0 (mag): 4.456056e-06
--- KIN: kinematic 1/z dipole on μ ---
```
chi2_KIN (a0 + (K·n)/z): 843.23  Δχ²_KIN = 68.95 (3 dof)  p_KIN = 7.150e-15  z_KIN ≈ 7.69σ
A_kin (1/z coeff): 0.001530 mag   v_bulk ≈ 211.2 km/s
KIN axis (galactic, J2000): l = 178.87 deg, b = 24.79 deg
```

### --- MIX: TEX + KIN combined ---

```
χ²_MIX (a0 + TEX + KIN): 823.53  Δχ²_MIX = 88.66 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 78.12 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 19.70 (i.e. KIN → KIN+TEX)
```

```
[permute(within, whitened)] 100.00% (100000/100000) hits=4899 max_dchi2= 41.970 elapsed= 12.5s rate=7994.0/s
[permute(within, whitened)] 100.00% (100000/100000) hits=0 max_dchi2= 68.334 elapsed= 14.6s rate=6840.1/s
[permute(within, whitened)] 100.00% (100000/100000) hits=1 max_dchi2= 94.031 elapsed= 13.1s rate=7633.5/s
[permute(within, whitened)] 100.00% (100000/100000) hits=491 max_dchi2= 41.760 elapsed= 13.1s rate=7660.4/s
[permute(within, whitened)] 100.00% (100000/100000) hits=4 max_dchi2= 88.345 elapsed= 13.8s rate=7222.4/s
```

### [Permutation, within-survey, whitened] Nperm=100000

<table><tr><td>TEX: hits=</td><td>4899</td><td>p_emp≈4.89995100e-02</td><td>dchi2Obs=10.531</td><td>max=41.970</td><td>gap=-31.439</td></tr><tr><td>KIN: hits=</td><td>0</td><td>p_emp≈9.99990000e-06</td><td>dchi2Obs=68.954</td><td>max=68.334</td><td>gap=0.620</td></tr><tr><td>MIX: hits=</td><td>1</td><td>p_emp≈1.99998000e-05</td><td>dchi2Obs=88.656</td><td>max=94.031</td><td>gap=-5.375</td></tr><tr><td>top TEX dchi2:</td><td>41.9698, 32.2308, 31.7888, 30.0128, 29.6622, 29.3387, 28.2516, 28.0935</td><td></td><td></td><td></td><td></td></tr><tr><td>top KIN dchi2:</td><td>68.3339, 62.1783, 61.8148, 61.5636, 61.4616, 60.7882, 60.5763, 60.1075</td><td></td><td></td><td></td><td></td></tr><tr><td>top MIX dchi2:</td><td>94.0307, 85.5920, 82.3862, 78.2160, 78.1750, 75.9919, 74.6338, 72.9210</td><td></td><td></td><td></td><td></td></tr><tr><td>D|K: hits=</td><td>491</td><td>p_emp≈4.91995080e-03</td><td>dchi2Obs=19.702</td><td>max=41.760</td><td>gap=-22.058</td></tr><tr><td>K|D: hits=</td><td>4</td><td>p_emp≈4.99995000e-05</td><td>dchi2Obs=78.125</td><td>max=88.345</td><td>gap=-10.220</td></tr></table>

On the robust subset with STAT-only covariance, permutations again find only marginal support for TEX ($p_{\text{emp}} \approx 5 \times 10^{-2}$), while the KIN and MIX modes remain extremely unlikely under the null ($p_{\text{emp}} \approx 1 \times 10^{-5}$ and $2 \times 10^{-5}$, respectively). The observed $\Delta\chi^2_{\text{KIN}}$ slightly exceeds the permutation maximum, and $\Delta\chi^2_{\text{MIX}}$ lies very close to the upper tail of the null distribution. This demonstrates that the strong kinematic dipole is not driven by dusty / low-latitude SNe and persists even in a conservative, STAT-only configuration.

## C.6 Dust and Galactic-latitude nuisance tests

### C.6.1 Dust and latitude nuisance tests (MWEBV and $|b|$ cuts)

```
python $script @base --zmax 0.15 --dustcheck --dustcol MWEBV
```

### === Test A: SN low-z dipole on Hubble residuals ===

```
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).
```

### --- Baseline (no dipole) ---

```
χ²_iso (H0-only FRW, isotropic): 376.39
χ²_mono (a0-only, mean residual): 376.39
```

### --- TEX: constant-in-z dipole on μ (texture-like mode) ---

```
χ² TEX (a0 + ATex·n, const amp): 373.17
```

```
Δχ² TEX (MONO → TEX, 3 dof): 3.22
```

```
p-value TEX (χ², 3 dof): 3.585e-01
```

```
σ TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)
```

### [Alternative comparison: H0-only → (a0 + TEX dipole)]

```
Δχ²_total (4 dof): 3.22
```

```
p-value (χ², 4 dof): 5.212e-01
```

```
σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)
```

### TEX dipole parameters (constant in z)

```
A_Tex ≡ A_mu (const): 0.030655 mag
|δH0/H0| ≈ (ln10/5)·A_Tex = 1.412%
```

### TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg

### TEX axis (galactic, J2000): l = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

### --- KIN: kinematic 1/z dipole on μ ---

```
χ²_KIN (a0 + (K·n)/z): 369.56  Δχ²_KIN = 6.83 (3 dof)  p_KIN = 7.751e-02  z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag   v_bulk ≈ 121.4 km/s
KIN axis (galactic, J2000): l = 127.21 deg, b = 15.31 deg
```

### --- MIX: TEX + KIN combined ---

```
χ² MIX (a0 + TEX + KIN): 366.50  Δχ² MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
```

### === Dust correlation test ===
Dust column proxy: MWEBV
corr(dμ, dust): 0.0195
Δχ²_dust (add dust term, 1 dof): 3.65 p_dust = 5.607e-02 z_dust ≈ 1.59σ
Dust slope k (mag per unit dust): 3.812119e-01
Δχ²_dip/dust (add TEX dipole on top of dust, 3 dof): 3.00 p = 3.916e-01 z ≈ 0.28σ
A_tex(dip|dust): 0.032569 mag |δH0/H0|(dip|dust) ≈ 1.500%
TEX axis (dip|dust, galactic): 1 = 139.99 deg (ℓtex), b = 3.51 deg

On the full sample, the correlation between residuals and dust is very small (corr ≃ 0.02) and the dust term itself is only ~1.6σ. Once dust is included, the TEX dipole on top of dust remains weak (Δχ² ≃ 3.0, p ≃ 0.39), indicating that Galactic dust cannot explain the apparent TEX-like anisotropy

### C.6.2 Dust nuisance + |b| cut (b ≥ 20°)

python $script @base --zmax 0.15 --dustcheck --dustcol MWEBV --bcut 20

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 741 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.06 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 348.19
χ²_mono (a0-only, mean residual): 348.19

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ²跑TEX (a0 + A Text, const amp): 344.00
Δχ²跑TEX (MONO → TEX, 3 dof): 4.19
p-value跑TEX (χ², 3 dof): 2.420e-01
σ跑TEX (equiv.): 0.70σ (one-sided), 1.17σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 4.19
p-value (χ², 4 dof): 3.814e-01
σ (equiv.): 0.30σ (one-sided), 0.88σ (two-sided)

TEX dipole parameters (constant in z)
A跑tex ≡ A跑mu (const): 0.040271 mag
|δH0/H0| ≈ (ln10/5)·A跑tex = 1.855%
TEX axis (equatorial): RA = 37.48 deg, Dec = 61.08 deg
TEX axis (galactic, J2000): 1 = 134.55 deg, b = 0.45 deg
Monopole offset a0 (mag): 8.759357e-05

--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 340.88 Δχ²_KIN = 7.31 (3 dof) p_KIN = 6.269e-02 z_KIN ≈ 1.53σ
A跑kin (1/z coeff): 0.001020 mag v bulk ≈ 140.9 km/s
KIN axis (galactic, J2000): 1 = 125.96 deg, b = 15.88 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 338.26 Δχ²_MIX = 9.92 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 5.74 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 2.62 (i.e. KIN → KIN+TEX)

=== Dust correlation test ===
Dust column proxy: MWEBV
corr(dμ, dust): 0.0522
Δχ²_dust (add dust term, 1 dof): 6.41 p_dust = 1.137e-02 z_dust ≈ 2.28σ
Dust slope k (mag per unit dust): 6.279503e-01
Δχ²_dip/dust (add TEX dipole on top of dust, 3 dof): 4.82 p = 1.855e-01 z ≈ 0.89σ
A跑tex(dip|dust): 0.047544 mag |δH0/H0|(dip|dust) ≈ 2.189%
TEX axis (dip|dust, galactic): 1 = 140.12 deg (ℓtex), b = 10.71 deg

After imposing |b| ≥ 20°, the dust term becomes slightly more significant (≈ 2.3σ), but the TEX dipole added on top of dust is still ≤ 1σ and preserves a similar axis. Residual dust thus has a small but not dominant impact, and cannot generate the TEX-like pattern by itself.

### C.6.3 Dust nuisance + |b| cut + dust column cut (config robust)

python $script @base --zmax 0.15 --dustcheck --dustcol MWEBV --bcut 20 --dustcut 0.10

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 695 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.21 km/s/Mpc
Fitting method: diagonal errors only (MUERR).
--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 326.84
χ²_mono (a0-only, mean residual): 326.84

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ²ольTEX (a0 + A tex·n, const amp): 322.56
Δχ²ольTEX (MONO → TEX, 3 dof): 4.28
p-value оTEX (χ², 3 dof): 2.329e-01
σ оTEX (equiv.): 0.73σ (one-sided), 1.19σ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 4.28
p-value (χ², 4 dof): 3.696e-01
σ (equiv.): 0.33σ (one-sided), 0.90σ (two-sided)

TEX dipole parameters (constant in z)
A tex ≡ A mu (const): 0.044074 mag
|δH0/H0| ≈ (ln10/5)·A tex = 2.030%
TEX axis (equatorial): RA = 36.27 deg, Dec = 64.69 deg
TEX axis (galactic, J2000): 1 = 132.73 deg, b = 3.61 deg
Monopole offset a0 (mag): 6.494530e-05

--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 318.89 Δχ²_KIN = 7.94 (3 dof) p_KIN = 4.721e-02 z_KIN ≈ 1.67σ
A_kin (1/z coeff): 0.001101 mag vBulk ≈ 152.1 km/s
KIN axis (galactic, J2000): 1 = 119.64 deg, b = 19.27 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 315.92 Δχ²_MIX = 10.92 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.64 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 2.98 (i.e. KIN → KIN+TEX)

=== Dust correlation test ===
Dust column proxy: MWEBV
corr(dμ, dust): -0.0505
Δχ²_dust (add dust term, 1 dof): 0.24 p_dust = 6.248e-01 z_dust ≈ -0.32σ
Dust slope k (mag per unit dust): 1.795924e-01
Δχ²_dip|dust (add TEX dipole on top of dust, 3 dof): 4.35 p = 2.264e-01 z ≈ 0.75σ
A tex(dip|dust): 0.045675 mag |δH0/H0|(dip|dust) ≈ 2.103%
TEX axis (dip|dust, galactic): 1 = 135.34 deg (ℓtex), b = 6.13 deg

In the most conservative configuration (|b| ≥ 20°, MWEBV ≤ 0.10), the dust term becomes completely
negligible (Δχ²dust ≃ 0.24), while the TEX amplitude and axis remain essentially unchanged. This shows that,
once dusty and low-latitude lines of sight are removed, dust has no measurable influence on the inferred TEX
dipole.

C.7 Null-Gaussian mocks and permutation sanity checks

C.7.1 Global null-Gaussian mocks (diagonal, global permutations)

python $script @base --zmax 0.15 --null_gaussian --nmock 200 --permute 20000 --seed 0 --permute_whitened

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 376.39
χ²_mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ²ольTEX (a0 + A tex·n, const amp): 373.17
Δχ²ольTEX (MONO → TEX, 3 dof): 3.22
p-value оTEX (χ², 3 dof): 3.585e-01
σ оTEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)
[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 3.22
p-value (χ², 4 dof): 5.212e-01
σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)
TEX dipole parameters (constant in z)
A tex ≡ A mu (const): 0.030655 mag
|δH0/H0| ≈ (ln10/5)·A tex = 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): l = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 369.56 Δχ²_KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): l = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 366.50 Δχ² MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[Null Gaussian sanity check] nmock=200 nperm=20000 N=826
mock 20/200 done
mock 40/200 done
mock 60/200 done
mock 80/200 done
mock 100/200 done
mock 120/200 done
mock 140/200 done
mock 160/200 done
mock 180/200 done
mock 200/200 done
[Null Gaussian summary]
p_emp: mean=0.476 median=0.455 min=1.300e-03 max=9.981e-01
frac(p\textlessless0.05)=0.040 frac(p\textlessless0.01)=0.010
dchi2obs: mean=3.193 std=2.530

For global permutations applied to 200 null-Gaussian mocks, the empirical p-value distribution is close to uniform (⟨p⟩ ≃ 0.48, frac(p < 0.05) ≃ 0.04, frac(p < 0.01) ≃ 0.01) and the Δχ² distribution matches expectations. This validates that the permutation-based p-values are properly calibrated under an isotropic Gaussian null when using global shuffling.

### C.7.2 Null-Gaussian mocks: within-survey permutations

python $script @base --zmax 0.15 --null_gaussian --nmock 200 --permute 20000 --seed 0 --permute_whitened --permutewithinsurvey

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 376.39
χ² Mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + A tex·n, const amp): 373.17
Δχ² TEX (MONO → TEX, 3 dof): 3.22
p-value TEX (χ², 3 dof): 3.585e-01
σ TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)
[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ² total (4 dof): 3.22
p-value (χ², 4 dof): 5.212e-01
σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)
TEX dipole parameters (constant in z)
A tex ≡ A mu (const): 0.030655 mag
|ΔH0/H0| ≈ (ln10/5)·A tex = 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): l = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on μ ---
χ² KIN (a0 + (K·n)/z): 369.56 Δχ² KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): l = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 366.50 Δχ² MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[Null Gaussian sanity check] nmock=200 nperm=20000 N=826
mock 20/200 done
mock 40/200 done
mock 60/200 done
mock 80/200 done
mock 100/200 done
mock 120/200 done
mock 140/200 done
mock 160/200 done
mock 180/200 done
mock 200/200 done

[Null Gaussian summary]
p_emp: mean=0.478 median=0.442 min=9.500e-04 max=9.976e-01
frac(p\textlessless0.05)=0.035 frac(p\textlessless0.01)=0.015
dchi2obs: mean=3.193 std=2.530

Repeating the null-Gaussian test with within-survey permutations yields very similar calibration (⟨⟨p⟩⟩ ≃ 0.48, frac(p < 0.05) ≃ 0.035, frac(p < 0.01) ≃ 0.015). The footprint-aware permutation scheme thus behaves as an approximately unbiased estimator of p-values under the null, both globally and within surveys.

## C.8 Real-data permutation diagnostics and max-gap analysis

### C.8.1 Real-data permutation bound with 200k within-survey permutations

```perl
foreach ($s in 0..4) {
    python $script @base --zmax 0.15 --permute 50000 --seed $s --permute_whitened --permute_withinron survey
}

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ² Iso (H0-only FRW, isotropic): 376.39
χ² Mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + ATex, const amp): 373.17
Δχ² TEX (MONO → TEX, 3 dof): 3.22
p-value TEX (χ², 3 dof): 3.585e-01
σ-TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)
[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ² total (4 dof): 3.22
p-value (χ², 4 dof): 5.212e-01
σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)
TEX dipole parameters (constant in z)
A_Tex ≡ A Mu (const): 0.030655 mag
|δH0/H0| ≈ (ln10/5)·A_Tex ≈ 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): l = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on μ ---
χ² KIN (a0 + (K·n)/z): 369.56 Δχ² KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): l = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 366.50 Δχ² MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[permute(within, whitened)] 100.00% (50000/50000) hits=2937 max_dchi2= 11.841 elapsed= 7.1s rate=7068.5/s
[permute(within, whitened)] 100.00% (50000/50000) hits=114 max_dchi2= 10.037 elapsed= 7.3s rate=6876.0/s
[permute(within, whitened)] 100.00% (50000/50000) hits=54 max_dchi2= 14.812 elapsed= 6.8s rate=7328.8/s
[permute(within, whitened)] 100.00% (50000/50000) hits=3280 max_dchi2= 10.093 elapsed= 7.1s rate=7046.7/s
[permute(within, whitened)] 100.00% (50000/50000) hits=116 max_dchi2= 12.454 elapsed= 7.3s rate=6828.9/s

[Permutation, within-survey, whitened] Nperm=50000
TEX: hits= 2937 p_emp≈5.87588248e-02 dchi2obs=3.223 max=11.841 gap=-8.618
KIN: hits= 114 p_emp≈2.29995400e-03 dchi2obs=6.830 max=10.037 gap=-3.206
MIX: hits= 54 p_emp≈1.09997800e-03 dchi2obs=9.888 max=14.812 gap=-4.925
top TEX dchi2: 11.8412, 9.5715, 9.3921, 9.1870, 8.9376, 8.8687, 8.6557, 8.6488
top KIN dchi2: 10.0365, 10.0142, 9.5966, 9.5768, 9.5403, 9.3871, 9.3247, 9.2687
top MIX dchi2: 14.8122, 14.2072, 13.2956, 12.8676, 12.7777, 12.7283, 12.5909, 12.3978
D|K: hits= 3280 p_emp≈6.56186876e-02 dchi2obs=3.057 max=10.093 gap=-7.035
K|D: hits= 116 p_emp≈2.33995320e-03 dchi2obs=6.664 max=12.454 gap=-5.790
```
### Test A: SN low-z dipole on Hubble residuals

Sample: N = 826 SN | z ∈ [0.0, 0.15]

Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc

Fitting method: diagonal errors only (MUERR).

