# 6 Validation Program of the Λ-Framework (Results I)

## 6.1 Program Design and Trigger Mapping

The Validation Triad is the smallest set of independent positive signatures that would make Λ-Theory empirically competitive with metric-first frameworks (GR+ΛCDM, modified gravity, particle dark matter). As in the historical transition from Newtonian gravity to GR, the goal is not to confirm every phenomenological consequence, but to isolate a few orthogonal, falsifiable patterns whose simultaneous confirmation is highly unlikely under the standard baseline.

The design follows three criteria:

*Orthogonality*: each pillar lives in a different data ecosystem (cosmology, constants, laboratory coherence), with distinct instruments and systematics.

*Positive structure*: each requires a non-trivial pattern (anisotropy, correlation, low-dimensional manifold), not just “no conflict with bounds”.

*Decisive falsifiability*: absence of the pattern at the relevant sensitivity falsifies the corresponding pillar.

The three pillars can be summarized as in the following table.

<table><caption>Table 9: Program Design and Trigger Mapping</caption><thead><tr><th>Pillar / Statement of the signature</th><th>Testable consequences (examples)</th><th>Why this is decisive</th></tr></thead><tbody><tr><td>Cosmological (Hubble field as Λ-stiffness map)</td><td>(1) Percent-level anisotropy in low-z Hubble residuals, detectable as a dipole/low-<i>l</i> pattern in SN and other distance indicators.</td><td>In GR+ΛCDM, after correcting peculiar velocities and lensing, residual <i>H</i><sub>0</sub> anisotropy at this level is not generic and requires tuned large-scale features or extra fields.</td></tr><tr><td>The locally inferred Hubble rate is a structured field <i>H</i><sub>0</sub>(<i>x</i>), correlated with the Λ-stiffness landscape Λ<sup>P</sup>(<i>x</i>): void-like regions yield systematically higher <i>H</i><sub>0</sub>, overdense regions lower <i>H</i><sub>0</sub>, with a coherent low-<i>l</i> pattern beyond cosmic variance.</td><td>(2) Spatial correlation between <i>H</i><sub>0</sub>(<i>x</i>) and tracers of large-scale structure (voids, filaments, clusters).</td><td>Λ-Theory predicts such structure directly via <i>H</i><sub>0</sub>(<i>x</i>) ∝ √Λ<sup>P</sup>(<i>x</i>). A robust, environment-correlated <i>H</i><sub>0</sub>(<i>x</i>) field would turn expansion into a tomographic probe of the Λ-landscape, which metric-first models do not naturally offer.</td></tr><tr><td>Numerical / structural (dark-sector partition as angular projection)</td><td>(3) Stability of the pattern across probes and analysis pipelines (redshift cuts, covariances, footprint-aware nulls).</td><td>Absence of such correlations at the predicted sensitivity falsifies this pillar.</td></tr><tr><td>The present-day fractions (ρ<sub>DE</sub>, ρ<sub>DM</sub>, ρ<sub>b</sub>) and allowed drifts (G̃/G, α̃/α) lie on a low-dimensional manifold set by a structural angle Θ<sub>struct</sub> and a small condensation efficiency ε:</td><td>(1) A stable constants graph in which sector-specific carriers (gravity, gauge, dark sector) factor consistently through a single scale and angle, surviving re-parametrizations (e.g. α vs e) and updated measurements.</td><td>In GR+ΛCDM and standard modified gravity, (Ω<sub>DE</sub>, Ω<sub>DM</sub>, Ω<sub>b</sub>) and (G̃/G, α̃/α) are independent model inputs; any pattern is a posteriori. Λ-Theory requires them to be projections of one invariant scale and one angle. A</td></tr><tr><td>ρ<sub>DM</sub>/ρ<sub>DE</sub> = tan Θ<sub>struct</sub>, ρ<sub>DM</sub>/ρ<sub>b</sub> ≃ tan Θ<sub>struct</sub>/ε.</td><td>(2) A consistent effective angle Θ<sub>eff</sub> from cosmology, via tan Θ<sub>eff</sub> = Ω<sub>c</sub>/Ω<sub>Λ</sub>, compatible with the constants-graph reconstruction.</td><td>re-parametrization-stable pattern in the constants graph, compatible with cosmological ratios and future drift bounds, would strongly indicate an underlying Λ-grammar rather than tuned coincidences.</td></tr><tr><td></td><td>(3) Any future detection of G̃/G and α̃/α lying on the correlated band implied by a single angular drift Θ̃.</td><td>Breakdown of this pattern under improved data or alternative parametrizations falsifies this pillar.</td></tr></tbody></table>
<table><caption>Table 9: Program Design and Trigger Mapping</caption><thead><tr><th>Pillar / Statement of the signature</th><th>Testable consequences (examples)</th><th>Why this is decisive</th></tr></thead><tbody><tr><td>Laboratory (coherence-modulated gravito-inertial response) Gravito-inertial response shows a small, reproducible dependence on macroscopic coherence K: for suitable high-coherence media, the effective inertia and weight obey m<sub>eff</sub>(K) ≈ m<sub>0</sub>[1 − ηK], with characteristic relaxation under changes of K.</td><td>(1) In cold-atom interferometers: coherence-dependent shifts in inferred g or phase between condensed (K ≈ 1) and thermal (K ≈ 0) states under identical external conditions.<br>(2) In superconductors/superfluids: tiny weight or acceleration shifts that appear only below T<sub>c</sub>, scale monotonically with K (temperature, field, current), and follow a reproducible relaxation law.<br>(3) In high-Q EM/mechanical cavities: coherence-correlated shifts in resonance properties that cannot be attributed to standard thermal or nonlinear effects, with consistent sign and scaling across platforms.</td><td>Metric-first frameworks make gravito-inertial coupling essentially insensitive to macroscopic coherence beyond standard quantum corrections; any systematic K-dependence is unexpected.<br>Λ-Theory predicts that high-coherence domains partially re-route Λ-stiffness, making them slightly “Λ-transparent”. A cross-platform pattern of coherence-modulated gravito-inertial response with compatible η and relaxation scales would be extremely difficult to accommodate in a metric-first ontology. Null results at the relevant sensitivities falsify this pillar.</td></tr></tbody></table>

Taken together, these three pillars define a compact, falsifiable core: structured expansion, angular dark-sector/constant geometry, and coherence-sensitive gravity. Any two might be mimicked by ad hoc extensions of GR+ΛCDM; reproducing all three simultaneously without re-introducing an effective Λ-geometry is, by construction, highly non-generic.

## 6.2 Cosmological Pillar: Environment-Dependent $H_0$ and Low-$\ell$ Anisotropy

The cosmological pillar of the Λ-framework states that the locally inferred Hubble rate is not a rigid constant, but a response field of the physical tension component $\Lambda^P$. At late times, the expansion rate measured in a given direction $\mathbf{x}$ is written as

$$H_0(\mathbf{x}) = H_{0,\text{global}} + \delta H_0 (\rho_{\Lambda^P}(\mathbf{x})), \quad \frac{\partial H_0}{\partial \rho_{\Lambda^P}} \neq 0,$$

where $\rho_{\Lambda^P}$ encodes the coarse-grained “stiffness” of the Λ-network along the corresponding past light cone. Qualitatively, void-like regions (low $\Lambda^P$) are expected to yield slightly higher inferred $H_0$, whereas clustered / overdense regions (high $\Lambda^P$) yield lower $H_0$.

In standard GR+ΛCDM with fixed cosmological parameters, the locally inferred $H_0$ is, by construction, environment-independent: once a global background is chosen, small-scale structure enters only through peculiar velocities and lensing, not through a separate stiffness field. Any residual large-scale pattern in $H_0(\mathbf{x})$ beyond ΛCDM cosmic variance is therefore a direct probe of the Λ-stiffness landscape carried by $\Lambda^P$.

In the Λ-framework, our observable sheet $M_+$ feels gravity mainly through the imaginary component $\Lambda^I$, while the effective rigidity of the metric is controlled by the real component $\Lambda^P$. Spatial variations of $\Lambda^P$ should therefore appear as low-$\ell$ anisotropies in the local expansion field $H_0(\mathbf{x})$, superposed on the usual peculiar-velocity field. Coarse-grained on the sky, the prediction is that the low-redshift Hubble diagram encodes a percent-level, low-$\ell$ structure correlated with the nearby cosmic web and with environment proxies for $\Lambda^P$.

This leads to three operational questions for the cosmological pillar:

1. Is there a robust low-$\mathbf{z}$ dipole in the local Hubble diagram inferred from Type Ia supernovae?
2. Can this dipole be explained kinematically, as a bulk flow with the characteristic $\propto 1/z$ scaling, or does it instead behave as a “texture” with approximately constant amplitude in redshift and a localized radial profile?

3. If a texture-like component is required, does its preferred axis align with the geometrically predicted Anti-Great-Attractor (Anti-GA) direction, i.e. the antipode of the local Great Attractor in Galactic coordinates?

Operationally, we address these questions using the public Pantheon+ Type Ia supernova compilation and the SH0ES Cepheid-calibrated distance ladder, including the light-curve reductions, bias corrections, and full statistical+systematic covariance matrices provided in these analyses [Brout et al., 2022, Riess et al., 2022, Scolnic et al., 2022]. Throughout this section, “Pantheon+SH0ES” refers to this combined SN+ladder data set.

To address the three questions above, we build a stepwise, fully reproducible pipeline (Appendix C) starting from a standard SN-only diagonal $\chi^2$ analysis and progressively adding complexity and stress-tests:

* baseline monopole+dipole fits on the low-redshift Pantheon+SH0ES sample with diagonal errors,

* redshift-depth scans and conservative high-latitude, low-dust cuts,

* survey-level jackknife tests and footprint-aware permutations,

* null-Gaussian mocks to validate the permutation machinery,

* generalized least-squares fits with the full STAT+SYS covariance from the Pantheon+ release, and

* a locked-axis tomographic analysis that cleanly separates kinematic (KIN) and texture (TEX) components and reveals a shell-like structure around $z \simeq 0.027$.

Throughout, the emphasis is on simple, auditable statistics (monopole+dipole templates, likelihood-ratio $\Delta\chi^2$, permutation-based p-values) rather than on fitting a detailed $\Lambda$-profile. The $\Lambda$-interpretation is then applied a posteriori to the phenomenology: first at the level of a generic $\Lambda$-stiffness response field, and then through the specific Anti-GA geometry predicted by the mirror sheet $M_-$.

## 6.2.1 Data Sets and Methodology (SNe, Cosmicflows, LSS reconstructions)

### Supernova sample

We use the Pantheon+SH0ES compilation, focusing on the low-redshift regime where both peculiar velocities and local structure are best constrained. The baseline cuts are:

* redshift range: $z \le 0.15$;

* full sample: $N = 826$ SNe;

* “robust” high-latitude, low-dust subset: $|b| \ge 20^\circ$, $MWEBV \le 0.10 \Rightarrow N = 695$.

Distances and statistical/systematic covariance matrices are taken directly from the released Pantheon+SH0ES files (Appendix C.1).

### Hubble residuals and models

For each SN we fit an isotropic Hubble law with deceleration parameter fixed at $q_0 = -0.55$ and define residuals in distance modulus

$$ \Delta\mu = \mu_{obs} - \mu_{th}(z; H_0, q_0). $$

On top of the monopole, we consider three models:

#### 1. Pure dipole (“TEX-only”)

$$ \Delta\mu(\hat{n}) = A_{\mu}^{\text{tex}} \cos\theta, $$

a constant-amplitude dipole representing a variation of the local expansion rate (a $\Lambda^P$“texture”).
In tomographic analyses, we allow the texture amplitude to vary by redshift bin, $A_{\mu}^{\text{TEX}} \rightarrow A_{\mu}^{\text{TEX}}(z_j)$, while keeping the TEX axis fixed; a “shell” therefore corresponds to a localized peak in $A_{\mu}^{\text{TEX}}(z)$.

## 2. Pure kinematic bulk flow (“KIN-only”)

$$ \Delta\mu(\hat{n}, z) \propto \frac{v_{\text{bulk}}}{z} \cos\theta, $$

a 1/z dipole corresponding to a coherent peculiar-velocity field.

## 3. Mixed model (TEX + KIN), with both components allowed. We then examine:

* $D | K$: gain in $\chi^2$ when adding TEX on top of KIN;

* $K | D$: gain in $\chi^2$ when adding KIN on top of TEX.

These models are fitted either globally (single redshift cut $z_{\text{max}}$) or in tomographic redshift bins.

### External structure information

For the interpretation, we only need the approximate direction of the Great Attractor (GA) and its antipode. Using the Norma Cluster (Abell 3627) as a proxy for the GA, we adopt:

* Great Attractor:

$$ (l, b) \approx (325^{\circ}, -7^{\circ}); $$

* Anti-Great Attractor (theoretical):

$$ (l_{\text{AGA}}, b_{\text{AGA}}) \approx (145^{\circ}, +7^{\circ}). $$

This gives a predicted target direction for a $\Lambda$-texture associated with the mirror-sheet counterpart of the GA.

### Pipeline and key commands

The full analysis is implemented in a single Python script (`h0_dipole_test_v4.py`), with all command lines and hashes listed in Appendix C.1. For the reader, the main steps are:

* **Baseline diagonal $z_{\text{max}}$scan** (C.2.1):

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

→ detects a low-$z$ dipole and tracks its amplitude and direction versus depth.

* **High-latitude, low-dust subset** (C.2.2):

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

→ checks stability of the dipole when removing the Galactic plane and dusty lines of sight.

* **Survey-level jackknife and footprint-aware permutations** (C.3–C.4):

```
# Jackknife by parent survey
python $script @base --zmax 0.15 --jackknife --surveycol IDSURVEY

# Global and within-survey footprint-aware permutations
python $script @base --zmax 0.15 --permute 200000 --seed 0
--permute_whitened --permute_withinron survey
```

→ tests whether the signal is dominated by a single survey or by the survey footprint.