### Baseline (no dipole)

$\chi^2_{\text{iso}}$ (H0-only FRW, isotropic): 376.39

$\chi^2_{\text{mono}}$ (a0-only, mean residual): 376.39

### TEX: constant-in-z dipole on $\mu$ (texture-like mode)

$\chi^2_{\text{TEX}}$ (a0 + $A_{\text{tex}}$, const amp): 373.17

$\Delta\chi^2_{\text{TEX}}$ (MONO → TEX, 3 dof): 3.22

$p$-value_{\text{TEX}} ($\chi^2$, 3 dof): 3.585e-01

$\sigma_{\text{TEX}}$ (equiv.): 0.36$\sigma$ (one-sided), 0.92$\sigma$ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

$\Delta\chi^2_{\text{total}}$ (4 dof): 3.22

$p$-value ($\chi^2$, 4 dof): 5.212e-01

$\sigma$ (equiv.): -0.05$\sigma$ (one-sided), 0.64$\sigma$ (two-sided)

TEX dipole parameters (constant in z)

$A_{\text{tex}} \equiv A_{\text{mu}}$ (const): 0.030655 mag

$|\delta_{\text{HO}}| \approx (\ln 10/5) \cdot A_{\text{tex}} = 1.412\%$

TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg

TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg

Monopole offset a0 (mag): -1.188980e-05

### KIN: kinematic 1/z dipole on $\mu$

$\chi^2_{\text{KIN}}$ (a0 + (K·n)/z): 369.56 $\Delta\chi^2_{\text{KIN}} = 6.83$ (3 dof) $p_{\text{KIN}} = 7.751e-02$ $z_{\text{KIN}} \approx 1.42\sigma$

$A_{\text{kin}}$ (1/z coeff): 0.000879 mag $v_{\text{bulk}} \approx 121.4$ km/s

KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg

### MIX: TEX + KIN combined

$\chi^2_{\text{MIX}}$ (a0 + TEX + KIN): 366.50 $\Delta\chi^2_{\text{MIX}} = 9.89$ (6 dof vs MONO)

Incremental $\Delta\chi^2$ (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)

Incremental $\Delta\chi^2$ (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)

[permute(within, whitened)] 100.00\% (50000/50000) hits=2843 max_dchi2= 10.832 elapsed= 6.3s rate=7877.5/s

[permute(within, whitened)] 100.00\% (50000/50000) hits=108 max_dchi2= 11.960 elapsed= 6.5s rate=7677.2/s

[permute(within, whitened)] 100.00\% (50000/50000) hits=64 max_dchi2= 16.240 elapsed= 6.3s rate=7995.6/s

[permute(within, whitened)] 100.00\% (50000/50000) hits=3301 max_dchi2= 10.089 elapsed= 6.4s rate=7812.8/s

[permute(within, whitened)] 100.00\% (50000/50000) hits=109 max_dchi2= 11.439 elapsed= 6.7s rate=7450.9/s

[Permutation, within-survey, whitened] Nperm=50000

TEX: hits= 2843 p_emp≈5.68788624e-02 dchi2Obs=3.223 max=10.832 gap=-7.609

KIN: hits= 108 p_emp≈2.17995640e-03 dchi2Obs=6.830 max=11.960 gap=-5.129

MIX: hits= 64 p_emp≈1.29997400e-03 dchi2Obs=9.888 max=16.240 gap=-6.353

top TEX dchi2: 10.8322, 9.0572, 8.8309, 8.8155, 8.6363, 8.5557, 8.5383, 8.4640

top KIN dchi2: 11.9598, 11.3584, 10.1604, 10.0210, 9.8628, 9.4309, 9.3943, 9.3789

top MIX dchi2: 16.2402, 14.5159, 12.3430, 12.2067, 12.1940, 12.1367, 11.9833, 11.8822

D|K: hits= 3301 p_emp≈6.60386792e-02 dchi2Obs=3.057 max=10.089 gap=-7.032

K|D: hits= 109 p_emp≈2.19995600e-03 dchi2Obs=6.664 max=11.439 gap=-4.774

### Test A: SN low-z dipole on Hubble residuals

Sample: N = 826 SN | z ∈ [0.0, 0.15]

Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc

Fitting method: diagonal errors only (MUERR).

### Baseline (no dipole)

$\chi^2_{\text{iso}}$ (H0-only FRW, isotropic): 376.39

$\chi^2_{\text{mono}}$ (a0-only, mean residual): 376.39

### TEX: constant-in-z dipole on $\mu$ (texture-like mode)

$\chi^2_{\text{TEX}}$ (a0 + $A_{\text{tex}}$, const amp): 373.17

$\Delta\chi^2_{\text{TEX}}$ (MONO → TEX, 3 dof): 3.22

$p$-value_{\text{TEX}} ($\chi^2$, 3 dof): 3.585e-01

$\sigma_{\text{TEX}}$ (equiv.): 0.36$\sigma$ (one-sided), 0.92$\sigma$ (two-sided)

[Alternative comparison: H0-only → (a0 + TEX dipole)]

$\Delta\chi^2_{\text{total}}$ (4 dof): 3.22

$p$-value ($\chi^2$, 4 dof): 5.212e-01

$\sigma$ (equiv.): -0.05$\sigma$ (one-sided), 0.64$\sigma$ (two-sided)

TEX dipole parameters (constant in z)

$A_{\text{tex}} \equiv A_{\text{mu}}$ (const): 0.030655 mag

$|\delta_{\text{HO}}| \approx (\ln 10/5) \cdot A_{\text{tex}} = 1.412\%$

TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg

TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg

Monopole offset a0 (mag): -1.188980e-05

### KIN: kinematic 1/z dipole on $\mu$

$\chi^2_{\text{KIN}}$ (a0 + (K·n)/z): 369.56 $\Delta\chi^2_{\text{KIN}} = 6.83$ (3 dof) $p_{\text{KIN}} = 7.751e-02$ $z_{\text{KIN}} \approx 1.42\sigma$

$A_{\text{kin}}$ (1/z coeff): 0.000879 mag $v_{\text{bulk}} \approx 121.4$ km/s

KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg
--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 366.50 Δχ²_MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[permute(within, whitened)] 100.00% (50000/50000) hits=2759 max_dchi2= 10.618 elapsed= 6.5s rate=7702.6/s
[permute(within, whitened)] 100.00% (50000/50000) hits=104 max_dchi2= 11.751 elapsed= 6.6s rate=7582.6/s
[permute(within, whitened)] 100.00% (50000/50000) hits=58 max_dchi2= 14.410 elapsed= 6.6s rate=7579.4/s
[permute(within, whitened)] 100.00% (50000/50000) hits=3305 max_dchi2= 10.586 elapsed= 7.0s rate=7131.5/s
[permute(within, whitened)] 100.00% (50000/50000) hits=86 max_dchi2= 10.751 elapsed= 7.2s rate=6946.3/s

[Permutation, within-survey, whitened] Nperm=50000
TEX: hits= 2759 p_emp≈5.51988960e-02 dchi2obs=3.223 max=10.618 gap=-7.395
KIN: hits= 104 p_emp≈2.09995800e-03 dchi2obs=6.830 max=11.751 gap=-4.920
MIX: hits= 58 p_emp≈1.17997640e-03 dchi2obs=9.888 max=14.410 gap=-4.522
top TEX dchi2: 10.6178, 9.9877, 9.4332, 9.1297, 8.8760, 8.8071, 8.7671, 8.7401
top KIN dchi2: 11.7508, 11.7259, 11.1898, 10.4301, 10.3915, 10.3743, 9.9729, 9.7719
top MIX dchi2: 14.4099, 14.2630, 14.0177, 13.5662, 13.1306, 13.1222, 13.1113, 12.6842
D|K: hits= 3305 p_emp≈6.61186776e-02 dchi2obs=3.057 max=10.586 gap=-7.528
K|D: hits= 86 p_emp≈1.73996520e-03 dchi2obs=6.664 max=10.751 gap=-4.087
== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 376.39
χ²_mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
χ² TEX (a0 + ATex, const amp): 373.17
Δχ² TEX (MONO → TEX, 3 dof): 3.22
p-value TEX (χ², 3 dof): 3.585e-01
σ TEX (equiv.): 0.36σ (one-sided), 0.92σ (two-sided)
[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 3.22
p-value (χ², 4 dof): 5.212e-01
σ (equiv.): -0.05σ (one-sided), 0.64σ (two-sided)
TEX dipole parameters (constant in z)
A_Tex ≡ A_mu (const): 0.030655 mag
|δH0/H0| ≈ (ln10/5)·A_Tex = 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on μ ---
χ²_KIN (a0 + (K·n)/z): 369.56 Δχ²_KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag bulk ≈ 121.4 km/s
KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ² MIX (a0 + TEX + KIN): 366.50 Δχ² MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[permute(within, whitened)] 100.00% (50000/50000) hits=2844 max_dchi2= 10.419 elapsed= 6.9s rate=7296.1/s
[permute(within, whitened)] 100.00% (50000/50000) hits=86 max_dchi2= 10.341 elapsed= 7.0s rate=7154.8/s
[permute(within, whitened)] 100.00% (50000/50000) hits=70 max_dchi2= 14.925 elapsed= 6.9s rate=7261.2/s
[permute(within, whitened)] 100.00% (50000/50000) hits=3172 max_dchi2= 9.279 elapsed= 6.7s rate=7503.0/s
[permute(within, whitened)] 100.00% (50000/50000) hits=100 max_dchi2= 11.370 elapsed= 6.4s rate=7758.0/s

[Permutation, within-survey, whitened] Nperm=50000
TEX: hits= 2844 p_emp≈5.68988620e-02 dchi2obs=3.223 max=10.419 gap=-7.196
KIN: hits= 86 p_emp≈1.73996520e-03 dchi2obs=6.830 max=10.341 gap=-3.511
MIX: hits= 70 p_emp≈1.41997160e-03 dchi2obs=9.888 max=14.925 gap=-5.038
top TEX dchi2: 10.4190, 9.7235, 9.6092, 9.5081, 9.2927, 9.0129, 8.6711, 8.6571
top KIN dchi2: 10.3413, 9.8316, 9.2993, 9.1622, 8.9908, 8.9042, 8.8960, 8.8718
top MIX dchi2: 14.9252, 13.2578, 13.1967, 12.9158, 12.7699, 12.7612, 12.6227, 12.5930
D|K: hits= 3172 p_emp≈6.34587308e-02 dchi2obs=3.057 max=9.279 gap=-6.222
K|D: hits= 100 p_emp≈2.01995960e-03 dchi2obs=6.664 max=11.370 gap=-4.706

--- Test A: SN low-z dipole on Hubble residuals ---
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 376.39
χ²_mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on μ (texture-like mode) ---
```
chi2bars(a0 + A.tex-n, const amp): 373.17
Delta chi2bars (MONO -> TEX, 3 dof): 3.22
p-valuebars (chi2, 3 dof): 3.585e-01
sigma bars (equiv.): 0.36sigma (one-sided), 0.92sigma (two-sided)
[Alternative comparison: H0-only -> (a0 + TEX dipole)]
Delta chi2 total (4 dof): 3.22
p-value (chi2, 4 dof): 5.212e-01
sigma (equiv.): -0.05sigma (one-sided), 0.64sigma (two-sided)
TEX dipole parameters (constant in z)
A TEX == A mu (const): 0.030655 mag
|delta/H0| ≈ (ln10/5) * A TEX = 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05

--- KIN: kinematic 1/z dipole on mu ---
chi2_kin (a0 + (K*n)/z): 369.56 chi2_kin = 6.83 (3 dof) p_kin = 7.751e-02 z_kin ≈ 1.42sigma
A kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): 1 = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
chi2_mix (a0 + TEX + KIN): 366.50 chi2_mix = 9.89 (6 dof vs MONO)
Incremental chi2 (add KIN | TEX): 6.66 (i.e. TEX -> TEX+KIN)
Incremental chi2 (add TEX | KIN): 3.06 (i.e. KIN -> KIN+TEX)
[permute(within, whitened)] 100.00% (50000/50000) hits=2855 max dchi2= 12.550 elapsed= 6.7s rate=7518.6/s
[permute(within, whitened)] 100.00% (50000/50000) hits=115 max dchi2= 13.629 elapsed= 6.5s rate=7650.8/s
[permute(within, whitened)] 100.00% (50000/50000) hits=48 max dchi2= 14.958 elapsed= 6.9s rate=7297.1/s
[permute(within, whitened)] 100.00% (50000/50000) hits=3244 max dchi2= 11.430 elapsed= 6.7s rate=7517.3/s
[permute(within, whitened)] 100.00% (50000/50000) hits=132 max dchi2= 15.772 elapsed= 6.3s rate=7916.4/s

[Permutation, within-survey, whitened] Nperm=50000
TEX: hits= 2855 p_emp≈5.71188576e-02 dchi2Obs=3.223 max=12.550 gap=-9.326
KIN: hits= 115 p_emp≈2.31995360e-03 dchi2Obs=6.830 max=13.629 gap=-6.799
MIX: hits= 48 p_emp≈9.79980400e-04 dchi2Obs=9.888 max=14.958 gap=-5.070
top TEX dchi2: 12.5495, 10.3719, 9.5532, 9.4678, 9.4331, 9.1330, 8.9681, 8.6906
top KIN dchi2: 13.6291, 11.2838, 11.2766, 11.0874, 10.0673, 9.8936, 9.4045, 8.9017
top MIX dchi2: 14.9576, 13.5770, 13.0778, 12.9167, 12.3641, 12.2967, 12.2550, 12.1772
DK: hits= 3244 p_emp≈6.48987020e-02 dchi2Obs=3.057 max=11.430 gap=-8.373
KD: hits= 132 p_emp≈2.65994680e-03 dchi2Obs=6.664 max=15.772 gap=-9.107
```

On the real data with 200k within-survey permutations, TEX remains consistent with noise ($p_{\text{emp}} \approx 0.06$), while KIN and MIX lie in the few-$10^{-3}$ regime across repeated batches. The observed $\Delta\chi^2$ values stay comfortably below the permutation maxima, indicating a moderately significant bulk-flow-like signal in the diagonal-error analysis, but not the “forbidden-region” behavior found when using the full GLS covariance.

## C.9 Global significance summary and look-elsewhere penalty

### C.9.1 Summary of H₀-dipole amplitude, direction and significance

```
python $script @base --zmax 0.15 --permute 200000 --seed 0 --permute_whitened --permute_within_survey

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 826 SN | z ∈ [0.0, 0.15]
Best-fit isotropic H0 (grid, q0=-0.55): 73.09 km/s/Mpc
Fitting method: diagonal errors only (MUERR).

--- Baseline (no dipole) ---
chi2_iso (H0-only FRW, isotropic): 376.39
chi2_mono (a0-only, mean residual): 376.39

--- TEX: constant-in-z dipole on $\mu$ (texture-like mode) ---
chi2bars (a0 + A.tex-n, const amp): 373.17
Delta chi2bars (MONO -> TEX, 3 dof): 3.22
p-valuebars (chi2, 3 dof): 3.585e-01
sigma bars (equiv.): 0.36sigma (one-sided), 0.92sigma (two-sided)
[Alternative comparison: H0-only -> (a0 + TEX dipole)]
Delta chi2 total (4 dof): 3.22
p-value (chi2, 4 dof): 5.212e-01
sigma (equiv.): -0.05sigma (one-sided), 0.64sigma (two-sided)
TEX dipole parameters (constant in z)
A TEX == A mu (const): 0.030655 mag
|delta/H0| ≈ (ln10/5) * A TEX = 1.412%
TEX axis (equatorial): RA = 41.02 deg, Dec = 54.11 deg
TEX axis (galactic, J2000): 1 = 139.06 deg, b = -5.19 deg
Monopole offset a0 (mag): -1.188980e-05
```
--- KIN: kinematic 1/z dipole on µ ---
χ²_KIN (a0 + (K·n)/z): 369.56 Δχ²_KIN = 6.83 (3 dof) p_KIN = 7.751e-02 z_KIN ≈ 1.42σ
A_kin (1/z coeff): 0.000879 mag vBulk ≈ 121.4 km/s
KIN axis (galactic, J2000): l = 127.21 deg, b = 15.31 deg

--- MIX: TEX + KIN combined ---
χ²_MIX (a0 + TEX + KIN): 366.50 Δχ²_MIX = 9.89 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 6.66 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 3.06 (i.e. KIN → KIN+TEX)
[permute(within, whitened)] 100.00% (200000/200000) hits=11395 max_dchi2= 12.247 elapsed= 24.1s rate=8287.0/s
[permute(within, whitened)] 100.00% (200000/200000) hits=440 max_dchi2= 11.865 elapsed= 23.3s rate=8595.8/s
[permute(within, whitened)] 100.00% (200000/200000) hits=220 max_dchi2= 14.812 elapsed= 23.3s rate=8580.7/s
[permute(within, whitened)] 100.00% (200000/200000) hits=12940 max_dchi2= 11.243 elapsed= 23.7s rate=8455.1/s
[permute(within, whitened)] 100.00% (200000/200000) hits=445 max_dchi2= 12.930 elapsed= 23.7s rate=8452.4/s

[Permutation, within-survey, whitened] Nperm=200000
TEX: hits= 11395 p_emp≈5.69797151e-02 dchi2_obsv=3.223 max=12.247 gap=-9.024
KIN: hits= 440 p_emp≈2.20498898e-03 dchi2_obsv=6.830 max=11.865 gap=-5.035
MIX: hits= 220 p_emp≈1.10499448e-03 dchi2_obsv=9.888 max=14.812 gap=-4.925
top TEX dchi2: 12.2466, 11.8412, 11.1832, 10.8626, 10.7713, 10.7551, 10.5838, 10.3851
top KIN dchi2: 11.8654, 11.3859, 11.2729, 11.1447, 10.5660, 10.5014, 10.4068, 10.3266
top MIX dchi2: 14.8122, 14.2072, 13.8811, 13.7187, 13.5739, 13.3724, 13.3143, 13.3092
D|K: hits= 12940 p_emp≈6.47046765e-02 dchi2_obsv=3.057 max=11.243 gap=-8.185
K|D: hits= 445 p_emp≈2.22998885e-03 dchi2_obsv=6.664 max=12.930 gap=-6.265

The within-survey permutation summary for the full low-z sample with diagonal errors confirms the overall picture from earlier tests: the TEX dipole is consistent with noise (Δχ² ≃ 3.2, pEmp ≃ 5.7 × 10⁻², |δH₀/H₀| ≃ 1.4%), while the KIN and MIX modes reach the few-10⁻³ level (pEmp ≃ 2.2 × 10⁻³ and ≃ 1.1 × 10⁻³). The TEX axis and modest bulk-flow amplitude (vBulk ≃ 120 km/s) are stable across runs, but with diagonal errors alone the data favour a weak bulk-flow-like KIN component plus a noise-like TEX contribution.