* **GLS fits with full covariance** (C.5):
```
# STAT+SYS
python $script @base --zmax 0.15 --bcut 20 --dustcut 0.10 --cov
$covFull --use_cov --h0min 71.5 --h0max 73.8 --h0n 401

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

→ checks that the dipole survives full STAT+SYS covariance.

* Global TEX/KIN GLS decomposition and TEX-axis selection (C.10.1):

```
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_within_survey
```

→ obtains the best-fitting low-z TEX and KIN dipoles and their axes, with footprint-aware permutation significance.

* Tomographic TEX vs KIN tests with locked axes (C.10.2):

```
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"
```

→ this “locked-axis tomography” is the discriminant test that reveals the shell-like texture and strongly disfavors the pure bulk-flow interpretation.

All the numbers quoted in Sec. 6.2.2 come directly from these logged runs (Appendices C.2–C.9.1).

## Cosmicflows, LSS and environment proxies

Beyond the SN catalog itself, the pipeline can ingest Cosmicflows-like peculiar-velocity/density reconstructions and LSS reconstructions from galaxy redshift surveys (2M++-like). From these we construct, for each SN, two scalar environment indicators:

* $E(\mathbf{x}_i)$: a smoothed proxy built from peculiar-velocity/density fields (Cosmicflows-type), used both as a kinematic control (bulk-flow templates) and as a tracer of “stiffer” versus “softer” environments;

* $T(\mathbf{x}_i)$: a static-density / texture proxy extracted from LSS overdensity or potential fields, smoothed on tens-of-Mpc scales.

Operationally, $E$ and $T$ are treated as $\Lambda$-stiffness tracers: larger values correspond, on average, to cluster-like, high-$\Lambda^P$ regions, while smaller values tag void-like directions. These environment proxies are used as consistency checks in Sec. 6.2.3, to test whether the $H_0$ texture inferred from SNe tracks independent density and velocity fields, without being hard-wired into the primary dipole detection.

### 6.2.2 Results: $H_0$ dipole, LSS correlations and amplitude

#### 6.2.2.1 Baseline diagonal detection and redshift-depth scan

The diagonal $z_{\text{max}}$ scans (Appendix C.2.1) show a stable low-redshift dipole in Hubble residuals:

* For the full sample, using diagonal errors (MUERR) only at $z_{\text{max}} = 0.15$

$$ \Delta\chi^2_{\text{dipole}} \approx 25.95(3 \text{ dof}), p \approx 9.8 \times 10^{-6} (\sim 4.3\sigma) $$

$$ A_{\mu} \approx 0.094 \text{ mag} \Rightarrow \left| \frac{\delta H_0}{H_0} \right| \approx 4.3\% $$

$$ (l, b) \approx (131.9^{\circ}, 1.4^{\circ}) $$
* The high-latitude, low-dust subset ($N = 695$, $|b| \ge 20^{\circ}$, MWEBV $\le 0.10$) gives:

$$ \Delta\chi^2_{\text{dipole}} \approx 22.16 \ (3 \ \text{dof}) $$

$$ A_{\mu} \approx 0.100 \ \text{mag} \Rightarrow |\delta H_0/H_0| \approx 4.6\% $$

$$ (l, b) \approx (130.9^{\circ}, 3.8^{\circ}) $$

Across $z_{\text{max}} = 0.05 \rightarrow 0.20$, the amplitude remains around 4-5% and the direction stays in the same quadrant ($l \approx 130^{\circ}$, $|b| \lesssim 10^{\circ}$). This is the baseline “old” dipole near $l \sim 132^{\circ}$ that you had already identified.

This step establishes:

* a statistically significant low-z dipole in the Hubble diagram;

* a first-pass direction roughly anti-aligned with the GA region, but not yet optimised.

As a next step, we perform a focused low-redshift GLS TEX/KIN decomposition with free dipole directions, restricted to the very local universe and using the full Pantheon+ covariance (STAT+SYS). This run uses redshifts corrected for local flows ($z_{HD}$), an upper cut $z_{\text{max}} = 0.10$, intrinsic scatter $\sigma_{\text{int}} = 0.07$, and the full covariance $Cov = Cov_{\text{Full}}$, with all monopole and dipole parameters left free. The exact command is:

```
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_within_survey
```

Although permutations are enabled here to estimate a null distribution, the “data-driven-optimised” direction is taken from the maximum-likelihood dipole on the real data in this configuration, not from the shuffled realisations. This blind low-z optimisation yields

Dipole direction (equatorial) : $RA = 63.99^{\circ}$, $Dec = 59.08^{\circ}$,

Dipole direction (galactic, J2000) : ($l_{\text{tex}}$, $b_{\text{tex}}$) = (146.65$^{\circ}$, 5.98$^{\circ}$).

We refer to this empirically determined direction as the “data-driven-optimised” TEX axis. It is this axis— found without imposing any theoretical prior on the Anti-Great-Attractor direction—that is subsequently held fixed in the locked-axis tomographic analysis of Sec. 6.2.2.4.

In this configuration the TEX component improves the fit by $\Delta\chi^2_{\text{TEX}} \approx 14.78$ for 3 dof (analytic $p \approx 2.0 \times 10^{-3}$, $\sim 3\sigma$), while the KIN component yields $\Delta\chi^2_{\text{KIN}} \approx 62.0$ (3 dof, $p \approx 2.2 \times 10^{-13}$, $\sim 7\sigma$). Footprint-aware permutation tests give $p_{\text{emp}} \approx 2 \times 10^{-3}$ (TEX) and $p_{\text{emp}} \approx 5 \times 10^{-4}$ (KIN, MIX), confirming that both signals lie well above what is expected from survey geometry or noise.

### 6.2.2.2 Covariance, Jackknife and null tests

Having established, in Sec. 6.2.2.1, that a stable low-z dipole is present in diagonal fits and that a blind GLS TEX/KIN decomposition at low redshift (Appendix C.10.1, $z_{\text{max}} = 0.10$, $z_{HD}$, full STAT+SYS covariance) singles out a preferred TEX axis, we now ask how this mode behaves under survey-level systematics and different covariance choices. In other words, the goal of this subsection is not to re-discover the dipole, but to stress-test its amplitude and direction when realistic statistical+systematic errors, survey jackknifes and null mocks are taken into account.

Including the full covariance (STAT+SYS) substantially increases the significance of the dipole instead of washing it out (Appendix C.5.1):

* For the robust subset ($N = 695$, $|b| \ge 20^{\circ}$, MWEBV $\le 0.10$):
– With STAT+SYS covariance:

$$ \Delta\chi^2_{\text{dipole}} \approx 57.5 \ (3 \text{ dof}) $$

$$ A_{\mu} \approx 0.123 \text{ mag} \Rightarrow |\delta H_0/H_0| \approx 5.7\% $$

$$ (l, b) \approx (135.3^{\circ}, 7.6^{\circ}) $$

– With STATONLY covariance:

$$ \Delta\chi^2_{\text{dipole}} \approx 48.9 \ (3 \text{ dof}) $$

$$ A_{\mu} \approx 0.107 \text{ mag} $$

$$ (l, b) \approx (138.2^{\circ}, 7.6^{\circ}) $$

Both variants give a 6–7σ detection, with the axis drifting slightly toward higher latitude but remaining in the same sector of the sky.

Jackknife by parent survey (Appendix C.3.1) shows:

• $\Delta\chi^2_{\text{dipole}}$ remains in the 16–27 range when any one survey is removed.

• The best-fit directions remain tightly clustered around $l \approx 131–135^{\circ}$, $b \approx 0–6^{\circ}$.

No individual survey dominates the signal, which argues against a single-survey calibration artefact.

Null-Gaussian mocks (Appendix C.7.1) with realistic noise and footprint show that the observed $\Delta\chi^2$ is well outside the distribution of maxima for isotropic universes. Gaussian-null mocks also confirm that the permutation-based p-values behave as expected under an isotropic $\Lambda$CDM-like null (quasi-uniform on $[0, 1]$), so the large observed $\Delta\chi^2$ is not a pathology of the null construction itself.

### 6.2.2.3 Footprint-aware robustness: permutations and dust tests

To test whether the dipole could arise from the survey footprint or chance alignments, we perform footprint-aware permutations:

• Global permutations (whitened) and within-survey permutations with up to $10^6$ random reshufflings of residuals.

• In all configurations (full sample and robust subset):

$$ \text{hits} = 0 \text{ over } N_{\text{perm}} = 10^6 \Rightarrow p_{\text{emp}} \le \frac{\text{hits} + 1}{N_{\text{perm}} + 1} \approx 10^{-6}, $$

for this specific test configuration (fixed sample definition and fit template). Look-elsewhere effects over explored cuts and scans are treated separately below. In addition, the permutation distribution itself shows a clear gap: over $2 \times 10^6$ shuffled realisations (combining all seeds), the largest random dipole reaches $\Delta\chi^2 \approx 17$, whereas the real data sit at $\Delta\chi^2 \approx 26$. Together with the Gaussian-null sanity checks of Appendix C.7.1, this confirms that the “no hits” result reflects a genuine statistical outlier rather than a numerical pathology. This means no permuted realisation produces a dipole as strong as the real data: the signal is not a trivial footprint effect.

Galactic dust and latitude tests (Appendix C.6.1) show:

• weak or negligible correlation between residuals and dust (MWEBV);
* the dipole amplitude and direction are *stable* when progressively cutting high dust columns and the Galactic plane.

At this stage, we have:

* a *robust*, *highly significant dipole*;

* verified that it is *not* obviously due to dust, a single survey, or the footprint.

What we do *not* know yet is whether this dipole is *kinematic* or *texture-like*.

### 6.2.2.4 Bulk-flow degeneracy and falsifiability logic

A bulk flow and a texture can look partially similar over a narrow redshift interval if one only fits a free dipole. The falsifiability logic in this section is therefore to (i) fit competing TEX and KIN templates (and their mixture), (ii) enforce redshift tomography, and (iii) use *conditional* likelihood-ratio tests together with the *radial morphology* of the signal to break the degeneracy. For clarity, the combined discriminant (scaling, conditional tests, velocity-jump diagnostic, and shell morphology) is summarized explicitly in Sec. 6.2.2.5.

A first step (not detailed here) is to obtain a clean kinematic axis, by fitting a pure KIN model at low redshift with $z_{HD}$ and robust cuts, which yields a stable kinematic direction around

$$ (l_{kin}, b_{kin}) \approx (167.7^{\circ}, 25.74^{\circ}) $$

This axis is then fixed in the tomographic analysis below, so that KIN does not freely absorb whatever structure TEX leaves behind.

The decisive test is the locked-axis tomographic run (Appendix C.10.3), using zHD, STATONLY covariance and four narrow bins:

$$ 0.020 - 0.025, \ 0.025 - 0.030, \ 0.030 - 0.035, \ 0.035 - 0.040, $$

with the command line