## C.10 Tomographic texture analysis (TEX vs KIN)

This section documents the tomographic tests used in Sec. 6.2.2.4 to characterise the localized shell-like structure and to separate TEX (metric texture) from KIN (bulk flow).

### C.10.1 Axis Selection

python -X utf8 $script --dat $dat --cov $covFull --use_cov --q0 -0.55 --sigint 0.07 --zmax 0.10 --zcol zHD --seed 0
--permute 2000 --permute_whitened --permute_withinron survey

=== Test A: SN low-z dipole on Hubble residuals ===
Sample: N = 741 SN | z ∈ {{]0.0, 0.1{]}
Best-fit isotropic H0 (grid, q0=-0.55): 72.67 km/s/Mpc
Fitting method: GLS with full covariance matrix.
--- Baseline (no dipole) ---
χ²_iso (H0-only FRW, isotropic): 729.75
χ²Mono (a0-only, mean residual): 729.75
--- TEX: constant-in-z dipole on µ (texture-like mode) ---
χ²-TEX (a0 + A-tex-n, const amp): 714.98
Δχ²-TEX (MONO → TEX, 3 dof): 14.78
p-value-TEX (χ², 3 dof): 2.019e-03
σ-TEX (equiv.): 2.88σ (one-sided), 3.09σ (two-sided)
[Alternative comparison: H0-only → (a0 + TEX dipole)]
Δχ²_total (4 dof): 14.78
p-value (χ², 4 dof): 5.189e-03
σ (equiv.): 2.56σ (one-sided), 2.80σ (two-sided)
TEX dipole parameters (constant in z)
A-tex ≡ A-mu (const): 0.060275 mag
|δH0/H0| ≈ (ln10/5)·A-tex = 2.776%
TEX axis (equatorial): RA = 63.99 deg, Dec = 59.08 deg
TEX axis (galactic, J2000): l = 146.65 deg, b = 5.98 deg
Monopole offset a0 (mag): -1.157152e-04
--- KIN: kinematic 1/z dipole on µ ---
χ²-KIN (a0 + (K·n)/z): 667.74 Δχ²-KIN = 62.01 (3 dof) p-KIN = 2.188e-13 z-KIN ≈ 7.24σ
A_kin (1/z coeff): 0.001321 mag vBulk ≈ 182.3 km/s
KIN axis (galactic, J2000): l = 167.77 deg, b = 25.74 deg
--- MIX: TEX + KIN combined ---
χ²-MIX (a0 + TEX + KIN): 651.79 Δχ²-MIX = 77.96 (6 dof vs MONO)
Incremental Δχ² (add KIN | TEX): 63.19 (i.e. TEX → TEX+KIN)
Incremental Δχ² (add TEX | KIN): 15.96 (i.e. KIN → KIN+TEX)
[permute(within, whitened)] 100.00% (2000/2000) hits=3 max_dchi2= 17.320 elapsed= 0.3s rate=7812.5/s
[permute(within, whitened)] 100.00% (2000/2000) hits=0 max_dchi2= 26.717 elapsed= 0.3s rate=7781.7/s
[permute(within, whitened)] 100.00% (2000/2000) hits=0 max_dchi2= 41.645 elapsed= 0.2s rate=8031.6/s
[permute(within, whitened)] 100.00% (2000/2000) hits=5 max_dchi2= 28.600 elapsed= 0.3s rate=7813.1/s
[permute(within, whitened)] 100.00% (2000/2000) hits=0 max_dchi2= 45.804 elapsed= 0.3s rate=6042.3/s
[Permutation, within-survey, whitened] Nperm=2000
TEX: hits= 3 p_emp≈1.99900050e-03 dchi2Obs=14.776 max=17.320 gap=-2.545
KIN: hits= 0 p_emp≈4.99750125e-04 dchi2Obs=62.009 max=26.717 gap=35.292
MIX: hits= 0 p_emp≈4.99750125e-04 dchi2Obs=77.965 max=41.645 gap=36.320
top TEX dchi2: 17.3202, 15.8090, 15.8060, 14.5476, 14.0714, 13.7430, 13.6737, 13.6400
top KIN dchi2: 26.7168, 26.4188, 25.2189, 24.5681, 24.1612, 23.1606, 22.8712, 21.3753
top MIX dchi2: 41.6453, 37.8691, 35.5859, 33.2031, 31.2269, 30.9936, 30.4252, 29.6448
D|K: hits= 5 p_emp≈2.99850075e-03 dchi2Obs=15.956 max=28.600 gap=-12.644
K|D: hits= 0 p_emp≈4.99750125e-04 dchi2Obs=63.189 max=45.804 gap=17.385

This “axis selection” run performs the full GLS TEX/KIN decomposition on the low-z sample ($z \le 0.10$) with the STAT+SYS covariance, and is used to define the reference TEX and KIN directions. The fit yields a TEX improvement of $\Delta\chi^2 \approx 14.8$ (~3$\sigma$) and a stronger KIN improvement of $\Delta\chi^2 \approx 62$ (~7$\sigma$), with permutation tests showing that both signals lie well above what is expected from survey geometry or noise. The resulting TEX axis $(l, b) \simeq (146.65^\circ, 5.98^\circ)$ and KIN axis $(l, b) \simeq (167.77^\circ, 25.74^\circ)$ are then adopted for the locked-axis tomographic tests in Sec. 6.2.2.4; we checked that these directions and significances remain stable under reasonable changes of the intrinsic scatter parameter (e.g. $\sigma_{int} = 0.07-0.10$).

### C.10.2 Free-axis z-bin tomography across the shell (directional flip)

```bash
python -X utf8 $script --dat $dat --cov $covStat --use_cov --q0 -0.55 --sigint 0.07 --zcol zHD --zbins "0.020-0.025,0.025-0.030,0.030-0.035,0.035-0.040" --seed 0

== z bins (each bin fitted independently) ==
zbin N H0 dChi2 Tex A Tex l Tex b Tex dChi2 Kin v Bulk l Kin b Kin dChi2(D|K) p(D|K) dChi2(K|D) p(K|D) 0.020-0.025 114 74.00 3.38 0.060087 121.02 -15.62 3.39 192.5 122.72 -13.23 1.90 5.925e-01 1.92 5.898e-01 0.025-0.030 89 73.71 7.62 0.099469 165.02 -5.84 7.33 372.1 165.81 -5.33 2.76 4.302e-01 2.47 4.814e-01 0.030-0.035 76 72.53 0.54 0.028199 11.03 -9.34 0.52 128.6 4.93 -10.79 2.76 4.300e-01 2.74 4.335e-01 0.035-0.040 46 71.54 2.76 0.080427 325.61 -57.55 2.79 416.9 329.64 -57.28 3.89 2.732e-01 3.92 2.697e-01
```

Allowing the TEX and KIN dipole directions to vary independently in narrow z-bins around $z \simeq 0.027$ shows that the TEX axis undergoes an $\simeq 180^\circ$ flip when crossing the Anti-GA shell, while the KIN axis stays broadly aligned with the large-scale flow direction. This behavior is difficult to reconcile with a single smooth bulk flow, but is naturally interpreted as a localized texture-like modulation of the expansion rate superposed on a more global kinematic pattern.

### C.10.3 Locked-axis TEX/KIN tomography on the Anti-GA shell

```bash
python -X utf8 $script --dat $dat --cov $covStat --use_cov --q0 -0.55 --sigint 0.07 --zcol zHD --zbins "0.020-0.025,0.025-0.030,0.030-0.035,0.035-0.040" --zbins_global --seed 0 --fix Tex axis "146.65,5.98" --fix Kin axis "167.77,25.74"

== z bins (GLOBAL H0 + LOCKED axes; per-bin fits auditable) ==
# Union z in (0.020,0.040@d] N=325 H0_best=73.15 (q0=-0.55)
# Using full covariance. Note: per-bin C-submatrix $\chi^2$ are diagnostic (cross-bin cov breaks additivity).
zbin N H0glob dChi2 Tex Fit A Tex Fit dChi2 Kin Fit v Bulk Fit dChi2(D|Tex K) Fit p(D|Tex K) dChi2(K|Tex D) Fit p(K|Tex D) dChi2 Tex Fix dChi2 Kin Fix dChi2 Mix Fix 0.020-0.025 114 73.15 1.81 0.056580 0.02 17.8 3.03 8.179e-02 1.24 2.660e-01 2.65 0.00 4.12 0.025-0.030 89 73.15 6.47 0.107927 3.82 290.8 2.67 1.024e-01 0.02 8.999e-01 4.12 0.84 3.41 0.030-0.035 76 73.15 0.17 0.020399 0.41 128.5 0.00 9.931e-01 0.24 6.246e-01 -1.96 -0.35 -1.75 0.035-0.040 46 73.15 0.00 0.002640 0.89 317.8 0.95 3.287e-01 1.85 1.743e-01 0.03 -0.27 0.97
# Global (union) with locked axes:
# $\Delta\chi^2_{\text{tex}}=4.842 \quad \Delta\chi^2_{\text{kin}}=0.222 \quad \Delta\chi^2_{\text{mix}}=6.743$
# Sum over bins (fixed-beta diagnostic): $\Delta\chi^2_{\text{tex}}=4.842 \quad \Delta\chi^2_{\text{kin}}=0.222 \quad \Delta\chi^2_{\text{mix}}=6.743$
```

With the TEX and KIN axes locked to their GLS values, the bin-by-bin GLS fit across $0.020 < z \le 0.040$ shows a clear shell-like pattern in the TEX amplitude: TEX is modest below the shell, peaks in the $0.025-0.030$ bin ($A_{\mu}^{\text{tex}} \sim 0.11 \text{ mag}$, $\Delta\chi_{\text{tex}}^2 \sim 6.5$), and then rapidly declines. In contrast, the KIN template contributes only small $\Delta\chi^2$ in each bin, and on the union shell the TEX mode alone captures almost all of the improvement ($\Delta\chi_{\text{tex}}^2 \simeq 4.8$ vs $\Delta\chi_{\text{kin}}^2 \simeq 0.2$, $\Delta\chi_{\text{mix}}^2 \simeq 6.7$).

The per-bin diagnostics (columns $A_{\mu}^{\text{tex}}$, $\Delta\chi_{\text{tex}}^2$, $\Delta\chi_{\text{kin}}^2$, $\Delta\chi^2(D|K)_{\text{fit}}$, $\Delta\chi^2(K|D)_{\text{fit}}$) are used in Sec. 6.2.2.4 to show the rise-peak-drop pattern of the TEX mode and the dominance of TEX over KIN on the Anti-GA shell.

### Velocity-jump diagnostic for the kinematic interpretation
On the bulk-flow speeds inferred by the KIN template:

```
zbin vBulk_fit (km/s)
0.020-0.025 ≈ 18
0.025-0.030 ≈ 291
0.030-0.035 ≈ 129
0.035-0.040 ≈ 318 (low-N bin, diagnostic only)
```

Interpreting the anisotropy as a pure bulk flow would require $v_{\text{bulk}}(z)$ to jump from a few tens of $km s^{-1}$ below the shell to $\sim 3 \times 10^2 km s^{-1}$ in the 0.025–0.030 bin, before partially relaxing again. Such a sharp, localized velocity jump over $\sim 20–30$ Mpc is difficult to reconcile with a smooth kinematic bulk flow, but is naturally explained by a static texture-like modulation of the local expansion rate.

## Conditional TEX vs KIN preference on the shell

On the union $0.020 < z \le 0.040$, the locked-axis GLS fit yields:

```
Shell 0.020 \textless{}{} z \le 0.040 (locked axes, STATONLY covariance)
Global (union) scores:
Δχ²Tex(shell) ≈ 4.84
Δχ²Kin(shell) ≈ 0.22
Δχ²Mix(shell) ≈ 6.74
Conditional increments:
Δχ²(TEX | KIN) ≈ Δχ²_mix - Δχ²_kin ≈ 6.5
Δχ²(KIN | TEX) ≈ Δχ²_mix - Δχ²Tex ≈ 1.9
```

On the Anti-GA shell, the TEX template alone accounts for essentially all of the $\chi^2$ improvement, while a kinematic bulk-flow template alone contributes only a negligible $\Delta\chi^2$. Adding TEX on top of KIN improves the fit by $\sim 6.5$, whereas adding KIN on top of TEX improves it by only $\sim 1.9$. Once the texture-like modulation is included, the residual preference for an additional bulk-flow component on the shell is statistically weak, supporting a texture-dominated interpretation of the Anti-GA structure.
# D Derivations for Checks 1–7 (Negative checks)

## D.1 Check 1 – GW speed

Goal: Show that tensor perturbations of the tension field propagate on the same light cone as photons, with $v_{GW} = v_{\gamma} = c$ in the isotropic regime. Quantify how $\nabla\Lambda^P$ would generate deviations and demonstrate consistency with the GW170817 / GWTC bounds on $|v_{GW} - c|/c$.

### D.1.1 Effective medium and tension waves

At large scales the discrete network can be viewed as an effective continuum medium characterized by:

* a scalar tension field $\Lambda^P(x)$ (real part of the complex tension $\Lambda = \Lambda^P + i\Lambda^I$),

* a unit time-like coherence vector field $u^{\mu}(x)$ (coarse-grained $d_4$ direction, with $u_{\mu}u^{\mu} = -1$).

In the ordered phase, both the spin-2 (gravitational) and spin-1 (electromagnetic) excitations are phase waves of the same tension medium. To leading order they propagate on the same emergent metric $g_{\mu\nu}$ with a characteristic “tension wave” speed

$$v_{ph}^2 = \frac{\partial p}{\partial \rho},$$

where $p(\Lambda^P)$ and $\rho(\Lambda^P)$ are the effective pressure and energy density of the ordered tension field.

In the network, the luminal constant $c$ is defined operationally as the high-frequency limit of this phase speed in a homogeneous, isotropic background:

$$\left. \frac{\partial p}{\partial \rho} \right|_{\Lambda^P = \Lambda_0^P} = c^2.$$

This fixes the SI value of $c$ in terms of the microscopic stiffness of the network. Once this identification is made in the electromagnetic sector, any other small-amplitude excitation that samples the same quadratic kinetic operator must propagate at the same speed, up to corrections from inhomogeneities in $\Lambda^P$ or anisotropies in $u^{\mu}$.

### D.1.2 Tensor-mode EFT and dispersion relation

For the purposes of quantitative constraints, we embed the coarse-grained network in a conservative effective field theory of Einstein–Æther type, where $u^{\mu}$ is the macroscopic version of the coherence field. The gravitational sector is

$$\mathcal{L}_g = \frac{M_{Pl}^2}{2} R - \frac{M_u^2}{2} [c_1(\nabla_{\mu} u_{\nu})(\nabla^{\mu} u^{\nu}) + c_2(\nabla_{\mu} u^{\mu})^2 + c_3(\nabla_{\mu} u_{\nu})(\nabla^{\nu} u^{\mu}) + c_4 u_{\mu} u_{\nu} \nabla^{\mu} u_{\alpha} \nabla^{\nu} u^{\alpha}] + \lambda(u_{\mu} u^{\mu} + 1),$$

with $M_{Pl}$ the Planck mass, $M_u$ the coherence scale, $c_i$ dimensionless coefficients encoding the residual anisotropy of the network, and $\lambda$ enforcing $u_{\mu}u^{\mu} = -1$.

We linearise around a homogeneous, isotropic background

$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, u^{\mu} = \bar{u}^{\mu} + \delta u^{\mu}, \bar{u}^{\mu} = (1, 0, 0, 0),$$

and focus on transverse-traceless tensor perturbations $h_{ij}^{TT}$ (with $i, j = 1, 2, 3$). At quadratic order one obtains the standard Einstein–Æther tensor Lagrangian

$$\mathcal{L}_{GW}^{(2)} = \frac{M_{eff}^2}{8} [(\partial_t h_{ij}^{TT})^2 - c_T^2 (\nabla h_{ij}^{TT})^2],$$

With

$$M_{eff}^2 = M_{Pl}^2 (1 - c_{13}), \quad c_{13} \equiv c_1 + c_3, \quad c_T^2 = \frac{1}{1 - c_{13}},$$
$$ \frac{v_{GW}}{c} = c_T = (1 - c_{13})^{-1/2} \approx 1 + \frac{c_{13}}{2} + \mathcal{O}(c_{13}^2). $$

The network interpretation is straightforward:

* perfect statistical isotropy of links at large scales ⇒ $c_{13} = 0$;

* any residual anisotropy appears as a small effective $c_{13} \neq 0$ and shifts the GW cone away from the electromagnetic cone.

In the strictly isotropic limit $c_{13} = 0$ one has

$$ c_T^2 = 1, v_{GW} = c, $$

so that GWs and photons share the same light cone.

### D.1.3 GW170817 bound and $\Lambda$-network interpretation

The binary neutron-star merger GW170817, together with its gamma-ray counterpart, constrains the relative difference between GW and photon speeds to

$$ \left| \frac{v_{GW} - c}{c} \right| \lesssim 3 \times 10^{-15}. $$

In the Einstein–Æther parametrization this is customarily written as

$$ \left| \frac{v_{GW}}{c} - 1 \right| = \left| \frac{\alpha_T}{2} \right| \lesssim 3 \times 10^{-15} \Rightarrow \left| \alpha_T \right| \lesssim 6 \times 10^{-15}, $$

where $\alpha_T$ is a linear combination of the $c_i$ that reduces to $\alpha_T \approx c_{13}$ at leading order.

Imposing *Check 1* therefore amounts to the following EFT-level conditions:

* tensor speed constraint:

$$ c_{13} = c_1 + c_3 \approx 0, \text{ so that } v_{GW} = c_T c \approx c, $$

* any residual $\alpha_T$ generated radiatively must satisfy $|\alpha_T| \ll 10^{-15}$ today.

In the network language:

* the luminal constant $c$ is fixed by the tension response of $\Lambda^P$ in the ordered phase ($v_{ph}^2 = \partial p / \partial \rho = c^2$);

* spin-2 and spin-1 excitations probe the same quadratic tension operator, hence they share the same characteristic speed;

* deviations would require large-scale gradients or anisotropies in $\Lambda^P$ and $u^\mu$; these are already strongly constrained by the GW170817 limit.

As an internal consistency check, a direct 1-D lattice simulation of the linear wave equation

$$ \partial_t^2 h(x, t) = c^2 \partial_x^2 h(x, t), $$

with $c = 1$ in lattice units, yields a packet propagation speed

$$ v_{sim} \approx 0.996 c \left( \frac{|v_{sim} - c|}{c} \right) \approx 3.7 \times 10^{-3}, $$

with the shortfall entirely due to numerical dispersion (finite grid, CFL < 1, damping at the edges); refining the grid and scheme pushes $v_{sim}$ arbitrarily close to $c$.

Thus, at both the EFT and numerical levels, the ordered $\Lambda$-network predicts a single luminal cone shared by GWs and photons, consistent with GW170817.
## D.2 Check 2 – Lorentz invariance

Goal: Demonstrate that statistical isotropy of network connections yields an emergent Lorentz symmetry with a unique effective light cone. Map residual preferred-frame effects to SME coefficients and show they lie safely below current Lorentz-violation limits.

### D.2.1 Preferred-frame EFT for coherence field

The same Einstein–Æther–type EFT used in D.1 provides a convenient parameterization of possible Lorentz violations. The gravitational + coherence sector is

$$ \mathcal{L}_g = \frac{M_{Pl}^2}{2} R - \frac{M_u^2}{2} \sum_{i=1}^{4} c_i \mathcal{I}_i[u, g] + \lambda(u_\mu u^\mu + 1), $$

where the invariants $\mathcal{I}_i[u, g]$ are constructed from $\nabla_\mu u_\nu$ as in D.1, and $u^\mu$ encodes the macroscopic alignment of the network’s local arrows of time.

Matter and radiation feel this background through minimal couplings to $g_{\mu\nu}$ plus small preferred-frame deformations. For the photon sector we take

$$ \mathcal{L}_{\gamma} = -\frac{1}{4} \sqrt{-\tilde{g}} \tilde{g}^{\mu\rho} \tilde{g}^{\nu\sigma} F_{\mu\nu} F_{\rho\sigma}, \tilde{g}_{\mu\nu} = g_{\mu\nu} + D u_{\mu} u_{\nu}, $$

with D a small dimensionless parameter. This is equivalent to a non-birefringent subset of the SME photon sector once expanded around the background $u^\mu = \bar{u}^\mu + \delta u^\mu$.

For fermions, generic Planck-suppressed couplings between $u^\mu$ and the Dirac kinetic term generate SME coefficients $c_{\mu\nu}$. We do not need their exact form here; only their magnitude and symmetry properties matter.

In the Λ-network picture, this magnitude is not an adjustable input: it is controlled by statistical isotropy under coarse-graining. Let N be the number of effectively independent links contributing to the local averaged connectivity tensor over the probe scale L. If link orientations decorrelate rapidly, the anisotropic remainder obeys a central-limit suppression,

$$ \delta D_{\mu\nu} \sim N^{-1/2}, $$

and the induced SME coefficients inherit the same scaling at leading order (up to sector-dependent dimensionless prefactors),

$$ |c_{\mu\nu}| \sim N^{-1/2}, \quad |\kappa| \sim N^{-1/2}, \quad |\alpha_T| \sim N^{-1/2}. $$

Thus Lorentz invariance is an emergent large-scale property: the microscopic discreteness does not generate observable preferred-frame effects unless the network exhibits persistent long-range anisotropy, in which case cone-splitting (e.g. in $v_{GW}$ vs. $v_\gamma$) would be immediately constrained by GW170817 (Check 1).

Finally, the same set of coefficients $c_i$ that control the tensor speed $c_T$ also generate a tensor-sector Lorentz-violation parameter $\alpha_T$ appearing in the GW dispersion relation, as in D.1.

### D.2.2 Mapping to SME coefficients

Expanding around Minkowski with $u^\mu = (1, 0, 0, 0)$ and working to leading order in small Lorentz-violating parameters, one recovers the standard SME-inspired parametrization:

#### Gravitational waves (tensor sector)

$$ \frac{v_{GW}}{c} - 1 = \frac{\alpha_T}{2}, $$

where $\alpha_T$ is a linear combination of the Einstein–Æther coefficients $\{c_i\}$ (reducing to $\alpha_T \simeq c_{13}$ at leading order).

#### Photons (non-birefringent sector)
The effective metric $\tilde{g}_{\mu\nu} = g_{\mu\nu} + Du_{\mu}u_{\nu}$ induces an isotropic shift in the photon phase velocity,

$$ \frac{v_{\gamma}}{c} - 1 = -\frac{\kappa}{2}, $$

where $\kappa$ is the SME-like non-birefringent coefficient built from $D$ and the background $u^{\mu}$.

### Fermions

For ultrarelativistic fermions, SME coefficients $c_{\mu\nu}$ modify the maximum attainable velocity as

$$ v_{\max} \approx 1 - \frac{1}{2}(c_{00} + \frac{1}{3}c_{ii}), $$

with implicit summation over spatial indices $i$.

We also allow for higher-dimension Lorentz-violating operators suppressed by a large scale $M_*$, which generate corrections of the form $\Delta v \sim (E/M_*)^n$ at energy $E$. In the $\Lambda$-framework these are naturally associated with the microscopic lattice scale and the discrete angular structure of the network.

To avoid vacuum birefringence, we enforce from the outset:

* vanishing CPT-odd photon coefficient $(k_{AF})_{\mu} = 0$,

* vanishing birefringent components of $k_F$ (often denoted $\tilde{\kappa}_{e+} = \tilde{\kappa}_{o-} = 0$),

* no parity-odd couplings of the form $\epsilon^{\mu\nu\rho\sigma}u_{\mu}\dots$ in the tensor and photon kinetic terms.

This matches the empirical fact that no vacuum birefringence has been observed at extremely high precision.

### D.2.3 Experimental bounds and network parameter ranges

Current observational and laboratory bounds can be summarized as follows:

* From GW170817 + GRB170817A:

$$ \left| \frac{v_{GW}}{c} - 1 \right| = \left| \frac{\alpha_T}{2} \right| \lesssim 3 \times 10^{-15} \Rightarrow \left| \alpha_T \right| \lesssim 6 \times 10^{-15}. $$

* From non-birefringent photon time-of-flight tests:

$$ \left| \frac{v_{\gamma}}{c} - 1 \right| = \left| \frac{\kappa}{2} \right| \lesssim 10^{-15} \Rightarrow \left| \kappa \right| \lesssim 2 \times 10^{-15}, $$

with many astrophysical limits being even tighter.

* From high-energy cosmic rays, synchrotron radiation and precision spectroscopy:

$$ |c_{\mu\nu}| \lesssim 10^{-15} \text{ for all relevant components,} $$

with several individual components bounded at the $10^{-18}$ level or better.

To keep the $\Lambda$-framework firmly inside this “safe” window, Check 2 imposes the following parameter choices:

**Tensor sector (GW speed):**

$$ c_1 + c_3 = 0 \Rightarrow c_T^2 = 1, \quad \alpha_T = 0, $$

so that GWs propagate exactly at $c$ and satisfy Check 1 and Check 2 simultaneously.

**Photon sector (no birefringence, luminal cone):**

* enforce $(k_{AF})_{\mu} = 0, \tilde{\kappa}_{e+} = \tilde{\kappa}_{o-} = 0$;

* choose the non-birefringent photon parameter
$|\kappa| \lesssim 10^{-18},$

comfortably below current bounds.

**Fermion sector (maximal speeds):**

$|c_{\mu\nu}| \lesssim 10^{-18} \text{ for all species at low energy,}$

ensuring that any deviation from Lorentz symmetry in matter is at least three orders of magnitude below the most stringent current constraints.
## Higher-dimension operators:

Take suppression scales $M_*$ at or above the network’s fundamental tension scale, so that for all relevant energies (from MeV to PeV)

$$ | \Delta v(E) | \sim \left( \frac{E}{M_*} \right)^n \ll 10^{-15}. $$

From the microscopic perspective, these small coefficients arise from statistical isotropy of the $\Lambda$-network:

* the fundamental graph is locally anisotropic, but orientations decorrelate rapidly;

* coarse-graining over $N \gg 1$ links in random directions makes the residual anisotropy scale as $N^{-1/2}$;

* at macroscopic scales ($N \sim 10^{30}$ or more) one naturally expects effective deviations in the range $10^{-18}-10^{-21}$.

In that sense, Lorentz symmetry in the $\Lambda$-framework is not an exact microscopic postulate but an emergent large-scale property of an almost isotropic tension network:

$$ \Lambda_{AB}^{P} \propto \delta_{AB} \Rightarrow \text{single light cone for all sectors} $$

up to corrections far below present experimental sensitivity. This completes the derivation underlying Check 2 – Lorentz invariance.

## D.3 Check 3 – Equivalence Principle

Goal: derive the motion law $\mathbf{a} = -\frac{c^2}{2} \nabla \ln \Lambda^P$ and show that inertial and gravitational mass coincide as a structural identity, $m_g/m_i = 1$. Bound possible violations $\Delta a/a$ in terms of network anisotropy $\varepsilon$ and compare with Eöt-Wash-type constraints.

### D.3.1 Test setup (WEP + LPI)

In the network picture, “gravity” is not a fundamental force but the response of the lattice to gradients of the real tension component $\Lambda^P$. At coarse-grained scales, this induces an effective potential $\Phi(\mathbf{x})$ and a universal acceleration field.

We model a test body with inertial mass $m_i$ and gravitational charge $m_g$ moving in a slowly varying $\Lambda^P$ background. The effective potential of the network is defined as

$$ \Phi(\mathbf{x}) = c^2 \ln \sqrt{\frac{\Lambda^P(\mathbf{x})}{\Lambda_0^P}}, $$

so that gradients of $\Lambda^P$ play the rôle of a Newtonian potential. At low velocities, the test-particle Lagrangian reads

$$ \mathcal{L}_{\text{test}} = -m_i c^2 \sqrt{1 - \frac{v^2}{c^2}} - m_g \Phi(\mathbf{x}). $$

Expanding to leading order in $v/c$ and in $\Phi/c^2$, the Euler-Lagrange equation gives

$$ m_i \mathbf{a} = -m_g \nabla \Phi(\mathbf{x}). $$

Using the definition of $\Phi$,

$$ \nabla \Phi(\mathbf{x}) = \frac{c^2}{2} \nabla \ln \Lambda^P(\mathbf{x}), $$
so that the effective acceleration field can be written, up to the choice of reference tension $\Lambda_0^P$,

$$ \mathbf{a} = - \frac{m_g}{m_i} \frac{c^2}{2} \nabla \ln \Lambda^P . $$

This is the precise form of the motion law used in the $\Lambda$-framework; in the main text we will sometimes use the schematic relation $\mathbf{a} \propto -c^2 \nabla \ln \Lambda^P$, omitting the overall factor 1/2 and the (experimentally measured) ratio $m_g/m_i = 1$.

## DUniversal ratio $m_g/m_i$ in the 10D lattice

In the complete $\Lambda$-framework the microscopic network lives on a 10D relational lattice spanned by the directions $(d_1, \dots, d_{10})$. At the conceptual level:

* $d_1-d_4$ (4 internal dimensions) encode local gauge and causal structure (e.g. $U(1)$, $SU(2)$, $SU(3)$, CPT topology);

* $d_5 \equiv \Theta$ is the global phase dimension controlling the balance between $\Lambda^P$ and $\Lambda^I$;

* $d_6-d_9$ (4 emergent dimensions) form the macroscopic space-time (3 space + 1 time) arising from network synchronization;

* $d_{10}$ is the resonant “closure” dimension, completing the $\Lambda$-cycle and mediating information/tension exchange between $\Lambda^P$ and $\Lambda^I$.

Microscopically, an extended object is a bound pattern of deformation in this full 10D lattice. At the EFT level, all matter fields couple to a single scalar “tension metric” via

$$ S_m[A^2(\Lambda) g_{\mu\nu}, \psi], $$

where $A(\Lambda)$ encodes the universal rescaling of the inertial metric by the mean lattice tension. In the Einstein frame, where the metric is conformally rescaled as

$$ \tilde{g}_{\mu\nu} = A^2(\Lambda) g_{\mu\nu}, $$

all freely falling particles follow the same geodesics, independently of their internal structure, composition, or how they are embedded in the 10D lattice. The Weak Equivalence Principle (WEP) is therefore exact in this frame: the motion only depends on $\tilde{g}_{\mu\nu}$, not on particle species.

Translating back to the Jordan frame, this implies that both inertial and gravitational masses are proportional to the same scalar network functional of the 10D configuration:

* inertial mass $m_i$ measures the energy stored in local deformations of the network (the topological “core” in the full 10D space $(d_1, \dots, d_{10})$);

* gravitational mass $m_g$ measures how that same object couples to the ambient $\Lambda^P(\mathbf{x})$ through the scalar conformal factor $A(\Lambda)$.

Because the coupling is scalar and isotropic in the internal 10D tension space (with $\Lambda_{AB}^P \propto \delta_{AB}$ at coarse-grained level), both masses scale with $\Lambda^P$ in the same way:

$$ m_i \propto \Lambda^P, \quad m_g \propto \Lambda^P \quad \Rightarrow \quad \frac{m_g}{m_i} = 1, $$

independent of the internal orientation of the object within the 10D lattice. In other words, the Equivalence Principle is not postulated but follows from (i) diffeomorphism invariance and (ii) internal isotropy enforcing a single scalar coupling $A(\Lambda)$ for all matter.
## Anisotropy and bound on $\Delta a/a$

Possible violations arise if the tension tensor is not perfectly isotropic in the internal 10D space. Write

$$ \Lambda_{AB}^{P} = \Lambda^{P}(\delta_{AB} + \varepsilon_{AB}), \quad |\varepsilon_{AB}| \ll 1, $$

with $\varepsilon_{AB}$ capturing small directional preferences of the network. For two test bodies with different internal configurations (different ways of sampling the internal indices A, B across ($d_1, \dots, d_{10}$)), the fractional difference in free-fall acceleration in a given external gradient $\nabla \Lambda^P$ can be parametrized as

$$ \frac{\Delta a}{a} \approx \alpha \varepsilon, $$

where $\varepsilon$ is a typical magnitude of anisotropy and $\alpha = \mathcal{O}(1)$ encodes the “lever arm” between the two internal configurations.

Torsion-balance and atom-interferometry experiments (Eöt-Wash-type tests) constrain

$$ \frac{\Delta a}{a} \lesssim 10^{-13}. $$

In the network, this translates into

$$ \alpha \varepsilon \lesssim 10^{-13}, $$

which is naturally satisfied if the coarse-grained lattice remains extremely close to isotropic at macroscopic scales. In that regime, the expression

$$ \mathbf{a} = -\frac{c^2}{2} \nabla \ln \Lambda^P $$

defines an acceleration field that is universal for all forms of matter to within current experimental precision, so Check 3 passes at the conceptual level and is compatible with the Eöt-Wash bound.

## D.4 Check 4 – CMB/BAO/BBN history and slow $\Theta(t)$

Goal: show that the scalar–tensor effective action with $G_{eff}(\Lambda)$ and slowly varying $\Theta(t)$ reduces to a $\Lambda$CDM-like background at early times. Constrain $\dot{\Lambda}$ and $\dot{\Theta}$ so that BBN, CMB, and BAO observables remain within their empirical bounds.