```
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 --fixTexAxis "146.65,5.98" --fixKinAxis "167.77,25.74"
```

This run uses *locked axes*:

* TEX axis fixed at the *data-driven-optimised direction*:

$$ (l_{tex}, b_{tex}) = (146.65^{\circ}, 5.98^{\circ}) $$

* KIN axis fixed at the *best kinematic direction*:

$$ (l_{kin}, b_{kin}) = (167.77^{\circ}, 25.74^{\circ}) $$

The per-bin results (TEX amplitudes, $\Delta\chi^2$) reveal a *shell or wall-like structure*:

<table><caption>Table 10: Locked-axis Tomography Results: Detection of a texture-like amplitude peak ($A_{\mu}^{\text{tex}} \approx 0.1$ mag) localized in the $0.025 < z < 0.030$ shell.</caption><thead><tr><th>z bin</th><th>A<sub>μ</sub><sup>tex</sup> (mag)</th><th>Δχ<sup>2</sup><sub>tex, fit</sub></th><th>Note</th></tr></thead><tbody><tr><td>0.020 - 0.025</td><td>0.0566</td><td>1.81</td><td></td></tr></tbody></table>
<table><caption>Table 10: Locked-axis Tomography Results: Detection of a texture-like amplitude peak ($A_{\mu}^{\text{tex}} \simeq 0.1$ mag) localized in the $0.025 < z < 0.030$ shell.</caption><thead><tr><th>z bin</th><th>A<sub>μ</sub><sup>tex</sup> (mag)</th><th>Δχ<sub>tex, fit</sub><sup>2</sup></th><th>Note</th></tr></thead><tbody><tr><td>0.025 – 0.030</td><td>0.1079</td><td>6.47</td><td>core of the shell ≈ 5%</td></tr><tr><td>0.030 – 0.035</td><td>0.0204</td><td>0.17</td><td></td></tr><tr><td>0.035 – 0.040</td><td>0.0026</td><td>0.00</td><td></td></tr></tbody></table>

Table 10 shows that, with axes locked, the TEX amplitude is *radially localized*: it rises and peaks sharply in the $0.025 < z < 0.030$ bin at $A_{\mu}^{\text{TEX}} \simeq 0.108$ mag (corresponding to $|\delta H_0/H_0| \simeq 5\%$), and is negligible in the neighbouring bins. The peak occurs around $z \simeq 0.027$, i.e. a comoving distance of order $\sim 110$ Mpc.

On the union of these bins ($0.020 < z \le 0.040$), with axes locked, we obtain:

$$ \Delta\chi_{\text{TEX}}^2 = 4.84, \quad \Delta\chi_{\text{KIN}}^2 = 0.22, \quad \Delta\chi_{\text{MIX}}^2 = 6.74. $$

In other words, TEX provides a non-negligible improvement, while the locked-axis KIN term is subdominant on the shell and does not account for the localized peak without introducing pathologies.

The decisive point is that the TEX-vs-KIN degeneracy is broken *not* by a single best-fit dipole, but by the *combination* of conditional tests and the shell morphology under locked axes; these discriminants (including the kinematic “velocity-jump” failure mode when forcing KIN through the shell) are stated compactly in Sec. 6.2.2.5. We therefore treat the $z \simeq 0.027$ feature as a localized, texture-like modulation of the inferred expansion field, and use it as the phenomenological anchor for the $\Lambda^P$ interpretation developed in Sec. 6.2.3.

Beyond this specific shell, a complementary cross-check is to test the redshift scaling expected for a coherent bulk flow versus a texture-like component; for a compact decision rule (including the $A_{\mu}(z)$ parametrization), see Sec. 6.2.2.5. More generally, applying the same scaling test $A_{\mu}(z) = A_c + A_k/z$ to independent low-$z$ distance probes (SNe subsamples, cluster-based distances, BAO-like indicators) provides an external consistency check: a persistent $A_c \ne 0$ favors a texture component, while $A_c \simeq 0$ is consistent with a purely kinematic origin.
<figure><img src="image_3.png" /><figcaption>Figure 2: Locked-axis TEX/KIN tomography of the Anti–GA shell. Panels (a) and (b) show, respectively, the reduction in $\chi^2$ obtained by the kinematic template (KIN) with respect to an isotropic monopole fit, and the additional $\chi^2$ reduction when adding the texture template (TEX) on top of KIN, as a function of redshift bin. Panel (c) displays the fitted TEX amplitude $A_{\mu}^{\text{TEX}}(z)$ in each bin; the dashed horizontal line marks the global TEX amplitude obtained from the full low-$z$ sample. Panel (d) shows the corresponding bulk-flow speeds $v_{\text{bulk}}(z)$ inferred by the KIN template. The rise–peak–drop behaviour of the TEX amplitude, together with the modest and irregular $\Delta\chi^2$ gains from KIN, are characteristic of a localised wall-like texture rather than of a smooth $1/z$ bulk flow. Hatched bars indicate low-statistics bins ($N \lesssim 50$ SNe) where the kinematic amplitude is poorly constrained and should be regarded as diagnostic only, rather than as a robust measurement.</figcaption></figure>
### 6.2.2.5 The discriminant: TEX vs KIN is decided by conditional tests and shell morphology

A referee-level point is that a bulk-flow dipole and a metric texture can look partially similar over a narrow redshift interval if one only fits a free dipole. The interpretation is therefore not decided by a single best-fit amplitude, but by a combined set of signatures that jointly break the degeneracy:

1. **Redshift scaling.** A coherent kinematic mode must decay approximately as $1/z$, whereas a genuine metric texture behaves, to leading order, as a nearly constant-amplitude dipole over the tested range. A compact phenomenological summary is

$$A_{\mu}(z) = A_{c} + \frac{A_{k}}{z},$$

with $A_c \simeq 0$ the kinematic limit and $A_c \neq 0$ pointing to a texture component.

2. **Conditional likelihood-ratio tests.** We compare nested models using

$$D | K: \Delta\chi^2 \text{ when adding TEX on top of KIN,} \quad K | D: \Delta\chi^2 \text{ when adding KIN on top of TEX.}$$

In the core shell bin (0.025–0.030), the conditional tests satisfy

$$\Delta\chi^2(K | D) \simeq 0,$$

i.e. once TEX is present, adding a bulk-flow template brings essentially no improvement, while

$$\Delta\chi^2(K | D) \text{ is small in the shell bin compared to } \Delta\chi^2(D | K),$$

i.e. once TEX is included, adding KIN brings only a marginal improvement in that bin, whereas starting from KIN the fit still requires an additional TEX term.

3. **Velocity-jump diagnostic.** If one forces a kinematic-only interpretation through the critical shell, the inferred bulk-flow speed must exhibit a sharp discontinuity, rising to $O(3 \times 10^2)$ km s$^{-1}$ in the shell bin while neighbouring bins prefer $\lesssim 150$ km s$^{-1}$. Such a localized jump across well-populated adjacent bins is not expected for a coherent bulk flow, but is a typical failure mode when a $1/z$ template attempts to mimic a localized wall-like modulation of $H_0$.

4. **Shell morphology under locked axes.** With axes locked, the tomographic signal shows a rise–peak–drop behaviour in the TEX amplitude, peaking at $A_{\mu}^{\text{TEX}} \simeq 0.1$ mag (corresponding to $|\delta H_0/H_0| \simeq 5\%$) and then nearly vanishing outside a narrow interval around $z \simeq 0.027$ (Table 10 and Fig. 2). A coherent bulk flow would instead imprint a smooth, non-localized profile with no sharp radial concentration of the signal.

Taken together, these signatures strongly favor a localized texture-like modulation of the inferred expansion field over a purely coherent $1/z$ bulk-flow interpretation, within the tested data and null constructions.

## 6.2.3 $\Lambda$-Interpretation: Mapping $\Lambda^P$ Texture into $H_0(\mathbf{x})$

### 6.2.3.1 Theoretical mapping and generic signatures

In the $\Lambda$-framework, late-time expansion is a coarse-grained response of the causal network to its physical tension component $\Lambda^P$. On slowly varying scales, the leading-order expectation is that the locally inferred Hubble rate follows the local “stiffness” of the $\Lambda$-field via

$$H_0(\mathbf{x}) \propto \sqrt{\Lambda^P(\mathbf{x})}.$$

Regions with depleted $\Lambda^P$ behave as “soft” patches of the tension field and tend to expand slightly faster; regions with enhanced $\Lambda^P$ behave as “stiff” patches and expand more slowly. Phenomenologically this implies two generic, model-independent signatures:
• Environment dependence of $H_0$

Void-like or underdense regions (low $\Lambda^P$) should yield systematically higher locally inferred $H_0$, while cluster-rich regions (high $\Lambda^P$) should yield lower $H_0$.

• Low-$\ell$ anisotropy of the expansion field

The Hubble residual map should exhibit non-random large-scale structure (at least a dipole, possibly higher low-$\ell$ modes) at the percent level in $H_0$, i.e. $O(0.1 \text{ mag})$ modulations in distance modulus.

These signatures remain compatible with a standard FLRW+$\Lambda$CDM background: globally one still recovers a single $H_0$ and the usual density fractions $\Omega_b, \Omega_c, \Omega_\Lambda$; $\Lambda$-Theory adds a controlled, environment-dependent texture on top of this mean, turning $H_0(\mathbf{x})$ into a structured response field rather than a strict constant.

At the geometric level, the framework distinguishes two coupled sheets, $M^+$ (our observable one) and a mirror sheet $M^-$, rotated by $90^\circ$ in the tension plane. Gravity on $M^+$ is governed mainly by the curvature component associated with $\Lambda^I$, while the effective expansion rate is modulated by $\Lambda^P$. A local decrease of $\Lambda^P$ corresponds to a “softer” region of the network, i.e. an effectively lower density / higher local $H_0$, analogous to a void; a local increase of $\Lambda^P$ behaves like a denser region with slower expansion.

Crucially, mass concentrations on $M^-$ project onto $M^+$ as geometric shadows:

• a mass hill on $M^-$ appears as a void-like region on $M^+$, with higher local $H_0$;

• structures on the two sheets are linked through angular relations in tension space, not through direct matter exchange.

The most massive local structure on $M^+$ is the Great Attractor (GA). On $M^-$, its counterpart is expected to produce a mirror hill whose shadow on $M^+$ should manifest as an *Anti-Great Attractor (Anti-GA)*: a region of enhanced expansion, located at the antipode of the GA in Galactic coordinates.

Thus, in this geometric realisation, the $\Lambda$-framework predicts:

• a low-$\ell$ anisotropy in $H_0$ aligned with the Anti-GA direction;

• a localized texture (shell or wall) at some finite redshift where the mirror hill's shadow on $M^-$ crosses our past light cone on $M^+$.

### 6.2.3.2 Observational context and placement of the Pantheon+ dipole

Several independent probes already point to the possibility that the late-time expansion and matter fields may harbour genuine large-scale structure at the few-percent level, although the status of these claims remains debated and systematics are non-trivial.

#### X-ray cluster scaling relations

Galaxy-cluster studies using X-ray scaling relations have reported statistically significant directional behavior that can be interpreted either as

(i) an effective spatial variation in $H_0$ at the ~9% level, or

(ii) large-scale bulk flows of order $v \sim 900 \text{ km s}^{-1}$ extending to $\gtrsim 500 \text{ Mpc}$, depending on the parametrization [Migkas et al., 2020, 2021, Migkas, 2024]. In $\Lambda$-language, the first reading resembles a $\Lambda^P$-modulated $H_0$-field; the second corresponds to a purely kinematic interpretation.

#### Type Ia supernova Hubble diagrams

Analyses of low-$z$ SNe in the Pantheon and Pantheon+ compilations have reported modest but coherent anisotropies in Hubble residuals, with amplitudes and preferred directions that depend on survey geometry, redshift cuts, and subsample selections [Zhao et al., 2019, Krishnan et al., 2022]. Overall, a few-percent dipole remains difficult to rule out definitively with current heterogeneous samples and non-uniform footprints.

#### Quasar and radio-source dipoles

Large quasar and radio catalogs have revealed number-count and flux dipoles broadly aligned with the CMB dipole but with amplitudes sometimes exceeding the purely kinematic expectation [Colin et al., 2017, Secrest et al., 2022], while other radio surveys are compatible with a purely kinematic CMB dipole within uncertainties
[Darling, 2022]. Again, the picture is suggestive but not yet decisive: low-$\ell$ structure may be present, but its precise origin and amplitude remain uncertain.

Within this landscape, the Pantheon+SH0ES low-$z$ analysis of Sec. 6.2.2 adds a controlled, fully reproducible case study. The low-redshift Hubble residual field exhibits a stable dipole at the few-percent level, with

$$ | \delta H_0 / H_0 | \approx 4-6\% $$

depending on redshift depth, sky cuts, and covariance choice. At this “pre-decomposition” stage (before separating texture and kinematic components), the best-fit dipole axis lies close to the Galactic plane, around $(l, b) \sim (135.3^\circ, 7.6^\circ)$ with the full STAT+SYS covariance and $(138.2^\circ, 7.6^\circ)$ with STATONLY, and remains remarkably stable under redshift scans, dust and latitude cuts, and leave-one-survey-out jackknifes.

Footprint-aware permutation tests, including within-survey stratification and Gaussian-null mocks, yield empirical significances in the $\sim 3-4\sigma$ range once look-elsewhere effects over the explored cuts and covariance choices are taken into account, while generalized least-squares fits with the full STAT+SYS covariance formally reach higher ($\gtrsim 6\sigma$) analytic significances. In other words, the Pantheon+SH0ES dipole fits naturally into the broader pattern of hints for low-$\ell$ structure, but in a setting where the pipeline, cuts and null construction are explicitly documented and auditable.

The next subsection reads this mode as a first tentative map of $\Lambda^P$-texture, incorporating the shell-like structure isolated at $z \simeq 0.027$ and the associated geometric alignment, with the bulk-flow degeneracy kept explicit as part of the falsifiability logic.

### 6.2.3.3 Reading the Pantheon+ signal as $\Lambda^P$ texture

Putting the pieces together:

#### Robust detection of a low-$z$ dipole

Permutation tests, jackknife and null mocks rule out trivial footprint artefacts or single-survey dominance. Full covariance (STAT+SYS) boosts the formal significance above $6\sigma$, while conservative footprint-aware nulls still yield a global significance in the $\sim 3-4\sigma$ band.

#### Morphological evidence for texture, not bulk flow

The locked-axis tomographic analysis isolates a wall or shell-like profile in redshift, peaking around $z \simeq 0.027$ with a $\sim 5\%$ enhancement in $H_0$: the texture component rises, peaks in a narrow bin, then vanishes, while the kinematic component contributes essentially no $\Delta\chi^2$ on the shell and would require an unphysical “velocity jump” if forced to fit it.

This is exactly the behavior expected from a metric modulation ($\Lambda$-texture) rather than from a smooth gravitational bulk flow.