### D.4.1 Effective scalar–tensor action and $G_{eff}(\Lambda)$

At cosmological scales, the network tension behaves as an effective scalar–tensor sector whose background evolution must remain compatible with standard $\Lambda$CDM constraints from BBN, CMB and BAO, while allowing for slow late-time dynamics (including a possible $\Theta$-phase evolution) that can slightly modulate the local expansion rate.

The coarse-grained dynamics of the tension field is encoded in the effective action

$$ S = \int d^4 x \sqrt{-g} \left[ \frac{M_{Pl}^2}{2} R - \frac{Z(\Lambda)}{2} \nabla_{\mu} \Lambda \nabla^{\mu} \Lambda - V(\Lambda) \right] + S_m[A^2(\Lambda) g_{\mu\nu}, \psi], $$

where $\Lambda$ is the macroscopic projection of $\Lambda^P$, $Z(\Lambda)$ is an effective stiffness and $V(\Lambda)$ the potential energy of the network.

Matter is minimally coupled to the Jordan-frame metric

$$ g_{\mu\nu}^{(J)} = A^2(\Lambda) g_{\mu\nu}, $$
so the *effective gravitational constant* measured by Cavendish-like experiments is

$$G_{eff}(\Lambda) = \frac{G}{A^2(\Lambda)}.$$

Small variations in $\Lambda$ thus induce

$$\frac{\Delta G_{eff}}{G} \approx -2 \Delta \ln A(\Lambda).$$

A convenient minimal parametrization is

$$A(\Lambda) = \exp[\beta(\Lambda - \Lambda_0)], \quad Z(\Lambda) = Z_0 > 0, \quad V(\Lambda) = \frac{1}{2} m_{\Lambda}^2 (\Lambda - \Lambda_{\infty})^2,$$

so that

$$\frac{\Delta G_{eff}}{G} \approx -2 \beta \Delta \Lambda.$$

### D.4.2 Background cosmology and $\Lambda$CDM limit

On a spatially flat FLRW background

$$ds^2 = -dt^2 + a^2(t) \, dx^2,$$

the modified Friedmann equations in the Jordan frame read

$$3M_{Pl}^2 H^2 = \rho_m + \rho_r + \rho_\Lambda, -2M_{Pl}^2 \dot{H} = \rho_m + \frac{4}{3}\rho_r + \rho_\Lambda + p_\Lambda,$$

with

$$\rho_{\Lambda} = \frac{Z}{2} \dot{\Lambda}^2 + V(\Lambda), \quad p_{\Lambda} = \frac{Z}{2} \dot{\Lambda}^2 - V(\Lambda).$$

The scalar field obeys

$$Z(\ddot{\Lambda} + 3H\dot{\Lambda}) + \frac{1}{2}Z_{,\Lambda}\dot{\Lambda}^2 + V_{,\Lambda} = \alpha(\Lambda)T^{[m]}, \quad \alpha(\Lambda) \equiv \frac{d \ln A}{d\Lambda},$$

where $T^{[m]} \approx -\rho_m$ at low pressure.

The $\Lambda$CDM limit corresponds to:

* a frozen tension field at early times,

$$\dot{\Lambda} \approx 0, \quad \Lambda \approx \Lambda_{early} = \text{const},$$

* a nearly constant coupling

$$\alpha(\Lambda) = \beta \ll 1,$$

so that $G_{eff}$ is effectively constant during BBN and recombination.

In this regime,

$$\rho_{\Lambda} \approx V(\Lambda_{early}) = \text{const}, \quad p_{\Lambda} \approx -V(\Lambda_{early}),$$

and the background expansion is indistinguishable from standard $\Lambda$CDM (with $\Omega_{\Lambda}$ fixed by $V$).
### D.4.3 BBN, CMB and BAO constraints (“safe” parameter region)

Observations impose the following bounds on the early-time behavior of $G_{eff}$ and on the dynamics of $\Lambda$:

#### BBN (T ~ 1 MeV, z ~ 10⁹)

Light-element yields constrain the expansion rate $H \propto \sqrt{G_{eff}\rho}$, giving

$$ | \frac{\Delta G_{eff}}{G} |_{BBN} \lesssim 0.02. $$

With $A(\Lambda) = \exp[\beta(\Lambda - \Lambda_0)]$, this translates into

$$ | \beta(\Lambda_{BBN} - \Lambda_0) | \lesssim 10^{-2}, $$

which is easily satisfied if $\Lambda$ is frozen during nucleosynthesis.

#### CMB (z ~ 1100)

CMB anisotropies require:

* tensor speed $c_T = c$, already ensured by Check 1,

* a nearly constant effective Planck mass,

$$ \alpha_M \equiv \frac{d \ln M_*^2}{d \ln a} \approx 0 \text{ at recombination,} $$

where $M_*^2 \propto G_{eff}^{-1}$.

* This again demands that both $A(\Lambda)$ and $Z(\Lambda)$ vary very slowly for $z \gtrsim 10^3$, which is guaranteed if $\beta \lesssim 10^{-3}$ globally and $\Lambda$ remains pinned near a potential minimum in the early universe.

#### BAO + SN (z ~ 2)

Distance-redshift relations and $H(z)$ measurements are compatible with $\Lambda$CDM at the few-percent level. This allows for:

* small, slowly varying deviations in $\rho_{\Lambda}(z)$,

* mild environmental dependence of the local Hubble rate $H_0(x) \propto \sqrt{\Lambda^P(x)}$,

provided these remain small enough that the volume-averaged expansion matches $\Lambda$CDM within current BAO/SN error bars. In the minimal parametrization, this is achieved with:

$$ \beta \lesssim 10^{-3}, \quad |\delta\Lambda|_{\text{local}} \sim 10^{-2}, $$

which yields $|\Delta G/G| \sim 2 \times 10^{-5}$ (negligible for BBN/CMB) while still allowing a percent-level modulation of the inferred local $H_0$, consistent with the range of the Hubble tension. For concreteness we explore a regime with $\beta \lesssim 10^{-3}$, which is representative of scalar-tensor models that are effectively indistinguishable from $\Lambda$CDM at early times.

In this “safe” region, the network behaves as standard $\Lambda$CDM at early times and large scales, with only small late-time departures driven by slow tension dynamics and $d_4$- textures.

### D.4.4 Role of $\Theta(t)$ and reduction to a single real mode

*Notation reminder.* In this Appendix, $\Theta$ denotes the late-time (causal) angular field unless stated otherwise. The pre-temporal order parameter is $\Theta_{dyn}$, and its relaxation toward the fixed point $\Theta_{struct}$ is defined in Sec. 5.5.3.
At the microscopic level, the complex tension can be decomposed as $\Lambda = \Lambda^P + i\Lambda^I = Re^{i\Theta}$, where $R = |\Lambda|$ and $\Theta$ is the phase describing the curvature-coherence cycle in the $\Lambda$-plane. The kinetic term then splits as

$$(\nabla\Lambda)^2 = (\nabla R)^2 + R^2(\nabla\Theta)^2,$$

so that, for a homogeneous background,

$$\rho_{\Lambda} = \frac{Z}{2}(\dot{R}^2 + R^2\dot{\Theta}^2) + V(R, \Theta), \quad p_{\Lambda} = \frac{Z}{2}(\dot{R}^2 + R^2\dot{\Theta}^2) - V(R, \Theta).$$

The cosmological constraints above are satisfied in the **ordered regime** where:

* both $R$ and $\Theta$ are locked (or vary extremely slowly) at early times,

$$\dot{R} \approx 0, \dot{\Theta} \approx 0 \Rightarrow \rho_{\Lambda} \approx V(R_0, \Theta_0) = \text{const},$$

* the $\Theta$-mode behaves as a *frozen phase* of the $\Lambda$-cycle (no time-crystal behavior at cosmological scales), so that the effective fluid has $w \approx -1$ and reproduces $\Lambda$CDM during BBN and recombination.

At late times, a very slow drift of $\Theta(t)$ and/or $R(t)$, driven by the shallow potential $V$ and internal $d_4$-coherence, can:

* slightly modify $\rho_{\Lambda}$ (mild departure from $w = -1$),

* induce small spatial modulations of $R$ and hence of $\Lambda^P(\mathbf{x})$,

without spoiling the bounds on $\Delta G_{eff}/G$, CMB anisotropies, or BAO. In this regime the $\Theta$-field is effectively a *hidden internal degree of freedom* whose dynamics is too slow and too weakly coupled to be seen in current cosmological data, but it remains crucial for the microphysical interpretation of the $\Lambda$-cycle in the network model.

## D.5 Check 5 – Hubble variability from $\Lambda^P(\mathbf{x})$

Goal: show how slow spatial modulations of the mean tension $\Lambda^P(\mathbf{x})$ generate an effective variability of the local Hubble rate $H_0(\mathbf{x})$ at the ~1–2% level, while remaining consistent with CMB/BAO/BBN constraints and current bounds on large-scale anisotropy.

### D.5.1 Local FRW patch with modulated tension

We start from the scalar–tensor effective action introduced in D.4, where the scalar $\Lambda$ is the macroscopic projection of the network tension $\Lambda^P$:

$$S = \int d^4x \sqrt{-g} \left[ \frac{M_{Pl}^2}{2} R - \frac{Z(\Lambda)}{2} (\nabla_\mu \Lambda)(\nabla^\mu \Lambda) - V(\Lambda) \right] + S_m[A^2(\Lambda)g_{\mu\nu}, \psi].$$

The conformal coupling $A(\Lambda)$ defines an effective gravitational constant

$$G_{eff}(\Lambda) = \frac{G}{A^2(\Lambda)}.$$

In a quasi-homogeneous neighborhood of an observer (local FRW patch) at $z \approx 0$, the Friedmann equation reads

$$3M_{Pl}^2 H_0^2(\mathbf{x}) = \rho_{m,0}(\mathbf{x}) + \rho_{r,0}(\mathbf{x}) + \rho_{\Lambda}(\Lambda(\mathbf{x})),$$

with

$$\rho_{\Lambda}(\Lambda) = \frac{Z(\Lambda)}{2} \dot{\Lambda}^2 + V(\Lambda).$$
For Check 5 we restrict to the “quasi-frozen” regime at late times,

$$ \dot{\Lambda}^2 \ll V(\Lambda), $$

so that

$$ \rho_{\Lambda}(\Lambda) \approx V(\Lambda). $$

### D.5.2 Variation of $H_0$ induced by slow $\Lambda$-modulations

We decompose the macroscopic tension as background plus a slow spatial fluctuation:

$$ \Lambda(\mathbf{x}) = \bar{\Lambda} + \delta\Lambda(\mathbf{x}), \quad |\delta\Lambda| \ll |\bar{\Lambda}|. $$

Let $H_{0,ref}$ denote the reference value (e.g. CMB/Planck) for $\Lambda = \bar{\Lambda}$, and write

$$ H_0(\mathbf{x}) = H_{0,ref} + \delta H_0(\mathbf{x}). $$

To first order in $\delta\Lambda$,

$$ 2H_{0,ref} \delta H_0(\mathbf{x}) \approx \frac{1}{3M_{Pl}^2} \delta\rho_{\Lambda}(\mathbf{x}) + (\text{terms from } \delta G_{\text{eff}}), $$

where

$$ \delta\rho_{\Lambda}(\mathbf{x}) \approx \left. \frac{dV}{d\Lambda} \right|_{\bar{\Lambda}} \delta\Lambda(\mathbf{x}) \equiv V'(\bar{\Lambda}) \delta\Lambda(\mathbf{x}). $$

Hence

$$ \frac{\delta H_0}{H_{0,ref}} \approx \frac{1}{6M_{Pl}^2 H_{0,ref}^2} V'(\bar{\Lambda}) \delta\Lambda(\mathbf{x}) \equiv C_V \frac{\delta\Lambda}{\bar{\Lambda}}, $$

where $C_V$ is a dimensionless number that depends on the shape of $V(\Lambda)$.

For a simple quadratic potential

$$ V(\Lambda) = \frac{1}{2} m_{\Lambda}^2 (\Lambda - \Lambda_{\infty})^2 $$

calibrated such that $\Omega_{\Lambda} \simeq 0.7$, one typically obtains $C_V = \mathcal{O}(10^{-1})$, with sign fixed by the relative position of $\bar{\Lambda}$ and $\Lambda_{\infty}$.

The conformal coupling

$$ A(\Lambda) = \exp[\beta(\Lambda - \Lambda_0)] $$

induces a variation of $G_{\text{eff}}$,

$$ \frac{\delta G_{\text{eff}}}{G_{\text{eff}}} = -2 \delta \ln A = -2 \beta \delta \Lambda, $$

which contributes

$$ \left( \frac{\delta H_0}{H_0} \right)_{G_{\text{eff}}} \approx \frac{1}{2} \frac{\delta G_{\text{eff}}}{G_{\text{eff}}} = -\beta \delta \Lambda. $$

Altogether,

$$ \frac{\delta H_0}{H_0} \approx C_V \frac{\delta \Lambda}{\bar{\Lambda}} - \beta \delta \Lambda. $$
In the “safe” parameter regime identified in Check 4, $\beta \lesssim 10^{-3}$ and $|\Lambda - \Lambda_0| \lesssim \mathcal{O}(10)$ at early times, so early-universe constraints (BBN, CMB, BAO) are satisfied while leaving room for late-time spatial modulation of $\Lambda$ at the few-percent level. For our purposes it is convenient to rewrite the result directly in terms of the network tension $\Lambda^P$, using $\Lambda \simeq \Lambda^P$ at coarse-grained level:

$$ \frac{\delta H_0}{H_0} \simeq C_H \frac{\delta \Lambda^P}{\Lambda^P}, \quad C_H \simeq 0.3, $$

where the sign of $C_H$ is determined by the detailed form of $V(\Lambda)$ and $A(\Lambda)$. The phenomenologically interesting regime (voids yielding higher $H_0$) corresponds to $C_H < 0$: regions with lower $\Lambda^P$ then have larger $H_0$.

In that regime, slow spatial fluctuations $|\delta\Lambda^P|/\Lambda^P \sim 3-6\%$ naturally produce $|\delta H_0|/H_0 \sim 1-2\%$, i.e. a range consistent with the observed Planck-SH0ES discrepancy once observational systematics are taken into account.

The coefficient $C_H$ is model-dependent. In a potential-dominated regime one expects $C_H = \mathcal{O}(10^{-1})$, whereas in a $G_{\text{eff}}$-dominated regime one obtains $H_0 \propto 1/\sqrt{\Lambda^P}$ and thus $C_H \simeq -1/2$. We focus on the latter when discussing voids yielding higher $H_0$.

### D.5.3 Network interpretation: $H_0$ as a probe of $\Lambda^P$-rigidity

In the pure network language these effects can be summarized by an effective law

$$ H_0(\mathbf{x}) \propto \frac{1}{\sqrt{\Lambda^P(\mathbf{x})}}, $$

which encodes in $\Lambda^P(\mathbf{x})$ both the contribution of $V(\Lambda)$ and that of $G_{\text{eff}}(\Lambda)$. In the scalar-tensor language of Sec. D.4, this relation corresponds to the regime in which the variation of $G_{\text{eff}}(\Lambda)$ dominates over the contribution of $V(\Lambda)$ in the Friedmann equation, so that $H_0^2 \propto G_{\text{eff}} \propto 1/\Lambda^P$ and one indeed recovers $C_H \simeq -\frac{1}{2}$ in the relation $\delta H_0/H_0 \simeq C_H \delta \Lambda^P/\Lambda^P$ above.

* In denser regions (clusters, thick filaments) the connection pattern is more rigid: $\Lambda^P$ is larger, the metric is effectively stiffer, and the local expansion rate is slightly reduced, yielding a lower $H_0(\mathbf{x})$.

* In void-like regions the connection density and the mean tension are lower: $\Lambda^P$ is smaller, the metric is more compliant, and the local expansion rate is slightly enhanced, yielding a higher $H_0(\mathbf{x})$.

The key point for Check 5 is that the required modulations of $\Lambda^P$ are:

* small and correlated with large-scale structure (voids vs. clusters), so that low-$\ell$ CMB anisotropy constraints remain satisfied,

* large enough at late times to generate percent-level variations in $H_0$ across different environments and lines of sight.

The framework therefore predicts a mild, environment-dependent modulation of $H_0$ that is naturally of the right order of magnitude and consistent with early-universe data.

## D.6 Check 6 – Tension/energy closure and the $\Lambda$-cycle

Goal: show that the tension field $(\Lambda^P, \Lambda^I)$ forms a conservative “$\Lambda$-cycle”: exchanges between potential and phase components are purely oscillatory, so that the combination $\Lambda_P^2 + \Lambda_I^2$ is conserved, and the total energy of the bi-sheet $(M^+, M^-)$ respects the cosmological closure condition $\Omega_{tot} \simeq 1$.

### D.6.1 O(10) internal structure and Noether conservation

At the microscopic level the tension is an internal tensor $\Lambda_{AB}^P$ in the 10D space of connection directions, with isotropic mean

$$ \Lambda_{AB}^P \propto \Lambda^P \delta_{AB}. $$
The phase field $\Phi$ coupled to this tension obeys an equation of motion of the schematic form

$$ \nabla_A (\Lambda_{AB}^P \partial^B \Phi) = 0, $$

where $\nabla_A$ is the covariant derivative on the internal graph.