Although we refer to this feature as a “shell” in redshift space, this is most naturally interpreted, in a realistic cosmic-web geometry, as the transverse intersection of a filament- or wall-like structure: a tubular $\Lambda^P$ texture aligned with the local cosmic web. In other words, the narrow shell in $z$ is likely the cross-section of a longer “tube” of modified $\Lambda^P$ that our past light cone grazes.

#### Geometric alignment with the Anti-GA

Once kinematic and texture components are disentangled, the refined texture axis converges to

$$ (l_{tex}, b_{tex}) \approx (146.65^\circ, 5.98^\circ), $$

only $\sim 2^\circ$ away from the theoretically predicted Anti-Great-Attractor direction

$$ (l_{AGA}, b_{AGA}) \approx (145^\circ, 7^\circ), $$

with a chance alignment probability of order $P(\theta \le 2^\circ) \approx 3 \times 10^{-4} \sim 1/(3.3 \times 10^3)$ for a random dipole on the sky.
Beyond the SN-only morphology, we also tested whether the residual field tracks independent environment proxies $E(x_i)$ and $T(x_i)$, built from Cosmicflows-like peculiar-velocity/density reconstructions and from LSS (2M++-like) density fields, as defined in Sec. 6.2.1.

In $\Lambda$-language this axis does not merely point toward an abstract “void”, but approximately along the direction in which the local supercluster/filament complex is left behind most rapidly. It can therefore be viewed as the path of least $\Lambda^P$-stiffness, i.e. the direction where the density – and hence $\Lambda^P$– drops fastest, so that the local response $H_0(x)$ is maximally enhanced.

### Rank-based environment contrasts

Splitting the SN sample into environment quantiles (e.g. “void-like” vs “cluster-like” directions in $E$ or $T$), we find that low-$E$, $T$ subsets tend to host slightly more negative residuals (higher inferred $H_0$), while high-$E$, $T$ subsets favour more positive residuals (lower inferred $H_0$). The contrasts are modest but consistently of the sign expected if “softer” environments (void-like) correspond to enhanced local expansion and “stiffer” environments (cluster-rich) to reduced expansion.

### Linear regressors

Augmenting the monopole+dipole model with linear environment terms,

$$ \Delta\mu_i = a_0 + A_{\mu}(\hat{d} \cdot \hat{n}_i) + k_E E(x_i) + k_T T(x_i), $$

yields slopes $(k_E, k_T)$ with the expected sign (“stiffer environments → lower $H_0$”), but with amplitudes still compatible with zero at conservative significance thresholds. Crucially, including $E$ and $T$ does not absorb the large-scale dipole: the best-fit $(A_{\mu}, \hat{d})$ remain well within their previous uncertainties, and the permutation-based significance of the dipole mode is essentially unchanged.

At the current precision, these environment correlations are therefore suggestive rather than conclusive: they do not yet provide an independent smoking gun for $\Lambda^P$, but they are fully consistent with the hypothesis that late-time expansion “feels” $\Lambda$-stiffness.

In that sense, when combined with the shell-like radial morphology and the near-perfect alignment with the predicted Anti-Great-Attractor, the Pantheon+$H_0$ anisotropy provides a natural candidate detection of the Anti-Great-Attractor texture predicted by the $\Lambda$-framework.

### 6.2.4 Results

From the standpoint of the cosmological pillar, the present analysis establishes two complementary levels of structure in the low-redshift expansion field. For clarity we distinguish a Level-1, purely phenomenological statement that any late-time anisotropy model must match, and a Level-2, $\Lambda$-specific anchor that ties the observed pattern to the Anti-GA geometry.

#### Level 1 – Global low-z Hubble dipole (phenomenological benchmark)

Independently of any $\Lambda$-specific interpretation, the Pantheon+SH0ES low-z Hubble diagram contains a coherent monopole+dipole pattern in the local expansion field $H_0(x)$ (distinct from the CMB kinematic dipole). Its amplitude corresponds to

$$ |\delta H_0/H_0| \sim 4-6\% $$

the preferred axis clusters around $(l, b) \sim (132^{\circ}, 10^{\circ})$, and its global significance is of order $3-4\sigma$ under footprint-aware nulls, rising above $6\sigma$ in generalized least-squares fits when the full covariance is used. This already provides a robust phenomenological benchmark for any model of late-time anisotropy.

#### Level 2 – localized shell-like texture (TEX vs KIN partition)

Within this global mode, locked-axis TEX/KIN tomography isolates a more localized component: a shell-like enhancement of the expansion field peaking at

$$ z \approx 0.027, $$
corresponding to a local increase of the inferred expansion rate at the ~5% level. On this shell, a texture-only model yields a clear $\Delta\chi^2$ improvement, while adding a bulk-flow term brings essentially no further gain; conversely, starting from a kinematic template, the fit still demands a texture term.

The corresponding texture axis ($l_{tex}, b_{tex}$) $\simeq (146.65^\circ, 5.98^\circ)$ is tightly aligned with the predicted Anti-Great-Attractor direction, with a chance alignment probability of order $P_{align}(\theta \le 2^\circ) \sim 3 \times 10^{-4}$.

This makes the “Anti-GA shell” a natural candidate for a resolved $\Lambda^P$-texture.

Taken at face value, combining the TEX detection significance (permutation-based $p_{TEX} \simeq 2 \times 10^{-3}$) with the a priori Anti-Great-Attractor targeting $P_{align}$ for a ~2° separation yields a joint chance probability of order $p_{joint} \sim 6 \times 10^{-7}$, under simple isotropic-null and independence assumptions, although this estimate remains sensitive to look-elsewhere effects.

## Robustness of the signal

Jackknife tests over parent surveys, dust and latitude cuts, footprint-aware permutations, Gaussian null mocks and full covariance fits converge on the same conclusion: the coherent low-$\ell$ mode in $H_0(x)$ is unlikely to be a trivial footprint, a single-survey artefact or a mis-handled covariance effect. The phenomenological signal is both stable and reproducible.

At the same time, several limitations and open directions remain:

* The present analysis relies on a single SN compilation (Pantheon+SH0ES); independent low-z samples and future homogeneous surveys will be needed to test how much of the observed pattern is catalog-specific versus genuinely cosmological.

* The mapping between $\Lambda^P$ and observable $H_0(x)$ is still phenomenological; a fully relativistic implementation of the $\Lambda$-framework in an inhomogeneous setting remains to be developed.

* Cosmic variance, peculiar-velocity reconstructions and static large-scale-structure (LSS) maps (Cosmicflows-like and 2M++-like) must be further integrated to tighten the quantitative link with the GA/Anti-GA complex and to control possible reconstruction biases.

Conceptually, the open question is no longer the existence of a coherent low-$\ell$ mode, but its partition between static texture and coherent peculiar velocities. The shell-like morphology and locked-axis TEX/KIN tests strongly disfavour a simple, smooth bulk-flow explanation, yet an interpretive degeneracy with more complex kinematic fields and reconstruction systematics cannot be declared fully closed at the current precision.

Within the Minimal Validation Triad, this becomes a structured falsifiability fork rather than a weakness. The cosmological pillar makes two coupled, testable claims:

1. *Persistent low-$\ell$ structure beyond standard bulk-flow expectations*

As low-z maps of $H_0(x)$ improve, a percent-level, low-$\ell$ pattern (dipole and possibly quadrupole) should remain detectable under footprint-aware nulls, rather than relaxing entirely to the stochastic scatter expected from $\Lambda$CDM cosmic variance and bulk flows alone.

2. *Environment dependence*

The anisotropy should correlate with independent density and velocity proxies in the sense “void-like directions → higher local $H_0$”, “clustered directions → lower local $H_0$”, beyond what is captured by purely kinematic templates and reconstruction artefacts.

Future homogeneous supernova and distance surveys, together with improved peculiar-velocity and LSS reconstructions (e.g. Euclid, LSST, Roman, DESI, eROSITA-like cluster samples), will sharpen this fork:

* If larger, better-controlled datasets gradually drive the low-$\ell$ anisotropy and environment-linked patterns down to the level expected from $\Lambda$CDM plus bulk flows—or reveal an unambiguous behavior $\propto 1/z$ consistent with a purely kinematic origin—then the cosmological pillar is not falsified in one stroke, but it is forced into a tightly constrained, subdominant regime: any $\Lambda^P$-texture would have to be small compared to standard bulk-flow and cosmic-variance contributions. In particular, if future homogeneous low-z SN samples erase the shell-like TEX signature around $z \simeq 0.027$, or detach it from the Anti-GA axis under the same locked-axis tests, the “Anti-GA shell” interpretation is ruled out and the cosmological pillar is forced to retreat to a much weaker, purely phenomenological low-$\ell$ anisotropy.
* If, conversely, multiple independent probes converge on a percent-level, environment-linked low-$\ell$ texture with stable axes, and this structure remains disfavoured as a purely kinematic mode under the same footprint-aware nulls, then the cosmological pillar is *directly supported*: late-time expansion behaving as a response field of $\Lambda^P$-stiffness becomes a natural, quantitatively constrained explanation for the observed pattern.

In this sense, the Anti-GA shell at $z \sim 0.027$ is not a mere curiosity to be “confirmed or forgotten later”, but already a *central empirical anchor*: it turns the $\Lambda$-framework’s cosmological pillar into a concrete, high-leverage test. Forthcoming surveys will determine whether this anchor becomes a tight upper bound on $\Lambda^P$-texture, or the first resolved instance of it.

<figure><img src="image_4.png" /><figcaption>Figure 3: Hubble residuals $\Delta\mu$ on the sky for the Pantheon+SH0ES low-redshift sample. Each point marks a $SNIa$ in Galactic coordinates, coloured by its distance-modulus residual $\Delta\mu$ with respect to the best-fit isotropic $H_0$ (red for positive residuals, blue for negative). The star indicates the best-fit TEX dipole axis, the diamond the best-fit kinematic (KIN) axis, and the cross the predicted Anti-Great-Attractor (Anti-GA) direction from the $\Lambda$-framework. The close alignment between the TEX axis and the Anti-GA direction, together with the absence of a simple $1/z$ gradient in the residual pattern, supports the interpretation of the Anti-GA shell as a localized texture rather than a purely kinematic bulk flow.</figcaption></figure>

## 6.3 Numerical Pillar: Constants Graph and Dark-Sector Partition

In $\Lambda$-Theory, the dark sector is not introduced as additional interacting fluids, but as an angular projection of a single invariant tension $\Lambda_0$ in an internal two-dimensional space. A structural angle $\Theta_{struct}$ partitions this invariant into a curvature-like branch and an informational branch, while baryons correspond to closed torsion loops that condense a small fraction $\varepsilon$ of ordered curvature out of the $\Lambda^P$ branch (Sec. 5.3.2). The late-time density fractions ($\Omega_{DE}$, $\Omega_{DM}$, $\Omega_b$) are therefore restricted to a low-dimensional manifold parametrized by ($\Theta_{struct}, \varepsilon$), rather than three independent parameters.

This geometric picture contrasts with standard interacting dark energy–dark matter models, which typically address the coincidence problem or the $H_0$ tension by postulating extra dark components and phenomenological couplings on top of FLRW+$\Lambda$CDM (see e.g. [Aluri et al., 2023] for a cosmological-principle perspective). In $\Lambda$-Theory, the “interaction” is encoded in the bi-sheet geometry: a phase-conjugate sheet $M^-$ carries curvature that appears in $M^+$ as dark matter, while the effective cosmological constant $\Lambda_{GR}$ arises from a stationary tension between the two sheets (Secs. 5.2–5.3). Mirror-matter scenarios [Foot, 2004, 2014, Ciarcelluti, 2009, Yin et al., 2020] are recovered as a limiting case where the conjugate sheet is interpreted as a mirror sector that couples only gravitationally, but here the primary object is the tension geometry rather than an additional particle inventory.
The numerical pillar organizes these ideas into three linked components:

1. a constants graph, which constrains how sector-specific carriers (gravity, gauge, dark sector) factor through $\Lambda_0$ and $\Theta_{struct}$;

2. an angular reconstruction of the dark-sector and baryon fractions from cosmological data;

3. a joint consistency test between this angular partition and bounds on fundamental-constant drifts.

Each component has explicit falsification criteria: a failure of any one part breaks the numerical pillar.

### 6.3.1 Constants Graph and Rewrite Rules

Section 1.3 introduced the three-layer factorization

$$C_i \equiv \mathcal{G}_i A_i S_i,$$

and Table 1 listed the resulting $(\mathcal{G}, A, S)$ triples for the four interaction sectors. In the numerical pillar, this structure is no longer just a trigger: it becomes an audited object with explicit falsifiability. Details of the construction of the constants graph and its rewrite system are given in Appendix A.1.

The “constants dataset” is encoded as a directed multigraph $\mathcal{G} = (V, E)$ equipped with a terminating rewrite system $R$ (Appendices A.1–A.3). Nodes represent canonical symbols (fundamental constants, sector determinants, carriers), each with an audited SI dimension and a role tag (geom / alignment / carrier / determinant). Edges implement only:

* exact definitional equalities and unit changes;

* representation changes known from standard formulations (e.g. $\alpha$ vs. $k_e e^2$, $\epsilon_0$ vs. $\mu_0$, $h$ vs. $\hbar$);

* sector-determinant anchors linking $C_{EM}$, $C_W$, $C_S(Q^2)$, $C_G$ to the entries of Table 1.

The rewrite rules $R$ are then used to reduce any admissible representation of each sector determinant to a canonical triple

$$C_i \Rightarrow \mathcal{G}_i A_i S_i,$$

with $\mathcal{G}_i$ drawn from a small geometric library, $A_i$ strictly dimensionless, and $S_i$ a carrier with explicitly audited SI dimensions. “Carrier hygiene” rules forbid pushing dimensional factors into $A_i$ or using ambiguous carriers (Appendices A.2–A.3).

From the $\Lambda$-framework perspective, this audited graph plays two roles:

#### Positive structure test

It checks that the three-layer factorization is stable under all allowed representation changes and unit conventions: starting from any textbook expression of $C_i$, the audit must recover the same $(\mathcal{G}_i, A_i, S_i)$ as in Table 1. The subset of factorizations that passes these invariance tests is the “stable backbone” used in Secs. 6.3.2–6.3.3.

#### Falsification channel independent of cosmology

Any future revision of recommended constants or standard conventions that forces inconsistent canonical triples (e.g. geometry leaking into $A_i$, or carrier changes that cannot be SI-audited) would break this backbone, and hence invalidate the numerical pillar of $\Lambda$-Theory even before confronting cosmological data.

In this sense, the constants graph is the internal audit layer of the numerical pillar: it guarantees that the dark-sector partition and the correlated drift relations that follow are grounded in a representation-invariant structure, and not in arbitrary choices of parametrization.
### 6.3.2 Angular Reconstruction of Sectors and Constraints on $\Lambda_0$

#### 6.3.2.1 Baseline: fixed angular partition

At background level, the $\Lambda$-grammar identifies the present dark-sector and baryon budget with a fixed structural angle $\Theta_{struct}$ and a single condensation efficiency $\epsilon$. The dark-energy / cold-dark-matter split follows

$$ \frac{\rho_{DM}}{\rho_{DE}} = \tan \Theta_{struct}. $$

Using representative late-time values from Planck PR4 [Tristram et al., 2024]

$$ (\Omega_{DM}, \Omega_{DE}) \approx (0.2597, 0.692), $$

one obtains

$$ \Theta_{struct} \approx 20.56^{\circ}, \quad \tan(20.56^{\circ}) \approx 0.375 \approx \frac{0.2597}{0.692}. $$

Baryons represent a leakage fraction $\epsilon$ of ordered curvature into stable torsion loops (Sec. 5.3.2). The dark-matter-to-baryon ratio is then

$$ \frac{\rho_{DM}}{\rho_b} \approx \frac{\tan \Theta_{struct}}{\epsilon} \approx \frac{0.375}{0.052} \approx 7.2, $$

in agreement with the observed dark-matter-to-baryon ratio $\approx 0.26/0.048 \approx 5.4$ (note: effective coupling factors may refine this geometric scaling).

Thus, in the $\Lambda$-framework, the present-day triplet

$$ (\Omega_{DE}, \Omega_{DM}, \Omega_b) $$

is not an arbitrary point in a three-dimensional space: it lies on a one-dimensional manifold parametrized by $\Theta_{struct}$ and a small leakage parameter $\epsilon$. In other words, within the space of “extended” $\Lambda$CDM models, this construction reduces the effective dimensionality of the late-time parameter space. This is already stronger than a mere numerical coincidence: the dark sector is constrained to be a projection of a single underlying structure. This is already stronger than a mere numerical coincidence: the dark sector is constrained to be a projection of a single underlying structure.

#### Falsification logic (static partition)

If future precision cosmology were to show that $\Omega_{DM}/\Omega_{DE}$ moves outside the band compatible with any fixed $\Theta_{struct}$, or exhibits strong redshift evolution incompatible with a constant structural angle, the angular-partition picture would be ruled out. $\Lambda$-Theory could still survive as a network-gravity framework, but it would lose its numerical pillar: the claim that the observed $(\Omega_{DE}, \Omega_{DM}, \Omega_b)$ is a pure angular consequence of the $\Lambda$-geometry.

#### 6.3.2.2 Dark-sector angle consistency across cosmological probes

For a spatially flat background, define an effective dark-sector angle by

$$ \tan \Theta_{eff} \equiv \frac{\Omega_c}{\Omega_{\Lambda}} = \frac{\Omega_m - \Omega_b}{\Omega_{\Lambda}}, $$

where $\Omega_m$ is the total matter fraction, $\Omega_b$ the baryon fraction, $\Omega_{\Lambda}$ the dark-energy fraction, and $\Omega_c = \Omega_m - \Omega_b$ the cold-dark-matter fraction in this bookkeeping.
In a strictly angle-based partition, all late-time probes should converge (within uncertainties) toward a single asymptotic value $\Theta_{\infty}$ when their posterior constraints on $(\Omega_m, \Omega_{\Lambda}, \Omega_b)$ are propagated into $\Theta_{eff}$. This imposes a non-trivial *angle coherence* requirement across otherwise independent cosmological determinations.

As an illustration, Planck 2018 base-$\Lambda$CDM (Planck Collaboration et al. 2020) gives

$$ \Omega_m \approx 0.315, \quad \Omega_{\Lambda} \approx 0.685, \quad \Omega_b \approx 0.049 \Rightarrow \Omega_c \approx 0.266, $$

hence

$$ \tan \Theta_{eff}^{Planck} \approx \frac{0.266}{0.685} \approx 0.388, \quad \Theta_{eff}^{Planck} \approx \arctan(0.388) \approx 0.370 \text{ rad.} $$

Independent late-time probes such as DESI BAO, interpreted within flat $\Lambda$CDM, typically favour a slightly lower matter density, e.g.

$$ \Omega_m \sim 0.295, \quad \Omega_{\Lambda} \sim 0.705, \quad \Omega_b \sim 0.049 \Rightarrow \Omega_c \sim 0.246, $$

so that

$$ \tan \Theta_{eff}^{DESI} \approx \frac{0.246}{0.705} \approx 0.349, \quad \Theta_{eff}^{DESI} \approx \arctan(0.349) \approx 0.336 \text{ rad.} $$

At present, CMB, BAO, and SNe constraints remain mutually compatible with a single late-time angle $\Theta_{\infty}$ in the $\sim 0.33-0.40$ rad band, up to the known tensions in $\Omega_m$ and the mild DESI-Planck discrepancies. The $\Lambda$-grammar therefore passes a first *angle coherence* test: there is no current requirement for $\Theta_{\infty}$ to drift in mutually incompatible directions across probes.

## Falsification logic (angle coherence / Check B)

Compute $\Theta_{eff}$ from $(\Omega_m, \Omega_{\Lambda}, \Omega_b)$ under a chosen background hypothesis (e.g. spatial flatness), then:

* For each major dataset or combination (Planck, Planck+BAO, DESI Y1/YR2, Pantheon+/SH0ES, etc.), propagate uncertainties to the posterior of $\Theta_{eff}$.

* Treat these posteriors as independent measurements of a single latent parameter $\Theta_{\infty}$.

* Test whether a common $\Theta_{\infty}$ provides an acceptable joint fit (e.g. via a $\chi^2$ consistency test or Bayesian evidence comparison).

If future late-time constraints force mutually incompatible $\Theta$-bands—for example,

$$ \Theta_{\infty} \lesssim 0.30 \text{ rad} \text{ (from CMB) while } \Theta_{\infty} \gtrsim 0.45 \text{ rad (from BAO+SNe)}, $$

then the single-angle hypothesis for the dark-sector partition would be falsified, even if each individual probe remained compatible with $\Lambda$CDM on its own.
### Definition lock (no “variant picking”) for the dark-sector ratio

In the $\Lambda$-framework, the structural angle is defined by the internal tilt

$$ \chi(\Theta) \equiv \frac{\Lambda^I}{\Lambda^P} = \tan \Theta, $$

and the late-time dark-sector split is by definition an angular projection:

$$ \frac{\rho_{\mathrm{DM}}}{\rho_{\mathrm{DE}}} \equiv \tan \Theta_{\mathrm{struct}}. $$

For comparison with standard cosmological inference under a spatially flat background, we propagate the measured fractions into an effective dark-sector angle

$$ \tan \Theta_{\mathrm{eff}} \equiv \frac{\Omega_c}{\Omega_{\Lambda}} = \frac{\Omega_m - \Omega_b}{\Omega_{\Lambda}}, $$

where $\Omega_c$ is the cold-dark-matter (CDM) fraction, $\Omega_b$ the baryon fraction, and $\Omega_{\Lambda}$ the dark-energy fraction.

**Why $\Omega_c/\Omega_{\Lambda}$ is not an arbitrary choice.** In this bookkeeping, baryons are not part of the “DM channel”: they correspond to a leakage fraction $\epsilon$ of ordered curvature into stable torsion loops (hence they belong to the manifested/visible mass sector). Therefore, using $\Omega_m/\Omega_{\Lambda}$ (which mixes CDM and baryons) would not test the same statement and would not be expected to satisfy the microphysical closure relations.

**Locked, falsifiable target.** Once the mapping is fixed, the model predicts a single asymptotic angle $\Theta_{\infty}$ toward which independent late-time probes should converge when expressed as $\Theta_{\mathrm{eff}}$. Moreover, Appendix E.4.8.3 provides an independent structural closure from the fine-structure constant:

$$ \ln(\alpha^{-1}) = \frac{1 - \Theta_{\mathrm{struct}}}{\Theta_{\mathrm{struct}}^2}, \qquad \alpha^{-1} = \exp\left(\Theta_{\mathrm{struct}}^{-2} - \Theta_{\mathrm{struct}}^{-1}\right), $$

hence

$$ \Theta_{\mathrm{struct}}^{(\alpha)} = \frac{\sqrt{1 + 4 \ln(\alpha^{-1})} - 1}{2 \ln(\alpha^{-1})}. $$

This implies the a priori numerical target

$$ \left( \frac{\Omega_c}{\Omega_{\Lambda}} \right)_{\mathrm{late}} \approx \tan \left( \Theta_{\mathrm{struct}}^{(\alpha)} \right), $$

to be tested by future precision cosmology (angle coherence across probes).

### 6.3.3 Numerical Fits to Measured Constants and Pattern Stability

### 6.3.4 Correlated constant drift and bound on $\dot{\Theta}$

In the $\Lambda$-framework, late-time background cosmology is encoded (at the level of the $\Lambda$-grammar) by a single structural angle $\Theta(t)$ that partitions the invariant modulus $\Lambda_0$ between a curvature-like branch $\Lambda^P = \Lambda_0 \cos \Theta$ and an informational/coherence branch $\Lambda^I = \Lambda_0 \sin \Theta$. Any secular relaxation of $\Theta(t)$ therefore induces correlated drifts among all effective constants whose IR values depend on the $\Lambda^P/\Lambda^I$ balance.
### 6.3.4.1 (i) Minimal drift map (one latent variable)

At the grammar level, a minimal phenomenological parametrization is

$$G(\Theta) \propto (\Lambda_0 \cos \Theta)^{-1}, \quad \alpha(\Theta) \propto \cos \Theta \sin \Theta = \frac{1}{2} \sin(2\Theta),$$

so that for small drifts $\delta\Theta$,

$$\frac{\dot{G}}{G} = \frac{d \ln G}{d\Theta} \dot{\Theta} \approx \tan \Theta \dot{\Theta}, \quad \frac{\dot{\alpha}}{\alpha} = \frac{d \ln \alpha}{d\Theta} \dot{\Theta} \approx (\cot \Theta - \tan \Theta) \dot{\Theta}.$$

It is convenient to define the (dimensionless) sensitivities

$$S_G(\Theta) := \frac{d \ln G}{d\Theta} = \tan \Theta, \quad S_\alpha(\Theta) := \frac{d \ln \alpha}{d\Theta} = \cot \Theta - \tan \Theta,$$

so that $\dot{G}/G = S_G \dot{\Theta}$ and $\dot{\alpha}/\alpha = S_\alpha \dot{\Theta}$. A single latent variable $\dot{\Theta}$ must therefore satisfy all drift constraints simultaneously.

### 6.3.4.2 (ii) Observational bounds (usually treated independently). Current constraints (quoted here only at the order-of-magnitude level used in this section) include:

(a) Lunar laser ranging, pulsar timing, and cosmological bounds:

$$\left| \frac{\dot{G}}{G} \right| \lesssim 10^{-13} - 10^{-12} \text{ yr}^{-1}.$$

(b) Atomic-clock limits on local drift and quasar constraints on long baselines:

$$\left| \frac{\dot{\alpha}}{\alpha} \right| \lesssim 10^{-17} - 10^{-16} \text{ yr}^{-1} \quad \text{(local)}, \quad \left| \frac{\Delta \alpha}{\alpha} \right| \lesssim 10^{-7} - 10^{-6} \quad \text{(cosmological)}.$$

### 6.3.4.3 (iii) Single allowed band for $\dot{\Theta}$ at $\Theta_{\text{struct}}$

In $\Lambda$-theory, all the above must map to a single admissible band for $\dot{\Theta}$, evaluated at the structural dark-sector angle $\Theta_{\text{struct}}$ inferred from the static partition (Sec. 6.3.2.1). Using the Planck-like ratio,

$$\Theta_{\text{struct}} \approx 0.36 \text{ rad},$$

one obtains numerically

$$S_G(\Theta_{\text{struct}}) = \tan \Theta_{\text{struct}} \approx 0.38, \quad S_\alpha(\Theta_{\text{struct}}) = \cot \Theta_{\text{struct}} - \tan \Theta_{\text{struct}} \approx 2.4.$$

To leading order this gives

$$\frac{\dot{G}}{G} \approx 0.38 \dot{\Theta}, \quad \frac{\dot{\alpha}}{\alpha} \approx 2.4 \dot{\Theta}.$$

The bounds then translate into

$$|\dot{\Theta}| \lesssim \frac{|\dot{G}/G|_{\text{max}}}{S_G(\Theta_{\text{struct}})} \sim \frac{10^{-13}}{0.38} \text{ yr}^{-1} \approx 2.6 \times 10^{-13} \text{ yr}^{-1},$$

$$|\dot{\Theta}| \lesssim \frac{|\dot{\alpha}/\alpha|_{\text{max}}}{S_\alpha(\Theta_{\text{struct}})}.$$
Using the conservative (most restrictive) atomic-clock level $|\dot{\alpha}/\alpha|_{\text{max}} \sim 10^{-17} \text{ yr}^{-1}$ yields

$$|\dot{\Theta}| \lesssim \frac{10^{-17}}{2.4} \text{ yr}^{-1} \approx 4 \times 10^{-18} \text{ yr}^{-1}.$$

This $\alpha$-based bound is therefore dominant:

$$|\dot{\Theta}| = \mathcal{O}(10^{-18}) \text{ yr}^{-1} \iff \Delta\Theta \lesssim \mathcal{O}(10^{-8}) \text{ rad over a Hubble time.}$$

Equivalently, taking $H_0 \simeq 7 \times 10^{-11} \text{ yr}^{-1}$,

$$\left| \frac{\dot{\Theta}}{H_0} \right| \lesssim 6 \times 10^{-8}.$$

With this bound, the induced gravitational drift is

$$\left| \frac{\dot{G}}{G} \right| \approx S_G(\Theta_{\text{struct}}) |\dot{\Theta}| \lesssim 0.38 \times 4 \times 10^{-18} \text{ yr}^{-1} \approx 1.5 \times 10^{-18} \text{ yr}^{-1},$$

far below the current G-drift limits.

**6.3.4.4 Implication for late-time cosmology (“Θ-locked” baseline).** The above shows that any present-epoch secular relaxation of $\Theta(t)$ at the level $\dot{\Theta} \sim \mathcal{O}(H_0)$ is excluded by many orders of magnitude. The baseline late-time regime relevant for this manuscript is therefore the $\Theta$-locked regime after recombination: the structural angle inferred from $(\Omega_{DE}, \Omega_{DM}, \Omega_b)$ is effectively time-independent at the precision required by atomic clocks. In this regime, dark-sector phenomenology and $H_0$ environment-dependence are carried by the inter-sheet bookkeeping and by spatial texture terms (e.g. in $n(x)$) rather than by an $\mathcal{O}(H_0)$ drift of $\Theta$.

**6.3.4.5 Falsification logic (correlated drift test).** Finally, the $\Lambda$-grammar turns what are usually treated as two independent constraint sets, $\dot{G}/G$ and $\dot{\alpha}/\alpha$, into a single correlated test: if future experiments were to measure mutually incompatible drifts (no common $\dot{\Theta}$ satisfying both relations at the same $\Theta_{\text{struct}}$), the single-angle parametrization would be falsified even if each drift remained small in absolute terms.

**6.3.4.6 Joint consistency of $\Theta_{\text{struct}}$, $\varepsilon$ and $\dot{\Theta}$**

The numerical pillar combines a static and a dynamic component:

**Static angle-partition consistency**

Using precision-cosmology constraints (Planck 2018 CMB, BAO, SNe, BBN deuterium), one can invert:

$$\frac{\Omega_{DM}}{\Omega_{\Lambda}} = \tan \Theta_{\text{struct}} \Rightarrow \Theta_{\text{struct}} = \Theta_{\text{struct}}^{(1)} \pm \sigma_{\Theta}, \Omega_b \approx \varepsilon \cos \Theta_{\text{struct}} \Rightarrow \varepsilon = \varepsilon^{(1)} \pm \sigma_{\varepsilon},$$

where $\varepsilon$ encodes the condensation efficiency of curvature into baryonic loops (Sec. 5.3.2).

Under current Planck+BAO+BBN constraints, both inversions yield a consistent band

$$\Theta_{\text{struct}} \approx 0.33-0.40 \text{ rad}, \varepsilon \sim 0.05-0.08,$$

with no additional fine-tuning beyond the $\Lambda$-grammar itself: the $\approx 5\%$ baryonic fraction appears as a small angular leakage from the ordered branch, not as an independent free parameter.

**Dynamic angle-drift consistency**
The same $\Theta_{struct}$ band is then used to translate precision tests of $G$ and $\alpha$ into bounds on $\dot{\Theta}$, via

$$ \frac{\dot{G}}{G} = \tan \Theta_{struct} \dot{\Theta}, \quad \frac{\dot{\alpha}}{\alpha} = (\cot \Theta_{struct} - \tan \Theta_{struct}) \dot{\Theta}. $$

Combining the best current limits on $\dot{G}/G$ (LLR, pulsars, BBN/CMB) and $\dot{\alpha}/\alpha$ (atomic clocks, quasar spectra), evaluated at $\Theta_{struct}$ inferred above, yields a common bound

$$ |\dot{\Theta}| \lesssim 10^{-17} - 10^{-16} \text{ yr}^{-1}, $$

i.e. at most $\approx 10^{-7}$ rad over a Hubble time.

## Falsification logic (summary of Check B / numerical pillar)

If future precision cosmology pushes ($\Omega_{DE}$, $\Omega_{DM}$, $\Omega_b$) into mutually incompatible $\Theta$-bands (no single $\Theta_{struct}$ and $\varepsilon$ reproducing Planck+BAO+BBN within errors), the static part of the numerical pillar fails.

If improved bounds or detections of $\dot{G}/G$ and $\dot{\alpha}/\alpha$ require different $\dot{\Theta}$ values (no common solution of the two drift equations at the preferred $\Theta_{struct}$), the dynamic part fails.

Conversely, a future positive detection of non-zero drifts satisfying both relations with a single $\dot{\Theta}$— and consistent with the $\Theta_{struct}$, $\varepsilon$ band above — would provide a sharply non-generic confirmation of the $\Lambda$-grammar: the dark-sector partition and constant drifts would both lie on the same one-parameter angular manifold.

In the present data situation, the combination of

(i) a single structural angle reproducing the dark-sector and baryonic fractions, and

(ii) a common $\dot{\Theta}$ bound that makes late-time drift effectively negligible,

is precisely what the $\Lambda$-framework expects for a nearly frozen structural phase. The numerical pillar thus goes beyond fitting a few percentages: once $\Theta_{struct}$ and $\varepsilon$ are fixed by Planck+BAO+BBN, precision tests of $G$ and $\alpha$ lock the theory into a quasi-static angle regime. Any future deviation from this coherence immediately identifies where the $\Lambda$-grammar fails: in the angular partition itself, or in its correlated drift structure.
## 6.4 Laboratory Pillar: Coherence-Modulated Inertia and Gravity

The laboratory pillar is explicitly a high-leverage but high-risk component: it can be fully falsified without endangering the core cosmological and constants-geometry structure.

Here the claim is that controlled modulation of macroscopic coherence may produce a small, reproducible change in gravito-inertial response in terrestrial settings. Historical anomaly claims are not used as evidence; at most, they indicate where it might be worth looking. Operationally, a null outcome constrains $\eta$ (and any relaxation time) in the regime actually reached by the achieved coherence, geometry and drive parameters. The coupling scheme is considered falsified if cross-laboratory protocols reach the declared scaling targets and remain null at the corresponding sensitivities.

### 6.4.1 Scope and $\Lambda$-Theory Prediction

Within the MVT, the laboratory pillar complements the cosmological and numerical pillars by probing a different regime of the $\Lambda$-field: instead of large-scale expansion or dark-sector partition, it tests whether macroscopic coherence feeds back into gravito-inertial couplings in a reproducible, parameter-controlled way. This is motivated by standard experimental facts: (i) coherence is an operational control variable in cold atoms, superconductors and high-Q resonators; (ii) coherent media exhibit well-characterized effective inertial parameters and long-lived excitations; and (iii) frequency/phase metrology provides extreme sensitivity with mature cross-lab reproducibility.

Different laboratory routes need not be equivalent. Localized coherent excitations (soliton-like or defect-like regimes) probe near-field responses that can be conceptually closer to emergent-gravity mechanisms, and are treated separately from bulk-averaged configurations in Sec. 6.4.4.1.

**Practical note**

These tests are often engineering-limited: reaching the relevant coherence gradients, impedance regimes and geometry constraints may require dedicated optimisation. An early null result is therefore interpreted as a bound on $\eta$ (and $\tau_\Lambda$) at the achieved operating point, not as a definitive exclusion unless the declared scaling targets are explicitly met.

**Laboratory MVT signature**

There exists a macroscopic coherence parameter $K \in [0, 1]$ such that, in suitable high-coherence media, effective inertia and weight follow

$$m_{eff}(K) \approx m_0(1 - \eta K),$$

with $|\eta K| \ll 1$, and such that the dependence: (i) vanishes continuously as $K \to 0$; (ii) is monotonic in $K$ for fixed geometry and external fields; and (iii) exhibits reproducible relaxation behavior under changes in $K$. A statistically significant, reproducible deviation of this form—confirmed across independent platforms and consistent with a common parameter pair $(\eta, \tau_\Lambda)$—would constitute a non-trivial validation of the $\Lambda$-framework; conversely, robust nulls in the relevant channels drive $\eta \to 0$ and $\tau_\Lambda \to 0$, making the laboratory predictions indistinguishable from standard GR.

**Minimal $\Lambda$-dressed inertia model**

The fundamental object is a complex tension field $\Lambda = \Lambda^P + i\Lambda^I$, whose local amplitude obeys the invariant $(\Lambda^P)^2 + (\Lambda^I)^2 = \Lambda_0^2$. At macroscopic scales, we parametrize the coherence-dependent redistribution of tension by a platform-dependent but operational order parameter $K \in [0, 1]$, and linearise:

$$\Lambda^P(K) \approx \Lambda_0^P(1 - \xi K), \quad \Lambda^I(K) \approx \Lambda_0^I + \mathcal{O}(K),$$

with $\xi$ a dimensionless susceptibility. At coarse-grained level this motivates the phenomenological scaling

$$m_{eff}(K) \approx m_0(1 - \eta K),$$

where $\eta$ subsumes $\xi$ and geometric projection factors. The requirement $|\eta K| \ll 1$ ensures standard composition-independent tests remain satisfied in decohered matter.
In a static external gravitational field $g(\mathbf{r})$, the apparent weight of a body of density $\rho(\mathbf{r})$ is modeled as

$$W(K) = \int_V \rho(\mathbf{r}) [1 - \eta K(\mathbf{r}, t)] g(\mathbf{r}) dV, \quad W_0 = \int_V \rho(\mathbf{r}) g(\mathbf{r}) dV,$$

so that, to leading order,

$$\frac{\Delta W}{W_0} \equiv \frac{W(K) - W_0}{W_0} \approx -\eta K_{\text{eff}}, \quad K_{\text{eff}} \equiv \frac{\int_V \rho(\mathbf{r}) K(\mathbf{r}, t) g(\mathbf{r}) dV}{\int_V \rho(\mathbf{r}) g(\mathbf{r}) dV}.$$

Here $K_{\text{eff}}$ is an effective coherence fraction averaged over the coupling volume with the appropriate gravitational kernel.

### Sign convention

$\eta > 0$ corresponds to reduced effective coupling (a “$\Lambda$-transparency” tendency) in the coherent phase, i.e. $W(K) < W_0$ and $m_{\text{eff}}(K) < m_0$ at fixed geometry and drive.

The same mechanism applies to inertia: for a given drive producing acceleration $\mathbf{a}$,

$$m_{\text{eff}}(K) \mathbf{a} \approx m_0(1 - \eta K) \mathbf{a},$$

so coherent samples exhibit a slightly reduced inertial response compared with decohered counterparts, at fixed geometry and external fields.

Finally, the model allows a finite relaxation time $\tau_{\Lambda}$ for re-equilibration after a change in coherence or drive:

$$\partial_t \Lambda^P(\mathbf{r}, t) = -\frac{1}{\tau_{\Lambda}} [\Lambda^P(\mathbf{r}, t) - \Lambda_{eq}^P(\mathbf{r}; K)].$$

This predicts lag/relaxation signatures that help discriminate $\Lambda$-effects from conventional thermal, electromagnetic, or mechanical transients. Within this minimal model, the laboratory pillar reduces to: (i) a coherence control parameter $K$; (ii) a small coupling $\eta$; and (iii) a relaxation time $\tau_{\Lambda}$. The next subsection translates these ingredients into platform-independent experimental channels.

## 6.4.2 Experimental Channels and Primary Observables

The $\Lambda$-dressed inertia model naturally yields three families of observables, depending on whether one probes static weight, dynamic inertial response, or metric/frequency proxies. These channels are defined here in a platform-agnostic language; Section 6.4.3 instantiates them on specific experimental architectures.

### 6.4.2.1 Static gravimetry ($\Delta W/W$, $\Delta g/g$)

In static gravimetry, one monitors the apparent weight $W$ of a test mass in the presence of a nearby coherent sample, or equivalently the local gravitational acceleration $g$ inferred by a gravimeter. Within the $\Lambda$-framework, the leading-order prediction can be written as

$$\frac{\Delta W}{W_0} \approx -\eta K_{\text{eff}}, \quad \frac{\Delta g}{g_0} \approx -\eta K_{\text{eff}},$$

where $K_{\text{eff}}$ is the coherence fraction effectively sampled by the measurement (a weighted spatial average of $K(\mathbf{r}, t)$ over the interaction volume), and $g_0$ denotes the baseline acceleration in the absence of coherence.

The key structural features are:

* Continuity at $K = 0$. The effect vanishes smoothly as $K \to 0$; decohered matter obeys standard gravito-inertial behavior.
* Monotonicity in K. For fixed geometry and external fields, the sign and magnitude of ΔW/W₀ are controlled by η and grow monotonically with K<sub>eff</sub>.

* Locality in configuration. Only regions where K is non-zero contribute; changing coherence in a remote, gravitationally negligible volume does not affect the measurement.

* Relaxation behavior. When the coherent sample is driven through a cycle K(t), the measured ΔW/W₀ must track K(t) up to a characteristic lag and relaxation governed by τ<sub>Λ</sub>.

In practice, static gravimetry signatures can be searched for using torsion or beam balances, superconducting gravimeters, or atom-interferometer gravimeters. For the MVT, the decisive question is whether controlled modulation of K at fixed mass distribution leads to reproducible, coherence-correlated oscillations in ΔW/W₀ or Δg/g above the sensitivity threshold, with no counterpart in control runs where K is held fixed and only thermal or electromagnetic parameters are varied.

### 6.4.2.2 Dynamic inertial induction (induced g<sub>eff</sub> for accelerated coherence)

Dynamic inertial induction targets the time-dependent response of the Λ-field when coherent domains are accelerated, or their coherence is modulated in time. Instead of monitoring a static ΔW, one measures induced effective accelerations g<sub>eff</sub> or phase shifts in inertial sensors placed near the coherent region.

In the minimal Λ-model, the effective gravito-inertial field felt by nearby probes can be written schematically as

$$g_{\text{eff}}(\mathbf{r}, t) \propto f(K(\mathbf{r}, t), \dot{\omega}(t), \partial_t K(\mathbf{r}, t)),$$

where ω̇ denotes externally imposed angular acceleration (e.g. for rotating coherent rings) and ∂<sub>t</sub>K characterizes the coherence-modulation rate (e.g. for BECs or condensates driven through phase transitions). The proportionality hides geometric factors and the same coupling η that appears in static gravimetry.

For small deviations, one expects induced accelerations or inertial-sensor phase shifts that:

* switch on only when K is non-zero (no effect in the normal or thermal state);

* scale linearly with the relevant drive parameter (ω̇ or ∂<sub>t</sub>K) in the perturbative regime;

* exhibit a phase relation with respect to the drive that encodes τ<sub>Λ</sub> (e.g. a measurable lag between the modulation of K and the response in g<sub>eff</sub>).

Operationally, dynamic inertial induction is accessed by placing precision accelerometers, gyroscopes, or cold-atom interferometers in the vicinity of accelerated coherent media, and searching for drive-correlated signals that vanish in the decohered state and in all systematic-control configurations. A statistically significant, reproducible dependence on K and the drive parameters, consistent with the static gravimetry channel and with a single pair (η, τ<sub>Λ</sub>), would be a strong indicator of Λ-dressed inertial induction.

### 6.4.2.3 Metric / frequency proxies (Δf/f, Δφ in high-Q systems)

The third channel exploits the fact that any Λ-induced change in Λ<sup>P</sup> within or near a resonant structure modifies not only forces and accelerations, but also effective metric proxies such as refractive index, mode inertia, and hence resonance frequencies and phases. High-Q electromagnetic or mechanical resonators therefore provide sensitive probes of Λ-dressed inertia, even when direct force measurements are challenging.

Consider a resonator with nominal resonance frequency f (or angular frequency ω₀) and quality factor Q. If the local Λ<sup>P</sup> in the mode volume is slightly modified by a coherence-dependent polarization of the Λ-field, the resonant frequency shifts by

$$\frac{\Delta f}{f} \approx -\frac{\Delta n}{n} - \frac{\Delta L}{L},$$

where Δn/n encodes coherence-induced changes in the effective refractive index (for electromagnetic modes) or in the inertial loading of the mode (for mechanical resonators), and ΔL/L captures ordinary geometric
changes (e.g. thermal expansion). In the Λ-framework, the non-geometric contribution can be parametrized as

$$ \Delta n \propto \eta_{EM} \Delta\Lambda^P(K_{EM}), $$

where $\eta_{EM}$ is an effective coupling in the electromagnetic sector and $K_{EM}$ quantifies the coherence of the stored field (e.g. through $Q$ and stored energy).

Similarly, in an interferometric configuration of arm length $L$ and laser wavelength $\lambda_{laser}$, any Λ-induced path-length modification manifests as a phase shift

$$ \Delta\phi \approx \frac{2\pi L}{\lambda_{laser}} \eta_{EM} \Delta\Lambda^P(K_{EM}). $$

The salient features of this channel are:

* *High raw sensitivity.* Frequency and phase can be measured at the $10^{-15}-10^{-18}$ level in state-of-the-art resonators and interferometers, provided thermal and mechanical noise are under control.

* *Complementarity.* Metric/frequency proxies are less exposed to some mechanical artefacts that affect static weight or inertial measurements, and they probe $\Lambda^P$ through light or mode propagation rather than through forces on test masses.