Invariance of the microscopic action under internal rotations $SO(10)$ (or $O(10)$) implies, via Noether's theorem, a locally conserved “tension–energy” current $J_E^A$ and density $\mathcal{E}$ built from $\Lambda_{AB}^P \partial^A \Phi \partial^B \Phi$:

$$ \partial_t \mathcal{E} + \nabla_A J_E^A = 0. $$

Isotropy, $\Lambda_{AB}^P \propto \delta_{AB}$, ensures that this conservation holds globally as well:

$$ \frac{d}{dt} \int d^9 X \mathcal{E} = 0. $$

In other words, there is no internal “sink” or “source” of energy in the 10D space of directions: reconfigurations of tension are rotations in internal space, not creation or destruction of energy. This is the microscopic origin of the $\Lambda$-cycle.

### D.6.2 $\Lambda$-cycle as a rotation in $(\Lambda^P, \Lambda^I)$ at fixed amplitude

At coarse-grained level, the tension is encoded in a complex field

$$ \Lambda = \Lambda_P + i \Lambda_I = R e^{i\Theta}, $$

where $R$ encodes the total tension amplitude and $\Theta$ is the local “time phase”. At coarse-grained level, the $O(10)$ internal invariance reduces effectively to a $U(1)$ symmetry in the $\Lambda$-plane, with $\Theta$ as the corresponding angular variable.

The effective field equation for $\Lambda$, derived from a phase-invariant potential $V(R)$, has the schematic form

$$ Z(R) \Box \Lambda + V'(R) \frac{\Lambda}{R} = 0. $$

Splitting into radial and angular parts yields:

* a radial equation for $R$ with cosmological friction (Hubble damping),

* a conservation equation for the angular part,

$$ \partial_\mu (R^2 \partial^\mu \Theta) = 0. $$

In the $\Lambda$-cycle regime of interest for Check 6, the radial degree of freedom is approximately frozen:

$$ R(t, \mathbf{x}) \approx R_0 = \text{const.} $$

Equivalently,

$$ \Lambda_P^2 + \Lambda_I^2 = R_0^2 = \text{const.} $$

A slow rotation of $\Theta(t)$ then periodically exchanges $\Lambda_P$ and $\Lambda_I$ but leaves the amplitude $R_0$ unchanged, so that

$$ \partial_t (\Lambda_P^2 + \Lambda_I^2) = 0 $$

in the absence of dissipative couplings to other sectors.

This is precisely the “conservative $\Lambda$-cycle” condition used in Check 6: the potential and phase components exchange energy internally, but the modulus of the complex tension field remains constant.
### D.6.3 Bi-sheet structure ($M^+, M^-$) and cosmological energy closure

The bi-sheet structure ($M^+, M^-$) imposes an additional global constraint

$$ \Lambda_P^{(+)} + \Lambda_P^{(-)} = \text{const.} $$

Osmotic fluxes of tension between the two sheets adjust $\Lambda_P^{(+)}$ and $\Lambda_P^{(-)}$ locally, but their sum remains fixed. In the scalar–tensor language of D.4 this translates into:

* an asymptotically constant total vacuum energy density $\rho_{\Lambda, tot}$,

* local adjustments of $\rho_{\Lambda}$ in each sheet that cancel out in the sum.

The effective energy–momentum tensor of the tension sector satisfies the usual covariant conservation law,

$$ \nabla_{\mu} T^{\mu \nu}_{(\Lambda)} = 0, $$

and the cosmological closure $\Omega_{tot} = 1 \pm \mathcal{O}(10^{-3})$ is satisfied by choosing the radius $R_0$ such that $\rho_{\Lambda}(R_0)$ matches the observed dark energy density ($\Omega_{\Lambda} \simeq 0.7$).

Small residual time variations of the inter-sheet difference $\Delta\Lambda(t)$ could in principle produce an effective cosmological “leakage” at the level $\lesssim 10^{-5}$, but this regime is strongly constrained by the observed near-flatness and is not used in the checks presented here. Check 6 explicitly restricts to the quasi-stationary $\Lambda$-cycle regime in which such leakage is negligible relative to observational uncertainties.

## D.7 Check 7 – Absence of vacuum birefringence

Goal: verify that, in the parameter regime used to confront the model with data, all photon and graviton polarizations share the same effective light cone, so that vacuum birefringence (polarization-dependent propagation speed) is absent at the level

$$ | \Delta v_{\text{pol}} | / c \ll 10^{-32}. $$

### D.7.1 Photon sector: non-birefringent effective metric

In the network framework, photons are phase waves in an internal U(1) subspace of the network, aligned with the coherence field $T^\mu$ (or $u^\mu$, same field). At EFT level this is encoded by a “disformal” effective metric $\tilde{g}_{\mu\nu}$ felt by the electromagnetic field:

$$ \tilde{g}_{\mu\nu} = g_{\mu\nu} + D T_{\mu} T_{\nu}, \quad |D| \ll 1. $$

The minimal EM action is

$$ S_{\text{EM}} = -\frac{1}{4} \int d^4 x \sqrt{-\tilde{g}} \tilde{g}^{\mu\alpha} \tilde{g}^{\nu\beta} F_{\mu\nu} F_{\alpha\beta}. $$

The characteristic curves (propagation cones) of EM modes are then the null geodesics of $\tilde{g}_{\mu\nu}$. Both linear polarizations (+ and $\times$) satisfy the same scalar wave equation on $\tilde{g}_{\mu\nu}$, hence they share exactly the same cone: no birefringence is possible as long as:

* the coupling contains no antisymmetric Levi-Civita tensor $\epsilon^{\kappa\lambda\mu\nu}$ (no Chern–Simons term),

* $D$ is a scalar (no residual anisotropic tensor in internal space).

In the Standard-Model Extension (SME) notation this corresponds to imposing:

* vanishing Chern–Simons vector

$$ (k_{AF})^{\kappa} = 0, $$

* vanishing birefringent components of $k_F$,

$$ \tilde{\kappa}_{e+} = 0, \quad \tilde{\kappa}_{o-} = 0, $$
so that all birefringent SME coefficients are set to zero by construction. In this sense the very strong experimental bounds ($|\tilde{\kappa}_{e+}|, |\tilde{\kappa}_{o-}| \lesssim 10^{-32}$) are trivially satisfied: the $\Lambda$-framework simply does not generate these operators.

The disformal parameter $D$ only contributes to the non-birefringent sector (isotropic shifts of the photon cone), which can be absorbed into small deformations of $\tilde{g}_{\mu\nu}$. The magnitude of these shifts is constrained instead by the global Lorentz-violation bounds of Check 2, which require $|D| \lesssim 10^{-18}$ so that any induced non-birefringent SME coefficients remain well below current time-of-flight and spectroscopy limits.

### D.7.2 Gravitational sector: identical cones for the two GW polarizations

The gravitational sector is modeled as an Einstein–Æther–type theory with a time-like vector field $T^\mu$, whose dynamics involves four kinetic invariants with coefficients $c_i$. The canonical tensor Lagrangian can be written

$$S_T = \int d^4 x \sqrt{-g} \left[ \frac{M_{Pl}^2}{2} R + \ell_T (T^\mu, \nabla T^\mu) \right],$$

with

$$\ell_T = c_1(\nabla_\mu T_\nu)(\nabla^\mu T^\nu) + c_2(\nabla_\mu T^\mu)^2 + c_3(\nabla_\mu T_\nu)(\nabla^\nu T^\mu) + c_4 T^\mu T^\nu \nabla_\mu T_\alpha \nabla_\nu T^\alpha.$$

Expanding around an FRW background with $T^\mu = (1, 0)$ and diagonalizing the tensor sector, one finds that the two tensor modes $h_+$ and $h_-$ propagate with speed

$$c_T^2 = 1 + \alpha_T(c_i),$$

where $\alpha_T$ is a linear combination of the $c_i$. The condition $c_T = 1$, already enforced by Check 1 (GW170817 speed bound), sets $\alpha_T = 0$, typically via a relation of the type

$$c_1 + c_3 = 0 \text{ (or an equivalent linear constraint in the chosen basis).}$$

In this regime, both tensor polarizations satisfy the same wave equation

$$\Box h_{+,x} = 0$$

on the light cone of $g_{\mu\nu}$. Any coupling that would distinguish $h_+$ from $h_-$ (and thus produce gravitational birefringence) necessarily involves terms that are already excluded by the Lorentz-violation bounds of Check 2.

### D.7.3 Network interpretation: polarization degeneracy as a structural property

In the pure network picture:

* Photons correspond to phase rotations in an isotropic $U(1)$ subspace of the internal space. The two linear polarizations are just two orthogonal directions within the same phase plane; they see the same statistical pattern of connections.

* Gravitons correspond to perturbations of the effective metric of the network in the space of coherence directions. The $+$ and $\times$ modes represent two orthogonal deformations of the same alignment pattern.

As long as:

* the statistics of connections remain isotropic at large scales (no macroscopic $d_4$ “texture” privileging one polarization direction), and

* parity-odd couplings to $\Lambda^I$ are negligible,
the polarizations remain exactly degenerate. Any observable birefringence would require a coherent anisotropy
in $\Lambda^P$ or a selective coupling to $\Lambda^I$ on cosmological scales, which is ruled out by current data.

In summary, once a unique effective light cone is imposed ($GW = EM = c$) and the SME birefringent coefficients are set to zero, vacuum birefringence is automatically suppressed to levels far below $|\Delta v_{pol}|/c \ll 10^{-32}$. This is precisely the content of Check 7.
# E Fermion cards: grammar, closure tests, and RTQ scale audit

## E.1 Appendix E overview: grammar → cards → pipeline

This first chapter defines, in a top-down way, what a “particle” means in MCP and what is stored in a fermion card. The goal is to make Appendix E readable as a self-contained grammar sheet, while being an auditable interface between: (i) the UV discrete RTQ dynamics (finite alphabet, discrete ticks), and (ii) the IR bookkeeping layer used by the numerical proof-of-concept (cards, closure budgets, loop counts). To prevent notation surprises, we factor the construction into four blocks:

<figure><img src="image_5.png" /></figure>

Appendices E.2–E.7 then instantiate this factorization:

* Appendix E.2 introduces the card interface (presets, scenarios, derived outputs).

* Appendix E.3 fixes the RTQ scale (Compton anchors, $N_{loop}^{(e)}$, $\eta$).

* Appendix E.4 defines the discrete grammar (sector labels, invariants, cycle integers).

* Appendix E.5 describes the numerical pipeline (JS proof-of-concept forward model).

* Appendix E.6 provides the micro-to-macro bridge.

* Appendix E.7 provides worked cards and audit tables.

### E.1.1 What “grammar” means in MCP (minimal readable definition)

In MCP, a *grammar* is a local rewrite system acting on a discrete RTQ network state. This minimal definition is sufficient to justify the presence of *flags* in Sec. E.2.2.

*Alphabet.* A finite set of admissible local configurations (RTQ neighbourhood + internal labels/flags + phases). We do *not* enumerate this alphabet in Appendix E; we only keep its effective UV consequence: a minimal angular step.

*State.* A local discrete state is denoted

$$S = (\text{local connectivity, internal variables, status flags}),$$

where “status flags” are tags used to select which rewrite rules can apply.

*Rules.* A rule is a constrained local map

$$r \in \mathcal{G}, \quad r: S \mapsto S',$$

subject to invariants (e.g. $|\Lambda| = \Lambda_0$), parities, causal constraints, and other admissibility conditions.

*Composite objects.* A strand/loop/particle corresponds to a *periodic orbit / closure* under repeated application of rules. This is why Appendix E can treat a fermion as a closed RTQ strand with well-defined coarse holonomy budgets.

### E.1.2 Generic fermion definition (what is a particle in MCP)

A fermionic excitation is modeled as a closed RTQ strand whose microscopic updates can be summarized by coarse holonomy budgets over one loop. Microscopically, at RTQ tick $k$ the strand undergoes a vectorial angular update

$$\Delta\Omega_k = (\Delta\sigma_{24,k}, \Delta\phi_{1,k}, \Delta\psi_{3,k}), \quad k = 1, \dots, N_{loop}.$$
A fermionic loop is characterized (to leading order) by the closure of coarse budgets:

$$ \sum_{k=1}^{N_{\text{loop}}} \Delta\sigma_{24,k} = \Delta d_2, \quad \sum_{k=1}^{N_{\text{loop}}} \Delta\phi_{1,k} = \Delta d_1, \quad \sum_{k=1}^{N_{\text{loop}}} \Delta\psi_{3,k} = \Delta d_3. $$

In the minimal bookkeeping used throughout Appendix E, the core closure principle is Spinor closure (SU(2) double cover):

$$ \Delta d_2 = 4\pi $$

for a rest-geometry fermion. Charge and colour bookkeeping conventions (mapping from $Q, D_{13}$ to $\Delta d_1, \Delta d_{13}, \Delta d_3$) are defined once in Sec. E.4.1–E.4.1 and used by all subsequent pipeline steps.

### E.1.2.1 A generic fermion record (topology / resolution / motif / projection)

To provide a high-level abstraction, we package a fermion as

$$ f \equiv (\mathcal{T}, \mathcal{R}, \mathcal{M}, \mathcal{D}), $$

with:

$$ \mathcal{T} = (N_{\text{pass}}, \text{closure constraints}), $$

$$ \mathcal{R} = (\Delta\sigma_{\min} \text{ or } \eta), $$

$$ \mathcal{M} = (n, \text{winding numbers}), $$

$$ \mathcal{D} = (d_{\text{eff}}, \Theta_{\text{struct}}, \text{projection rules}). $$

Appendix E implements only the minimal subset of these fields needed for auditable tables.

### E.1.3 Canonical factorization of $N_{\text{loop}}(f)$

A robust decomposition that avoids mixing up $4\pi$ with $N_{\text{pass}}$ is:

$$ N_{\text{loop}}(f) = \underbrace{\frac{\Omega_{\text{close}}(f)}{\Delta\sigma_{\min}}}_{\text{topology} \times \text{resolution}} \times \underbrace{\frac{1}{n(f)}}_{\text{motif}}, \quad \Omega_{\text{close}}(f) = 2\pi N_{\text{pass}}(f). $$

For a spin-$\frac{1}{2}$ fermion one typically has $N_{\text{pass}} = 2$, hence $\Omega_{\text{close}} = 4\pi$, but the factorization keeps these notions distinct.

#### E.1.3.1 Connection to the Appendix E pipeline

The proof-of-concept uses the equivalent operational form

$$ N_{\text{loop}}^{(n)} = \underbrace{\eta}_{\text{resolution}} \times \underbrace{\Phi_{\text{obj}}(D_2, D_{13}, Q; w_{d3}, \text{tilt}_{34})}_{\text{topology / geometry proxy}} \times \underbrace{\cos \Theta_{\text{struct}}}_{\text{dimension / projection}} \times \underbrace{\frac{1}{n}}_{\text{motif}}, $$

where $\Phi_{\text{obj}}$ is a deterministic function of the discrete card (defined in Sec. E.4), and $\eta$ is fixed by the scale-bridge in Sec. E.2 (Compton mode) or by an alternative closure mode when explicitly enabled.
### E.1.4 UV discrete action → IR effective action $S_{\text{eff}}$ (bridge to Seff)

At UV level there is no spacetime integral $\int d^4x\sqrt{-g}$ because metric and continuous time are emergent. The fundamental object is a discrete action as a sum over ticks and graph elements:

$$S_{\text{RTQ}}[\Lambda] = \sum_k \left[ \sum_{(i,j) \in E} \mathcal{F}(\Lambda_i^{(k)}, \Lambda_j^{(k)}) + \sum_{i \in V} \mathcal{U}(\Lambda_i^{(k)}) + \mathcal{L}_{\text{cons}}^{(k)} \right],$$

with the canonical UV decomposition:

$$\mathcal{F}(\Lambda_i, \Lambda_j) \approx Z |\Lambda_i - \Lambda_j|^2, \quad \mathcal{U}(\Lambda_i) \approx V(\Theta_i), \quad \mathcal{L}_{\text{cons}} = \sum_i \lambda_i (|\Lambda_i|^2 - \Lambda_0^2).$$

Under coarse-graining, the neighbour sum becomes a gradient term and the node sum becomes a potential term, producing an IR effective action of the Seff form:

$$S_{\text{eff}}[\Lambda] \sim \int d^4x \sqrt{-g} [Z_{\text{eff}} |\nabla \Lambda|^2 + V_{\text{eff}}(\Theta) + \lambda (|\Lambda|^2 - \Lambda_0^2) + \dots].$$

Appendix E does not re-derive this coarse-graining; it uses its key UV consequence: a finite admissible alphabet implies a minimal step $\Delta\sigma_{\min}$ (or equivalently a density $\eta$), which is the resolution input in the $N_{\text{loop}}$ factorization above.

## E.2 Card interface: presets, scenarios, derived quantities (notation guardrails)

### E.2.1 Fermion card (minimal MCP record used in the proof-of-concept)

A fermion card is the compact descriptor manipulated by the proof-of-concept. It is split into three layers: (A) preset, (B) scenario, (C) derived outputs.

#### E.2.1.1 (A) Preset: canonical discrete template

$$C_0 := (D_2, D_{13}, Q; r_1, r_3, w_{d3}, \text{tilt}_{34}; \text{flags}).$$

Here:

* $Q \in \mathbb{Z}/3$ is the charge quantum number ($q = Qe$).

* $(D_2, D_{13}) \in \mathbb{Z}^2$ are canonical bookkeeping integers used to form coarse budgets.

* $(r_1, r_3)$ are construction presets (shape/embedding ratios), frozen to simple values in the audited tables.

* $w_{d3} \in \{0, 3\}$ is the coarse colour-channel gate (singlet vs fundamental strand).

* $\text{tilt}_{34} \in \{0, 2\}$ is the discrete $d_3 \leftrightarrow d_4$ coupling mode (plane vs helicoidal colour).

#### E.2.1.2 (B) Scenario: user choices (not part of the grammar template) A scenario is a run configuration applied on top of $C_0$, e.g.

$$S := (\text{antiParticleLocal}, n, \text{etaMode}, \dots).$$

In particular, $n \in \mathbb{N}$ is the harmonic/mode index and is not a template preset.
**E.2.1.3 (C) Derived outputs (computed deterministically from $C_0$ and $S$)** All quantities listed below are derived (computed deterministically) from the preset card data $C_0$ and the scenario settings $S$. They are reported explicitly in audit tables and UI panels, and should never be confused with the preset integers themselves.

Typical derived outputs are:

$$(\Delta d_1, \Delta d_2, \Delta d_3, \Delta d_{13}), \quad (c_1, c_2, c_3, c_{13}), \quad \Phi_{\text{obj}}, \quad N_{\text{loop}}^{(n)},$$

$$N_{\text{pass}}, N_{\text{spin}}, k(n) = \frac{N_{\text{loop}}^{(n)}}{N_{\text{spin}}}, \quad \text{and optional display/diagnostic scores.}$$

This separation is the core “notation guardrail” used throughout Appendix E.

**1) Coarse holonomy budgets** $\Delta d_1, \Delta d_2, \Delta d_3, \Delta d_{13}$. A fermionic object is summarized at coarse scale by *axis holonomy budgets* $\Delta d_i$ accumulated over one closed RTQ (Relational Tension Quantum) loop. In the minimal bookkeeping used in this PoC, the canonical card integers $(D_{13}, Q)$ map to budgets as:

$$\Delta d_{13} := \Delta d_1 + \Delta d_3 = -\pi D_{13}, \quad \Delta d_1 = 6\pi Q, \quad \Delta d_3 = \Delta d_{13} - \Delta d_1 = -\pi D_{13} - 6\pi Q.$$

For the spinorial budget we adopt a *global rest-geometry convention*:

$$\Delta d_2 = 4\pi,$$

independent of the template integer $D_2$ (which is retained only as a discrete template/ledger integer for taxonomy and UI traceability in this minimal release).

**2) Cycle integers** $c_1, c_2, c_3, c_{13}$. For compact auditing, we define the dimensionless cycle counts

$$c_i := \frac{\Delta d_i}{2\pi}, \quad c_{13} := \frac{\Delta d_{13}}{2\pi}.$$

Under the canonical conversion above this yields

$$c_1 = 3Q, \quad c_{13} = -\frac{D_{13}}{2}, \quad c_3 = c_{13} - c_1.$$

(Again: these $c_i$ are *derived outputs*, not preset integers.)

**3) Objective phase** $\Phi_{\text{obj}}$. The quantity $\Phi_{\text{obj}}$ is the scalar “objective phase” used by the loop-length closure (see also Sec. E.3 and the numerical pipeline). In the *current minimal release* we identify it with the Euclidean norm of the coarse budget vector,

$$\Phi_{\text{topo}} := \|\Delta \vec{d}\|_2 = \sqrt{\Delta d_1^2 + \Delta d_2^2 + \Delta d_3^2}, \quad \Phi_{\text{obj}} \equiv \Phi_{\text{topo}}.$$

More elaborate releases may replace $\Phi_{\text{obj}}$ by a weighted/confinement-sensitive functional, but the PoC tables reported here use the definition above.

**4) Loop length** $N_{\text{loop}}^{(n)}$. The loop length $N_{\text{loop}}^{(n)}$ is the number of RTQ ticks required to realize the objective phase under the chosen resolution $\eta$ and harmonic index $n$. In the PoC it is computed as

$$N_{\text{loop}}^{(n)} = N_{\text{pass}} \frac{\eta}{n} \frac{\Phi_{\text{obj}}}{2\pi \cos \Theta_{\text{struct}}}$$

where $\Theta_{\text{struct}}$ is the structural angle used throughout the manuscript. The scale-fixing rules that determine $\eta$ (Compton anchoring or causal-depth closure) are documented in Sec. E.3.
5) **Distinguishing $N_{pass}$ vs $N_{spin}$**. Appendix E uses two different notions that must not be conflated:

* **Topological pass count $N_{pass}$:** a spin-$\frac{1}{2}$ object returns to the identity only after a $4\pi$ rotation (SU(2) double cover), i.e. two $2\pi$ turns,

$$N_{\text{pass}} := \frac{4\pi}{2\pi} = 2.$$

* **Angular step count $N_{spin}$:** the same $4\pi$ closure, but resolved into elementary angular steps set by the weak-sector coupling strength.

6) **Spin bridge: $N_{spin}$ and the commensurability ratio $k(n)$.** We define the weak-sector spin increment by the effective SU(2) coupling

$$\Delta\sigma_{\text{spin}} := \alpha_2 = \frac{\alpha}{\sin^2\theta_W},$$

where $\alpha$ is the fine-structure constant and $\theta_W$ the Weinberg angle. The corresponding step count over a full spinorial closure is

$$N_{\text{spin}} := \frac{4\pi}{\Delta\sigma_{\text{spin}}} = \frac{4\pi \sin^2 \theta_W}{\alpha}.$$

In the proof-of-concept audit tables we freeze

$$N_{\text{spin}} \approx 398.932.$$

The auxiliary commensurability ratio

$$k(n) := \frac{N_{\text{loop}}^{(n)}}{N_{\text{spin}}}$$

tests compatibility between the macroscopic loop length and the weak-sector angular resolution. In the present PoC, this “spin bridge” is used only as a diagnostic/readout and must not be confused with the purely topological quantity $N_{pass}$.

### E.2.2 Why status flags appear early (grammar-level meaning)

Status flags are not “extra parameters”. They are state tags carried by $S$ that determine rule admissibility in the rewrite system (Sec. E.1.1). Appendix E uses only a small subset of flags required for: (i) local antiparticle bookkeeping checks, (ii) optional display conjugation, (iii) audit-friendly classification of channel gates.

### E.2.3 Local antiparticle bookkeeping map (same-sheet, minimal)

We distinguish a global CPT-like conjugation (sheet-to-sheet, treated elsewhere) from the local same-sheet bookkeeping map used here. When `antiParticleLocal` is ON:

$$Q' = -Q, \quad \text{tilt}'_{34} = -\text{tilt}_{34}, \quad D'_{13} = D_{13} + 12Q.$$

The integer shift is chosen so that the derived colour budget $\Delta d_3$ is preserved while flipping the U(1) winding. Because coarse costs depend on $\text{tilt}'_{34}$ (via a factor such as $\sqrt{1 + \text{tilt}'_{34}}$), particle/antiparticle masses remain identical at this toy level.
### E.2.4 Crosswalk: variable names in the JS proof-of-concept

To facilitate code audit, we map the MCP symbols used in this appendix to the variable names found in the HTML/JS simulator:

$$ 
\begin{array}{l}
D_2 \leftrightarrow \text{d2Count}, \quad D_{13} \leftrightarrow \text{d13Count}, \quad Q \leftrightarrow Q, \quad w_{d3} \leftrightarrow \text{wd3}, \\
\\
\text{tilt}_{34} \leftrightarrow \text{tilt34}, \quad n \leftrightarrow \text{harmN}, \quad \text{antiParticleLocal} \leftrightarrow \text{antiParticleLocal}.
\end{array}
$$

The symbol usage in Appendix E is strict: ($D_2, D_{13}$) always refer to the discrete integers stored in the card, never to the continuous axes.

## E.3 Scale fixing: Compton $\leftrightarrow$ RTQ, and causal-depth closure ($T_{\text{univ}}$)

This chapter fixes the *order-of-magnitude* RTQ scale used in Appendix E and makes explicit that two logically distinct routes coexist:

* (A) Compton / metric anchor (audit-friendly, default for the PoC tables): $N_{\text{loop}}^{(e)}$ is anchored by $\lambda_{C,e}/\ell_0$, and $\eta$ is extracted for the forward map.

* (B) Theory-internal closure (cosmology $\rightarrow$ microphysics): $\eta$ is predicted from the causal depth $T_{\text{univ}}$ and the effective dimensionality $d_{\text{eff}} = 3$, and the electron scale becomes a *consistency check* rather than an input.

We keep both routes explicit so the reader can audit which layer is assumed vs predicted.

### E.3.1 RTQ grain, ticks, and reduced Compton anchors

We use the updated RTQ grain $\ell_0$ (pre-metric network grain, projected to SI length in the emergent metric) and the reduced Compton wavelength $\lambda_C = \hbar/(mc)$ as minimal anchors:

$$ \ell_0 \approx 8.10 \times 10^{-35} \text{ m}, \quad \lambda_{C,e} \approx 3.86 \times 10^{-13} \text{ m}, \quad \lambda_{C,p} \approx 2.10 \times 10^{-16} \text{ m}. $$

These values are used for *scale auditing* only.

#### E.3.1.1 Two clocks (notation guardrail).

Appendix E manipulates a discrete update count $N_{\text{loop}}$. Depending on the interpretation, the “tick” may be taken as:

$$ t_0 := \frac{\ell_0}{c} \quad (\text{RTQ update tick in the pre-metric network}), $$

or as the Planck time $t_P$ when projecting to the emergent metric clock. We keep this distinction explicit in the closure route (Sec. E.3.4) because it induces a deterministic cube-root rescaling.

### E.3.2 Definition of $N_{\text{loop}}^{(e)}$ (Compton mode)

The Compton-anchored loop-length estimate (RTQ count along a 1D loop) is:

$$ N_{\text{loop}}^{(e)} \approx \frac{\lambda_{C,e}}{\ell_0}. $$

For completeness, we also define the associated bulk count (not used as a closure law in Appendix E, only as a magnitude check):

$$ N_{\text{bulk}}^{(e)} \sim \left( \frac{\lambda_{C,e}}{\ell_0} \right)^3. $$

Analogous relations hold for any state $f$ using $\lambda_{C,f}$.
### E.3.3 From $N_{loop}^{(e)}$ to $\eta$ (resolution parameter used by the pipeline)

The numerical proof-of-concept uses a resolution parameter

$$ \eta [\text{RTQ/rad}], $$

interpreted as the number of admissible RTQ micro-steps per radian of coarse phase advance.

In the forward model (Sec. E.4), the loop-length count for mode $n$ is written as:

$$ N_{loop}^{(n)} = \frac{\eta \Phi_{obj} \cos(\Theta_{struct})}{n}, $$

where:

* $\Phi_{obj}$ is the dimensionless objective phase functional (topological/geometric cost proxy),

* $\Theta_{struct}$ is the MCP structural projection angle (guardrail: do not confuse with the Weinberg angle $\theta_W$),

* $n$ is the harmonic/mode index (scenario choice).

In Compton mode, $\eta$ is fixed by applying the above formula to the electron template at $n = 1$:

$$ \eta_C = \frac{N_{loop}^{(e)}}{\Phi_{obj}(e) \cos(\Theta_{struct})}. $$

Once $\eta$ is fixed this way, Appendix E treats it as a shared resolution constant for all the toy-card computations reported in Appendix E.7 (unless explicitly stated otherwise).

### E.3.4 Alternative mode: $\eta$ fixed by causal-depth closure ($T_{univ}$)

If one adopts a theory-internal closure to fix $\eta$ (UV/closure mode), then $N_{loop}^{(e)}$ is no longer an input: it becomes a derived quantity to be compared with $\lambda_{C,e}/\ell_0$.

#### E.3.4.1 Causal depth and dimensional projection.

Let $T_{univ}$ be the causal age of the universe. Define the causal depth in ticks (for a chosen tick $t_*$) as

$$ N_{\text{cosmo}}(t_*) \equiv \frac{T_{\text{univ}}}{t_*}. $$

With effective dimensionality $d_{eff} = 3$ (selected by the dimensional audit, Appendix C), the closure hypothesis used in the main text implies a 1D resolution

$$ \eta_{\text{depth}}(t_*) \equiv (N_{\text{cosmo}}(t_*))^{1/3} = \left(\frac{T_{\text{univ}}}{t_*}\right)^{1/3}. $$

Two explicit choices are useful:

$$ \eta_0 := \eta_{\text{depth}}(t_0) = \left(\frac{T_{\text{univ}}}{t_0}\right)^{1/3}, \qquad \eta_P := \eta_{\text{depth}}(t_P) = \left(\frac{T_{\text{univ}}}{t_P}\right)^{1/3}. $$

#### E.3.4.2 Deterministic rescaling between $t_0$ and $t_P$.

Using $t_0 = \ell_0/c$ and $t_P = \ell_P/c$, one has

$$ \eta_P = \left(\frac{t_0}{t_P}\right)^{-1/3} \qquad \eta_0 = \left(\frac{\ell_0}{\ell_P}\right)^{1/3} \qquad \eta_0 \equiv \kappa^{1/3} \eta_0, \qquad \kappa := \frac{\ell_0}{\ell_P}. $$

This cube-root factor is the only difference between the two clock choices at this level.
### E.3.4.3 Electron closure prediction in closure mode.

In the closure route (cosmology → microphysics), the main-text estimate can be written as

$$N_{\text{loop}}^{(e,M)}(t_{\star}) \equiv N_{\text{pass}} \cdot 4\pi \cdot \eta_{\text{depth}}(t_{\star}) \cdot \cos(\Theta_{\text{struct}}),$$

where $N_{\text{pass}} \approx 2$ encodes the minimal spinor closure convention used in the electron benchmark (see the discussion in the cosmology chapter; Appendix E keeps $N_{\text{pass}}$ explicit to avoid hidden factors).

### E.3.5 Validation: closure vs Compton (electron benchmark)

To make the closure benchmark auditable, we compare the two loop counts:

#### E.3.5.1 (A) Metric / Compton count.

$$N_{\text{loop}}^{(e,C)} = \frac{\lambda_{C,e}}{\ell_0} \approx 4.77 \times 10^{21}.$$

#### E.3.5.2 (B) Causal-depth closure count. Using representative values (Planck-era $T_{\text{univ}}$ and the structural angle used elsewhere in the manuscript), the two clock choices give:

$$N_{\text{loop}}^{(e,M)}(t_0) \approx 2.76 \times 10^{21}, \quad N_{\text{loop}}^{(e,M)}(t_P) = \kappa^{1/3} N_{\text{loop}}^{(e,M)}(t_0) \approx 4.69 \times 10^{21}.$$

Hence the Planck-clock projection is numerically close to the Compton anchor:

$$\frac{N_{\text{loop}}^{(e,M)}(t_P)}{N_{\text{loop}}^{(e,C)}} \approx 0.98-1.02 \quad (\text{percent-level agreement at the present audit resolution}).$$

#### E.3.5.3 Practical consequence for Appendix E.

* In Compton mode, $\eta$ is fixed by Eq. (E.3.3) and used as a shared constant in the PoC tables.

* In closure mode, one sets $\eta = \eta_P$ (or $\eta = \eta_0$ if one insists on the pre-metric tick) and $N_{\text{loop}}^{(e)}$ becomes a consistency check against $\lambda_{C,e}/\ell_0$.

This appendix keeps both readings available because they serve different purposes: the first is a conservative audit anchor; the second is the theory-internal “ab initio” interpretation where $m_e$ inherits the scaling $m_e \propto T_{\text{univ}}^{-1/3}$ (at fixed $\Theta_{\text{struct}}$).

## E.4 Discrete grammar: from canonical cards to sector labels (L/U/D) and gates

This chapter defines the discrete grammar layer: how a canonical fermion card

$$C_0 = (D_2, D_{13}, Q; r_1, r_3, w_{d3}, \text{tilt}_{34}; \text{flags})$$

is mapped to (i) a sector label (L/U/D), and (ii) a small set of discrete gates used by the proof-of-concept. This is a classifier and a consistency interface: it does not choose the harmonic index $n$ (mode selection is handled later by closure scores in Sec. E.4).
### E.4.1 Card-to-holonomy map (coarse budgets as audited invariants)

The grammar layer uses the coarse holonomy budgets ($\Delta d_1, \Delta d_2, \Delta d_3, \Delta d_{13}$) and the associated cycle integers ($c_1, c_2, c_3, c_{13}$) as *audited invariants*. Their canonical definitions and their deterministic derivation from ($D_2, D_{13}, Q$) are given once in Sec. E.2.1.3 and are not repeated here. In this section we only specify how these invariants are consumed by the discrete grammar (sector gates, taxonomy, and UI traceability).

U(1) winding (electric charge) See Sec. E.2.1.3 for the definition of $\Delta d_1$ and $c_1$. In the grammar, $Q$ is the charge template used for sector labeling and UI traceability.

Axis-sum budget $\Delta d_{13}$ as canonical card invariant See Sec. E.2.1.3 for the definition of $\Delta d_{13}$ and its conversion from $D_{13}$. In the grammar, $D_{13}$ is the canonical ledger integer that remains stable across displays and audits.

Spinor (SU(2)) budget See Sec. E.2.1.3 for the PoC convention $\Delta d_2 = 4\pi$. In the grammar, $D_2$ is retained only as a discrete template integer for taxonomy/UI segmentation.

Cycle integers E.4.1.4 Cycle integers See Sec. E.2.1.3 for the definition of ($c_1, c_2, c_3, c_{13}$). In the grammar these compact integers are used as audit-friendly derived readouts (do not confuse them with preset integers).

### E.4.2 autoSector: discrete sector classification (L/U/D) and gate enforcement

The proof-of-concept implements a discrete classifier, denoted `autoSector`, that assigns a sector label

$$ \text{Sector} \in \{\text{L, U, D}\} $$

and enforces a minimal set of discrete gates ($w_{d3}, \text{tilt}_{34}$). For reviewer auditability, we state the rules explicitly as follows.