* *Shared parameters.* Any coherence-correlated $\Delta f/f$ or $\Delta\phi$ must be compatible with the same $(\eta, \tau_\Lambda)$ that govern static gravimetry and dynamic inertial induction, once geometry and sector-specific couplings $(\eta_{EM})$ are taken into account.

For the MVT, high-$Q$ frequency/phase experiments are thus not independent “exotic” tests, but part of a single, cross-checked structure: Λ-dressed inertia must appear, if at all, in a coherent way across the three channels. The following section (6.4.3) turns these abstract channels into concrete, transparent and auditable experimental designs.

### 6.4.3 Reference Λ-Lab MVT Experiments

the minimal Λ-dressed inertia model of Secs. 6.4.1–6.4.2 can be made operational only if it is tied to concrete, technologically realistic experiments with explicit falsifiability. For the laboratory pillar of the Minimum Validation Triad, we single out three families of reference experiments, labelled L1–L3:

* **L1:** BEC / cold-atom gravimetry and inertial sensing (coherence-tunable dilute gases).

* **L2:** Superconductors with gravito-inertial probes (bulk coherent solids, static and dynamic).

* **L3:** High-$Q$ electromagnetic and mechanical resonators (metric/frequency proxies).

All three are designed to probe the same two Λ-parameters, $\eta$ and $\tau_\Lambda$, through different observables and systematic environments. Each experiment is specified in terms of (i) geometry and coherence-control protocol; (ii) primary observables; (iii) the expected Λ-signature; and (iv) the null-result regime that would falsify the corresponding piece of the laboratory pillar.

#### 6.4.3.1 L1: BEC / cold-atom gravimetry and inertial sensing

MVT-L1 uses cold-atom interferometry, where both coherence and gravito-inertial sensitivity are state-of-the-art. Consider an atom-interferometer operated either:

* as a vertical gravimeter (atom fountain, optical lattice, or guided configuration), or

* as an inertial sensor for rotation / acceleration (Sagnac-type or multi-loop geometries).

In both cases, the atomic ensemble is prepared alternately in:

* a deeply condensed state with coherence fraction $K_{high} \approx 0.8-1$, and

* a thermal or weakly degenerate state with $K_{low} \approx 0-0.1$,
keeping the total atom number, trap geometry, and external fields (magnetic, optical, RF) fixed. In practice, the K-cycle is realised via controlled changes in temperature, interaction strength (e.g. Feshbach tuning), or trap ramping, with adiabatic protocols to avoid exciting unwanted collective modes.

The coherence order parameter K is defined in the standard way as the condensate fraction (corrected for finite-size effects and coherence length), so that $K \to 0$ in the thermal regime and $K \to 1$ in the deeply condensed limit.

### Static gravimetry channel ($\Delta g/g$)

In a vertical interferometer, the phase shift between arms obeys

$$ \phi_g \propto k_{\text{eff}} g T^2, $$

where $k_{\text{eff}}$ is the effective momentum transfer, g the local gravitational acceleration magnitude, and T the interrogation time. Within the Λ-framework, replacing a thermal cloud ($K_{\text{low}}$) by a condensate ($K_{\text{high}}$) at fixed geometry and external fields modifies the effective gravito-inertial coupling through

$$ \left. \frac{\Delta g}{g} \right|_{K_{\text{high}}-K_{\text{low}}} \approx -\eta \Delta K, \quad \Delta K \equiv K_{\text{high}} - K_{\text{low}} \sim \mathcal{O}(1). $$

The decisive features are:

* the shift must vanish continuously as $K_{\text{high}} \to K_{\text{low}}$;

* it must be monotonic in $\Delta K$ for fixed geometry and external fields;

* its sign must be fixed by $\eta$ and consistent across different realisations (different trap shapes, interrogation times, or atomic species).

Control runs where trap parameters are changed in ways that affect thermal clouds but scramble coherence (e.g. heating protocols that keep g-sensitivity intact) should yield null results, bounding systematic drifts.

### Inertial channel ($\Delta\phi_{\text{inertial}}$)

In rotation or vibration sensing, the interferometric phase encodes inertial effects. If

$$ m_{\text{eff}}(K) \approx m_0 [1 - \eta K], $$

then, for the same mechanical drive, a partially Λ-transparent condensate exhibits a slightly reduced inertial coupling compared with a thermal cloud. The corresponding phase difference scales as

$$ \Delta\phi_{\text{inertial}}(K) \propto \Delta m_{\text{eff}}(K) \approx -\eta m_0 \Delta K. $$

The expected Λ-signature is:

* same sign of the effect in gravimetric and inertial channels;

* same dependence on $\Delta K$ within experimental uncertainties;

* no correlation with magnetic-field gradients, trap deformations, or EM interference, once these are modeled and subtracted.

### Role in the MVT and null-result regime

Cold-atom interferometers already provide some of the tightest constraints on Equivalence Principle violations and Lorentz-symmetry breaking. A dedicated K-modulation protocol with sensitivity $|\Delta g/g|$, $|\Delta\phi_{\text{inertial}}|$ below the nominal $|\eta\Delta K|$ expected from the Λ-framework would immediately bound $|\eta|$ in the dilute-gas regime. A robust null result across multiple atomic species and geometries, with $\Delta g/g$ and $\Delta\phi_{\text{inertial}}$ consistent with zero at this level, falsifies Λ-dressed inertia for cold-atom systems at MVT precision. Conversely, a reproducible, K-monotonic, same-sign deviation in both channels would constitute a direct, high-credibility validation of the Λ-lab signature, independent of solid-state or EM-drive systematics.
### 6.4.3.2 L2: superconductors and gravito-inertial probes

Superconductors provide dense, high-$K$ coherent matter with a well-characterized order parameter and mature experimental control. MVT-L2 uses this platform in two complementary ways: static gravimetry near coherent bulk samples, and dynamic inertial induction near accelerated superconducting rings.

Historical claims of weight deficits above rotating granular YBCO disks Podkletnov and Nieminen [1992], partial replications and null results [Robertson, 1999, Fetisov, 2022], and tangential accelerations near rotating superconducting rings [Tajmar et al., 2007], together with theoretical “gravitational Meissner” proposals [Lano, 1996, Agop et al., 1999, Gallerati et al., 2022], have explored this regime in an ad hoc way. Here, they are used strictly as design templates and upper limits: no historical anomaly is taken as established, and all new experiments are framed as high-precision tests of the $\Lambda$-dressed inertia model.

#### (a) Static gravimetry near coherent bulk samples

##### Target systems and control parameters

The reference configuration uses bulk superconductors (e.g. niobium, YBCO-type cuprates) operated in a narrow band around their critical temperature $T_c$. The coherence fraction $K(T)$ follows a standard two-fluid behavior,

$$K_{sc}(T) \equiv \frac{n_s(T)}{n_{tot}} \approx 1 - (T/T_c)^\alpha,$$

with $\alpha \sim 2-4$, so that $K_{sc} = 0$ in the normal phase and $K_{sc} \to 1$ as $T \to 0$. Control knobs for $K(t)$ include:

* fine temperature modulation just below $T_c$;

* static magnetic fields that suppress or enhance the local superconducting fraction;

* weak transport currents that alter the superfluid density without introducing large Lorentz forces.

Crucially, the total mass and geometry of the sample remain fixed; only its coherence state is modulated.

##### Measurement protocol

A test mass is placed above or below the coherent sample and monitored by one of:

* a torsion or beam balance;

* a superconducting gravimeter;

* a nearby atom-interferometer gravimeter.

The protocol has two phases:

1. *Calibration phase ($K = 0$).* The sample is kept in the normal state, with all drives off, to establish baseline weight $W_0$ and local $g_0$, including environmental drifts and instrumental noise.

2. *Coherence-modulation phase ($K(t) \neq 0$).* The superconducting fraction is periodically modulated in time by cycling through $T_c$ or by small, harmonically varying magnetic fields. EM shielding and careful wiring are used to suppress direct electromagnetic forces on the test mass.

Within the $\Lambda$-framework, the apparent weight obeys

$$\frac{\Delta W(t)}{W_0} \approx -\eta K_{eff}(t),$$

where $K_{eff}(t)$ is the coherence fraction averaged over the coupling volume. When the modulation is switched off at time $t_{off}$, $\Lambda$-dynamics predicts a relaxation

$$\Delta W(t) \propto \exp[-(t - t_{off})/\tau_{\Lambda}],$$

with $\tau_{\Lambda}$ the $\Lambda$-relaxation time.
## Λ-signature and falsifiability

A Λ-consistent signal would show:

* oscillations of $\Delta W/W_0$ at the coherence-modulation frequency;

* a clear dependence on the amplitude of $K(t)$ (e.g. depth of temperature modulation);

* a relaxation curve whose time constant is inconsistent with purely thermal or EM transients, and reproducible across materials and geometries.

A null result at sensitivities below the $|\eta K_{\text{eff}}|$ implied by any MVT-compatible Λ-parameters would bound $|\eta|$ for dense coherent matter and constrain $\tau_{\Lambda}$. If static gravimetry remains null while cold-atom MVT-L1 shows a non-zero Λ-consistent signal, Λ-Theory would be forced into a strongly environment-dependent regime; if both are null, Λ-dressed inertia is excluded at MVT level for both dilute gases and bulk superconductors.

## (b) Dynamic inertial induction near rotating superconducting rings

### Geometry and drive

The dynamic counterpart to (a) employs superconducting rings cooled below $T_c$ and subjected to controlled angular acceleration $\dot{\omega}(t)$. Precision accelerometers and gyroscopes are placed outside the cryostat, in regions shielded from direct EM interference. The coherence fraction $K_{sc}(T, B)$ is again controlled via temperature and static fields, while the ring is driven through programmable acceleration profiles.

Historically, experiments of this type have reported tangential accelerations correlated with $\dot{\omega}$ at levels far above general-relativistic expectations [Tajmar et al., 2007], but with mixed reproducibility and strong systematic concerns [Gallerati et al., 2022]. Here, those results serve purely as *motivational upper limits* and as a guide for sensor placement; the new experiment is specified *a priori* in Λ-variables.

### Expected Λ-response

In Λ-Theory, cooling below $T_c$ drives $K_{sc} \to 1$ and partially suppresses $\Lambda^P$ inside the ring. Accelerating the supercurrent then induces a time-dependent reconfiguration of $\Lambda^P$, encoded schematically as

$$g_{\text{eff}}(\mathbf{r}, t) \propto \frac{\partial}{\partial t} [\Lambda^P(K_{sc}, \dot{\omega})],$$

so that inertial probes respond to the time-derivative of the local Λ-field shaped by the coherent motion.

The Λ-signature is therefore:

* *phase dependence*: induced accelerations or gyroscope drifts that appear only in the superconducting phase ($K_{sc} > 0$), and vanish in the normal state;

* *drive scaling*: amplitudes that are linear (or at least monotonic) functions of $\dot{\omega}$ in the small-signal regime;

* *relaxation*: a lag and relaxation consistent with the $\tau_{\Lambda}$ inferred from static gravimetry.

### Role in the MVT

Dynamic inertial induction provides a stringent cross-check on $\eta$ and $\tau_{\Lambda}$. If a Λ-consistent $\eta$ is inferred from static gravimetry in (a), then MVT-L2 demands that the same ($\eta, \tau_{\Lambda}$) predict detectable, coherence-dependent induced accelerations in realistic $\dot{\omega}$ ranges, or else that the model be revised. Conversely, a robust null result over a wide range of materials, temperatures, and acceleration profiles, at sensitivities below those implied by any Λ-consistent parameter set, excludes Λ-dressed inertial induction in dense coherent solids at MVT level.

#### 6.4.3.3 L3: high-Q cavities and mechanical resonators

The third reference experiment shifts focus from forces and accelerations to metric/frequency proxies. High-Q electromagnetic cavities and mechanical resonators can resolve $\Delta f/f$ and $\Delta\phi$ at levels that are extremely challenging for direct gravimetry, while being sensitive to any coherence-induced change in $\Lambda^P$ within the mode volume. This channel is technically close to existing work on transformation-optics cloaks and effective metrics [Pendry et al., 2006, Guenneau et al., 2012, Mackay and Lakhtakia, 2014, Smolyaninov, 2011, 2018, Tippett,
2011], on warp-like and exotic metric engineering [Alcubierre, 1994, Pfenning and Ford, 1997, Faraoni, 2023], and on vacuum engineering and dynamical Casimir effects in superconducting circuits [Puthoff et al., 2010, White, 2013, Woodward et al., 2020, Wilson et al., 2011]. None of these programs has produced an accepted demonstration of gravitational shielding or propulsion; in the present context, they delimit a technological landscape where Λ-induced metric proxies can be sought under well-characterized conditions.

(a) High-Q electromagnetic cavities with modulated $K_{EM}$

Setup and coherence parameter

Consider a superconducting or ultra-high-purity copper cavity operated at microwave or RF frequencies with quality factor $Q \gg 10^6$. Define an EM coherence parameter

$$K_{EM}(t) \propto Q P_{in}(t),$$

where $P_{in}(t)$ is the injected power; higher $K_{EM}$ corresponds to a larger number of coherent photons stored in the mode, and a longer interaction time with the local vacuum (Λ-field). The baseline resonance frequency $f_0$ and linewidth are measured at low power (small $K_{EM}$), with careful characterisation of thermal drifts, mechanical expansion, and standard nonlinearities.

Modulation protocol

The coherence parameter is then modulated in time by applying pulsed or amplitude-modulated RF power, on timescales short compared with thermal diffusion and mechanical response. Simultaneously, one monitors:

* the instantaneous resonance frequency $f(t)$, and

* where possible, the phase of a weak probe tone injected into the cavity or a coupled interferometer arm.

Within the Λ-framework, a coherence-induced polarization $\Delta\Lambda^P(K_{EM})$ inside the mode volume leads to a non-geometric contribution to the frequency shift:

$$\frac{\Delta f(t)}{f_0} \approx -\frac{\Delta n}{n} - \frac{\Delta L}{L},$$

$$\Delta n \propto \eta_{EM} \Delta\Lambda^P(K_{EM}(t)),$$

where $\Delta L/L$ subsumes ordinary thermal expansion, and $\eta_{EM}$ is an effective coupling in the electromagnetic sector.