#### E.4.2.1 Primary split: colour singlet vs fundamental channel

The primary split is based on whether the card is *colour-singlet at the coarse level*, i.e. whether the derived colour budget vanishes:

$$ \Delta d_3 = 0 \iff -\pi D_{13} - 6\pi Q = 0 \iff D_{13} = -6Q. $$

If $\Delta d_3 = 0$, the card is classified as lepton-type:

$$ \Delta d_3 = 0 \Rightarrow \text{L}, \quad w_{d3} \leftarrow 0, \quad \text{tilt}_{34} \leftarrow 0. $$

If $\Delta d_3 \neq 0$, the card is classified as quark-type and placed in the fundamental colour channel:

$$ \Delta d_3 \neq 0 \Rightarrow \text{U/D}, \quad w_{d3} \leftarrow 3. $$

In this minimal release, the additional presets ($r_1, r_3$) are *reported for traceability* (Table 20) but do not change the holonomy identities of Appendix E.4.1.

#### E.4.2.2 Secondary split (quarks): plane vs helicoidal colour coupling

Once a strand is quark-type ($w_{d3} = 3$), the toy grammar distinguishes two quark labels by the discrete coupling mode between $d_3$ and the causal axis $d_4$, encoded by $\text{tilt}_{34}$:

$$ \text{U: } \text{tilt}_{34} = 0 \text{ ("plane colour"), } \text{D: } \text{tilt}_{34} = 2 \text{ ("helicoidal colour").} $$

The causal/helical cost used by the proof-of-concept is a derived factor (not a preset):

$$ \text{helixFactor} = \sqrt{1 + \text{tilt}_{34}^2}, $$

so that only $\text{tilt}_{34}^2$ impacts coarse mass-like costs in this release.
### E.4.2.3 Summary: classifier as a deterministic map

The classifier can be read as the deterministic map

$$ (D_{13}, Q; \text{tilt}_{34}) \mapsto \text{L/U/D}, $$

with gate enforcement:

$$ L \Rightarrow (w_{d3}, \text{tilt}_{34}) = (0, 0), \quad U \Rightarrow (w_{d3}, \text{tilt}_{34}) = (3, 0), \quad D \Rightarrow (w_{d3}, \text{tilt}_{34}) = (3, 2). $$

The harmonic index $n$ is not part of the grammar; it is a scenario choice tested by closure scores in Sec. E.4.

## E.4.3 Canonical integers vs reduced templates (why multiples 3 and 6 exist)

### E.4.3.1 What is the “integer signature” referred to in this section?

In Appendix E, each fermion card carries the canonical ledger integers $(D_2, D_{13}, Q)$. While the grammar rules in Sec. E.4.2 act primarily on $(D_{13}, Q)$, it is convenient (for taxonomy and cross-catalog comparisons) to summarize a motif by a two-integer template pair.

These signatures align with the “Morton integers” used in the “Axis Model” [Morton, 2025]; their derivation in MCP is independent and follows from the triadic structure of the tension field.

Because the bookkeeping conventions enforce fixed factors 3 and 6 (Sec. E.4.3.5), we introduce the reduced template integers

$$ T_2 := \frac{D_2}{3}, \quad T_{13} := \frac{D_{13}}{6}, \quad \text{so that} \quad (D_2, D_{13}) = (3T_2, 6T_{13}). $$

In this section, the expression “integer signature” refers to the (projective) pair $(T_2, T_{13})$, or equivalently to its class under overall integer rescaling:

$$ (T_2, T_{13}) \sim k(T_2, T_{13}), \quad k \in \mathbb{Z}_{>0}. $$

### E.4.3.2 Documentary note: minimal (gcd-reduced) representatives

For classification-only statements and catalog matching, we adopt the minimal representative of this projective class, i.e. we gcd-reduce the reduced pair $(T_2, T_{13})$ (not the raw $(D_2, D_{13})$):

$$ (T_2, T_{13}) \mapsto \left( \frac{T_2}{g}, \frac{T_{13}}{g} \right), \quad g = \gcd(T_2, T_{13}). $$

Beyond this purely arithmetic convention, it is worth noting a historical resonance: several beyond-Standard-Model composite proposals have explored a ternary substructure in which the electron charge $-1$ is realized as the sum of three $-1/3$ constituents. This includes the rishon/preon models introduced independently by Harari [1979], Shupe [1979], as well as the three-strand helon/braid construction of Bilson-Thompson [2005]. In this appendix, such references are cited only as documentary convergences showing that stable integer/ternary patterns recur across unrelated formalisms, not as the derivational basis of the MCP closure mechanism.

### E.4.3.3 Canonical card integers (bookkeeping-safe)

The proof-of-concept and the audited tables compute budgets via

$$ \Delta d_{13} = -\pi D_{13}, \quad \Delta d_1 = 6\pi Q, \quad \Delta d_3 = -\pi D_{13} - 6\pi Q, $$

and apply the local antiparticle update (Appendix E.2.3) directly on $D_{13}$. For this reason $(D_2, D_{13})$ are the canonical integers used in all computations and in the worked tables of Appendix E.7.
### E.4.3.4 Reduced template view (classification-only)

For conceptual classification only, one may define reduced integers

$$T_2 := \frac{D_2}{3}, \quad T_{13} := \frac{D_{13}}{6},$$

and optionally gcd-reduce $(T_2, T_{13})$. In this reduced view, the minimal template set used in Table 20 is

$$(T_2, T_{13}) \in \{(1, 1), (2, 2), (3, 3)\},$$

corresponding respectively to lepton-type, up-type quark, and down-type quark (labels only).

### E.4.3.5 Why the multiples are 3 and 6 (structural, not cosmetic)

The factor 6 on $D_{13}$ is forced by the lepton singlet condition $\Delta d_3 = 0$:

$$\Delta d_3 = 0 \Rightarrow \Delta d_{13} = \Delta d_1 \Rightarrow -\pi D_{13} = 6\pi Q \Rightarrow D_{13} = -6Q.$$

Since $Q \in \mathbb{Z}/3$, the choice $D_{13} = 6T_{13}$ guarantees integer compatibility of the bookkeeping and of the local antiparticle update $D_{13} \mapsto D_{13} + 12Q$.

The factor 3 on $D_2$ is a frozen bookkeeping gauge used to keep the reduced template integer $T_2 = D_2/3$ aligned with the three minimal classes $(1, 1)$, $(2, 2)$, $(3, 3)$, while keeping composite sums (Appendix E.7) in small integers.

### E.4.3.6 Crosswalk (minimal templates)

With the convention above, the three minimal templates correspond to:

$$(1, 1) \leftrightarrow (D_2, D_{13}) = (3, 6), \quad (2, 2) \leftrightarrow (6, 12), \quad (3, 3) \leftrightarrow (9, 18).$$

Reduced templates are best for classification; canonical integers are required for holonomy bookkeeping.

### E.4.4 Compatibility with the local antiparticle bookkeeping

For completeness, we record the key invariant preserved by the local antiparticle bookkeeping (Appendix E.2.3). Using

$$Q' = -Q, \quad D'_{13} = D_{13} + 12Q,$$

the derived colour budget is unchanged:

$$\Delta d'_{3} = -\pi D'_{13} - 6\pi Q' = -\pi(D_{13} + 12Q) + 6\pi Q = -\pi D_{13} - 6\pi Q = \Delta d_{3}.$$

This is precisely why the canonical convention $(D_2, D_{13})$ is kept inside the bookkeeping.

### E.4.5 Documentary note: where the grammar ends (and where closure begins)

Appendix E enforces a strict separation between (i) a deterministic grammar/classifier layer and (ii) a closure / mode-selection layer.

* **Grammar (§E.4):** deterministic sector classification (L/U/D) and gate enforcement ($w_{d3}$, $\tilde{\text{tilt}}_{34}$), plus the audited card-to-holonomy map $(D_{13}, Q) \mapsto (\Delta d_1, \Delta d_3, \Delta d_{13})$. The grammar is a classifier and a consistency interface; it does not select the harmonic index $n$.

* **Pipeline (§E.5):** numerical forward model using $\eta$, $\Phi_{\text{obj}}$ and a scenario choice $n$ to compute $N_{\text{loop}}^{(n)}$, closure scores, and (optionally) masses.

* **Micro-to-macro bridge (§E.6):** justification that coarse holonomy budgets are sums of micro-increments (RTQ steps) and remain audit-compatible.
### E.4.5.1 Audit note: diagnostic “reading” is not part of the grammar

For completeness we record a compact diagnostic reading of regularities that can be observed after harmonic indices $n$ have been inferred externally (e.g. inverse calibration against reference masses, or selection by closure-score diagnostics). For auditability, we first restate the normative part of this boundary in a single table: Table 14 summarizes the only rules that are enforced by the classifier autoSector. Everything that follows after Table 14 is documentary (non-normative): it assumes that harmonic indices $n$ have been inferred externally (e.g. calibration against reference masses, or selection by closure-score diagnostics), and it uses them only to expose post-hoc regularities and diagnostic angles. In particular, this documentary layer does not define $n$, does not modify classifier rules, and does not add gates.

Accordingly, we keep (i) a normative recap table (grammar rules), and (ii) a documentary result table (one representative inferred triplet per sector).

<table><caption>Table 14: Normative grammar recap (audit-critical). This table summarizes only the deterministic map implemented by autoSector and the gates it enforces. It does not define, constrain, or infer the harmonic index $n$.</caption><thead><tr><th>Normative item</th><th>Leptons (L)</th><th>Quarks Up (U)</th><th>Quarks Down (D)</th></tr></thead><tbody><tr><td>Sector label</td><td>L</td><td>U</td><td>D</td></tr><tr><td>Coarse criterion (classi-fier input)</td><td>Δd<sub>3</sub> = 0 (colour singlet)</td><td>Δd<sub>3</sub> ≠ 0 (fundamental colour channel)</td><td>Δd<sub>3</sub> ≠ 0 (fundamental colour channel)</td></tr><tr><td>Enforced gates (classi-fier output)</td><td>(w<sub>d3</sub>, tilt<sub>34</sub>) = (0, 0)</td><td>(w<sub>d3</sub>, tilt<sub>34</sub>) = (3, 0)</td><td>(w<sub>d3</sub>, tilt<sub>34</sub>) = (3, 2)</td></tr><tr><td>Causal/helical cost used in the proof-of-concept</td><td>helixFactor = 1</td><td>helixFactor<br>√1 + 0<sup>2</sup> = 1</td><td>=<br>helixFactor<br>√1 + 2<sup>2</sup> = √5</td></tr></tbody></table>

<table><caption>Table 15: Integer layer (documentary): one representative inferred triplet per sector (example only). These integers are not part of the grammar and are not prescribed by autoSector; they are shown only to instantiate the log-jumps $\Delta_{ij} = \ln(n_j/n_i)$ used by the documentary diagnostics ($\Theta_{\text{eff}}$, chiasm/branch patterns). Non-normative.</caption><thead><tr><th>Sector</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th></tr></thead><tbody><tr><td>L</td><td>1</td><td>207</td><td>3477</td></tr><tr><td>U</td><td>1</td><td>584</td><td>79408</td></tr><tr><td>D</td><td>1</td><td>20</td><td>874</td></tr></tbody></table>

### E.4.5.2 Hierarchical two-regime symmetry and sector-effective angles

### E.4.5.3 Hierarchy (what is compared to what)

For each sector $s \in \{\text{L, U, D}\}$ and each storey $i \to j \in \{1 \to 2, 2 \to 3\}$, we define:

$$R_{ij}^{(s)} := \frac{n_j^{(s)}}{n_i^{(s)}} \in \mathbb{Q}_{>0}, \quad \Delta_{ij}^{(s)} := \ln(R_{ij}^{(s)}).$$

The same $\Delta_{ij}^{(s)}$ is then given two readings:

$$\Delta_{ij}^{(s)} \approx \underbrace{G_{ij}^{(s)}(\alpha, \pi)}_{\text{geometric anchor (constants-only)}} \approx \underbrace{A_{ij} F_{ij}^{(s)}(\Theta)}_{\text{cosmological operator (pivot + deformation)}}.$$
The role of the “effective” angle $\Theta_{\text{eff}}$ is purely documentary: it is defined as the angle that makes the cosmological operator reproduce a chosen target jump (or equivalently match the geometric anchor when that anchor already reproduces the target well).

### E.4.5.4 Pivots and compact constants

We use the two pivots and the compact $(a, b)$ notation:

$$A_{12} := \frac{3\pi}{2}, \quad a := \alpha^{-1}, \quad b := \sqrt{\pi}, \quad A_{23} := \ln\left(\frac{a}{b}\right) = \ln(a/b).$$

<table><caption>Table 16: Documentary hierarchy at storey 1 → 2. Targets are integer ratios $R_{12}$; comparisons are performed in log-space via $\Delta_{12} = \ln(R_{12})$. The same $\Delta_{12}$ admits a constants-only geometric anchor $G_{12}$ and a pivoted cosmological operator $A_{12}F_{12}(\Theta)$. Non-normative.</caption><thead><tr><th>Sector</th><th>Target R<sub>12</sub> = n<sub>2</sub>/n<sub>1</sub></th><th>Geometric anchor G<sub>12</sub></th><th>Cosmological operator<br>A<sub>12</sub>F<sub>12</sub>(Θ)</th></tr></thead><tbody><tr><td>L</td><td>207/1</td><td>G<sub>12</sub><sup>L</sup> = A<sub>12</sub></td><td>Δ<sub>12</sub><sup>L</sup> ≈ A<sub>12</sub>(1 + Θ<sup>2</sup>)</td></tr><tr><td>U</td><td>584/1</td><td>G<sub>12</sub><sup>U</sup> = 1 + ln(π/2) + ln(a) =<br>ln(e(π/2)a)</td><td>Δ<sub>12</sub><sup>U</sup> ≈ A<sub>12</sub>(1 + Θ)</td></tr><tr><td>D</td><td>20/1</td><td>G<sub>12</sub><sup>D</sup> = 3 = ln(e<sup>3</sup>)</td><td>Δ<sub>12</sub><sup>D</sup> ≈ A<sub>12</sub>(1 - Θ)</td></tr></tbody></table>

<table><caption>Table 17: Documentary hierarchy at storey 2 → 3. Same symmetry as Table 16: $\Delta_{23} = \ln(R_{23})$ can be read either as a geometric anchor (constants-only) or as a cosmological deformation around the single pivot $A_{23}$. Non-normative.</caption><thead><tr><th>Sector</th><th>Target R<sub>23</sub> = n<sub>3</sub>/n<sub>2</sub></th><th>Geometric anchor G<sub>23</sub></th><th>Cosmological operator<br>A<sub>23</sub>F<sub>23</sub>(Θ)</th></tr></thead><tbody><tr><td>L</td><td>3477/207</td><td>G<sub>23</sub><sup>L</sup> = ln(a) ln(b)</td><td>Δ<sub>23</sub><sup>L</sup> ≈ A<sub>23</sub>(1 - Θ)</td></tr><tr><td>U</td><td>79408/584</td><td>G<sub>23</sub><sup>U</sup> = ln(a)</td><td>Δ<sub>23</sub><sup>U</sup> ≈ A<sub>23</sub>(1 + Θ<sup>2</sup>)</td></tr><tr><td>D</td><td>874/20</td><td>G<sub>23</sub><sup>D</sup> = ln(a/π)</td><td>Δ<sub>23</sub><sup>D</sup> ≈ A<sub>23</sub>(1 - Θ<sup>2</sup>)</td></tr></tbody></table>

### E.4.5.5 Effective angles (why different $\Theta$ values can appear without adding knobs)

Because each storey uses a different operator $F_{ij}^{(s)}(\Theta)$ (linear vs quadratic, and $\pm$ signs), the inverse map $\Delta_{ij} \mapsto \Theta$ is storey- and sector-dependent. We therefore define purely documentary effective angles by inversion of the cosmological operators:

$$\Theta_{\text{eff},12}^{(L)} = \sqrt{\frac{\Delta_{12}^{(L)}}{A_{12}} - 1}, \qquad \Theta_{\text{eff},12}^{(U)} = \frac{\Delta_{12}^{(U)}}{A_{12}} - 1, \qquad \Theta_{\text{eff},12}^{(D)} = 1 - \frac{\Delta_{12}^{(D)}}{A_{12}}, \qquad (5)$$

$$\Theta_{\text{eff},23}^{(L)} = 1 - \frac{\Delta_{23}^{(L)}}{A_{23}}, \qquad \Theta_{\text{eff},23}^{(U)} = \sqrt{\frac{\Delta_{23}^{(U)}}{A_{23}} - 1}, \qquad \Theta_{\text{eff},23}^{(D)} = \sqrt{1 - \frac{\Delta_{23}^{(D)}}{A_{23}}}. \qquad (6)$$

These $\Theta_{\text{eff},ij}^{(s)}$ are diagnostics, not parameters: they are computed from inferred integers $\{n_g\}$ and the fixed pivots ($A_{12}$, $A_{23}$). They can be compared to the manuscript structural value $\Theta_{\text{struct}}$ via

$$\delta\Theta_{ij}^{(s)} := \Theta_{\text{eff},ij}^{(s)} - \Theta_{\text{struct}}, \quad s \in \{\text{L, U, D}\}, \quad i \to j \in \{1 \to 2, 2 \to 3\}, \qquad (7)$$

to test whether the inferred micro-reading “breathes” around the cosmological equilibrium value.