Λ-signature

A Λ-consistent signal is characterized by:

* coherent sidebands: frequency or phase modulation at the drive frequency of $K_{EM}(t)$, distinguishable from broadband drift;

* power scaling: a linear (or controlled nonlinear) dependence of $\Delta f/f$ and $\Delta\phi$ on the amplitude of the power modulation, after subtracting known nonlinearities;

* state dependence: absence of the effect when the cavity is operated in regimes with intentionally scrambled coherence (e.g. strong noise injection that reduces effective $Q$ without changing average power).

Null results at sensitivities $|\Delta f/f|$ and $|\Delta\phi|$ below those implied by Λ-compatible values of $\eta$ and $\tau_\Lambda$ place tight bounds on Λ-induced metric proxies in high-Q EM systems.

(b) Ultra-high-Q mechanical resonators near $T_c$

Geometry and control

As a mechanical counterpart, consider a resonator (e.g. a cantilever, membrane, or bulk acoustic mode) with quality factor $Q_{mech} \gtrsim 10^8$, fabricated from a superconducting material such as niobium. The resonator is operated just below the superconducting transition, so that its mass participates in a coherent state
characterized by a time-dependent $K(t)$. The mechanical resonance $\omega_0$ is monitored by laser interferometry or microwave readout, with sufficient resolution to track sub-part-per-billion shifts.

## Coherence modulation and $\Lambda$-response

If the effective mass scales as

$$m(t) = m_0[1 - \eta K(t)],$$

then, for small $\eta K$, the resonance obeys

$$\omega_0(t) = \sqrt{\frac{k}{m(t)}} \approx \omega_{0,\text{ref}}[1 + \frac{1}{2}\eta K(t)],$$

where $k$ is the mechanical stiffness and $\omega_{0,\text{ref}}$ the reference angular frequency. Modulating $K(t)$ at frequency $f_{\text{mod}}$ (e.g. with small, local heating, weak magnetic fields, or controlled quasi-static currents) generates parametric sidebands at $\omega_0 \pm 2\pi f_{\text{mod}}$, with amplitudes proportional to $\eta$.

The $\Lambda$-signature is:

* sidebands that track $K(t)$ and vanish when the resonator is driven in the normal (non-superconducting) state;

* a phase relation between the modulation of $K(t)$ and the sideband amplitude/phase compatible with the same $\tau_{\Lambda}$ inferred from other experiments;

* reproducibility across different resonator geometries and materials sharing similar coherence properties.

Because the bulk geometry is fixed and thermal drifts can be characterized independently (e.g. by running the same modulation with $K \equiv 0$), this configuration is particularly clean for falsifiability.

## Signature and role in the MVT

MVT-L3 demands a coherent modulation of $f$ or $\omega_0$ correlated with $K(t)$, with:

* no such modulation above $T_c$ or in deliberately decohered regimes;

* consistent scaling with $K(t)$ across both EM and mechanical resonators;

* quantitative compatibility with $\eta$ and $\tau_{\Lambda}$ inferred from MVT-L1 and MVT-L2.

If, for $\Lambda$-compatible parameters, the predicted $|\Delta f/f|$ and $|\Delta\omega_0/\omega_0|$ lie above the demonstrated sensitivity of high-$Q$ resonators, and yet no $K$-correlated modulation is observed in carefully controlled experiments, then MVT-L3 is falsified and the allowed $\Lambda$-parameter space is further squeezed. Conversely, any robust, cross-platform observation of coherence-correlated frequency or phase modulation, absent in all normal-state and scrambled-coherence controls, would provide a third, independent leg of the $\Lambda$-laboratory pillar, making a purely metric-first interpretation of gravito-inertial couplings extremely difficult to sustain.

### 6.4.4 Ancillary Hints and Analogues (Non-MVT)

The $\Lambda$-Lab pillar is defined exclusively by the MVT experiments of Sec. 6.4.3. A broader set of historical anomalies and analogue systems nonetheless played a useful role in shaping the design space and in identifying negative controls. To avoid overstating their evidential value, we group them as non-MVT ancillary hints: they motivate geometries and scaling laws, but they do not enter the minimum validation logic.

#### 6.4.4.1 Historical anomalies as design hints (Podkletnov, Tajmar, EM-drive claims)

Reports of anomalous gravito-inertial behavior near macroscopic quantum systems and high-$Q$ EM devices — weight deficits above rotating granular YBCO disks [Podkletnov and Nieminen, 1992], tangential accelerations near rotating superconducting rings [Tajmar et al., 2007], gravito-like or thrust-like signals in EM cavities and “EM-drive”-type setups [Puthoff et al., 2010, White, 2013, Woodward et al., 2020] — remain not accepted and, at present, non-reproducible in the sense required for inclusion in a minimum validation triad. Critical reviews
[Robertson, 1999, Gallerati et al., 2022, Fetisov, 2022] have highlighted multiple possible systematics and the lack of independent confirmation.

Within the Λ-framework, these claims are therefore used *only* as design hints:

* they highlight geometries where high coherence and large stored energy coexist with strong gradients (granular superconductors, accelerated rings, high-Q cavities);

* they suggest observable channels (static gravimetry, induced tangential accelerations, frequency/phase shifts) that are now systematically embedded in MVT-L1–L3;

* they provide *upper limits*: any Λ-induced effect must remain at or below the strongest null-result bounds derived from careful replications.

Crucially, Λ-Theory does *not* require large “gravity cancellation”, shielding, or exotic propulsion. The MVT predictions concern small, structured deviations in weight, inertia, and metric proxies, parametrized by η and τ_Λ. Historical anomalies are treated as a noisy exploratory pre-history; MVT-level tests are the only admissible evidence for or against Λ-dressed inertia.

### 6.4.4.2 Analogue benchmarks (levitated fluids, capillary solitons, metamaterials)

A second class of ancillary input comes from analogue gravity and metamaterial systems, where effective metric properties can be tuned at will. Hydrodynamic and optical analogues [Barceló et al., 2011], transformation-optics cloaks [Pendry et al., 2006, Guenneau et al., 2012, Mackay and Lakhtakia, 2014], and photonic/metamaterial structures [Smolyaninov, 2011, 2018] show that:

* a single control parameter (effective gravity, refractive index, stiffness) can generate curved-spacetime-like phenomena, horizons, or cloaking;

* solitons and complex wave turbulence arise naturally when competing “stiffness” contributions are tuned to comparable levels (e.g. gravity vs surface tension in capillary-gravity waves, including levitated-fluid experiments and capillary solitons).

These systems are *not* simulations of the Λ-field, but they provide *qualitative benchmarks* for what to expect when a curvature-bearing stiffness is gradually reduced:

* dispersion relations change in a structured way;

* localized, long-lived excitations (solitons, domain walls) become possible;

* fluctuation spectra develop non-Gaussian and intermittent features.

In Λ-language, Λ<sup>P</sup> plays the role of an effective curvature stiffness; increasing coherence K shifts weight into Λ<sup>I</sup>. Analogue experiments therefore suggest that, around high-*K* domains, one should look for:

* modified inertial-noise spectra and possible Λ-soliton-like features in gravimetric and inertial sensors;

* geometries where coherent matter or fields act as *gravitational metamaterials*, partially re-routing Λ-flux in analogy with optical cloaks.

These analogues *do not* contribute to the MVT evidence, but they help define realistic search strategies and negative-control patterns for the Λ-Lab program.

## 6.4.5 Summary and Falsification Logic for the Λ-Lab Pillar

### 6.4.5.1 Core laboratory prediction

The core prediction of Λ-Theory in the laboratory regime can be stated concisely:

Gravito-inertial response is not strictly universal, but acquires a controlled dependence on the macroscopic coherence parameter *K* through the balance between Λ<sup>P</sup> and Λ<sup>I</sup>. Highly coherent domains become partially Λ-transparent: they couple slightly less efficiently to curvature, and this manifests as small but structured deviations in weight, inertia, and metric/frequency proxies.
Formally, this is encoded in the invariant

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

and in an effective scaling

$$ m_{eff}(K) \approx m_0[1 - \eta K], $$

with a finite Λ-relaxation time $\tau_Λ$ controlling how quickly $\Lambda^P$ returns to its equilibrium configuration when coherence is changed.

Secs. 6.4.1–6.4.3 translate this into three observable channels — static gravimetry, dynamic inertial induction, and metric/frequency proxies — and into three platform tiers (Tier-A: superconductors / BECs; Tier-B: EM cavities / photonic condensates; Tier-C: ordered soft/biological matter). The MVT focuses on Tier-A/B platforms and on the reference experiments MVT-L1–L3 (see Sec. 7 for Tier definition).

### 6.4.5.2 Minimal positive signature (“win” condition for Λ-Lab)

For the Λ-Lab pillar to count as positively supported at MVT level, it is not necessary to validate all possible platforms and channels. A minimal positive signature requires:

**At least one Tier-A platform with a robust K-correlated effect**

A reproducible, coherence-dependent deviation in one primary channel (Δg/g in BEC/cold-atom gravimetry; static gravimetry near superconductors; inertial phase; or frequency/phase modulation in a high-Q resonator hosting coherent matter or fields).

The effect must:

* vanish continuously as $K \to 0$;

* grow monotonically with $K$;

* have a fixed sign consistent with Λ-transparency (reduced effective coupling).

**Consistent scaling with a single parameter pair ($\eta, \tau_\Lambda$).**

Within that platform, the same ($\eta, \tau_\Lambda$) must explain:

* the relative amplitudes of static and dynamic signals (where applicable);

* the phase-lag and relaxation behavior under on-off modulation of coherence or drive;

* the absence of signals in all normal-state and scrambled-coherence controls.

**Independent replication and cross-checks.**

* Independent groups, using different implementations (e.g. different superconductors, atomic species, interferometer geometries, cavity designs), must reproduce the K-correlated effect at compatible values of $\eta$.

* At least one second channel (e.g. inertial phase vs Δg/g, or frequency vs phase) should show a Λ-consistent pattern, even if at lower significance.

Under these conditions, Λ-dressed inertia becomes an empirical phenomenon. Λ-Theory then offers a concrete microphysical framework for it (via $\Lambda^P/\Lambda^I$ decomposition and Θ-dynamics), but any competing theory would also have to confront the same coherence-dependent gravito-inertial response.

Consistency with astrophysical and cosmological data must then be checked: laboratory-measured $\eta$ and $\tau_\Lambda$ cannot over-produce anisotropies in galaxy-cluster scaling relations, quasar/radio dipoles, or environment-dependent Hubble rates. Instead, they should either (i) remain compatible with current bounds, or (ii) help organize any robust large-scale anomalies as signatures of Λ-transparency in under- and over-dense regions.

Finally, a positive Λ-Lab result would establish an empirical bridge between high-coherence matter, coherent-information phases (Part 6), and gravitational engineering: the same Θ-controlled Λ-decomposition would be constrained both by cosmology and by laboratory-scale coherence experiments.
### 6.4.5.3 Null-result regime and falsification of the Λ-Lab pillar

Conversely, the Λ-Lab pillar is designed so that it can be sharply wrong. A concrete null-result regime that falsifies the laboratory pillar at MVT level is:

#### Static gravimetry nulls

Across superconductors and BEC/cold-atom platforms, dedicated K-modulation protocols in high-sensitivity gravimeters (torsion balances, superconducting gravimeters, atom interferometers) reveal no K-correlated Δg/g down to levels where any Λ-compatible | η | would have produced a detectable signal.

#### Dynamic inertial induction nulls

Around accelerated coherent domains (rotating superconducting rings, accelerated condensates), high-precision accelerometers and gyroscopes detect no coherence-dependent inertial induction beyond general-relativistic expectations and known systematics, again at or below the amplitudes implied by any Λ-compatible (η, τΛ).

#### Metric/frequency proxy nulls

In high-Q EM and mechanical resonators, controlled modulation of coherence K(t) (below and across Tc, or via EM-coherence control) fails to produce any reproducible K-correlated modulation in f, ω₀, or φ at sensitivities that would have resolved Λ-predicted effects.

If, after careful systematic studies and independent replications:

* all three MVT channels (static gravimetry, inertial induction, frequency/phase) remain null at these levels;

* and no consistent (η, τΛ) can be extracted from the data without violating cosmological and astrophysical constraints;

then the Λ-Lab pillar is falsified at its intended precision. Operationally, this corresponds to the effective limit

$$ \eta \to 0, \quad \tau_{\Lambda} \to 0 $$

in the laboratory regime, so that Λ-Theory collapses back to standard GR with universal coupling to stress-energy and no measurable role for macroscopic coherence beyond its conventional contribution to Tμν.

In that case, Λ-Theory can still be retained as a kinematical and informational framework (for example, for organizing cosmological constants, dark-sector partition, or coherent phases), but its claims about Λ-engineering of gravito-inertial couplings must be abandoned. The laboratory pillar thus plays a decisive role: either coherence-modulated gravito-inertial responses are detected at the predicted scales and scalings — in which case Λ-dressed inertia defines a new empirical domain — or they are ruled out across multiple platforms, and Λ-Theory is constrained to a purely metric-compatible regime with no exploitable laboratory Λ-transparency.

#### Laboratory audit manifest (acceptance criteria)

A laboratory claim is admissible only if it satisfies: (i) pre-registered analysis and blinding of the K-cycle labels; (ii) at least two negative controls that reproduce the thermal/EM/mechanical envelope while scrambling coherence; (iii) cross-platform parameter consistency (the same η sign and compatible magnitude after geometry mapping); (iv) independent replication with shared raw-data release.

### 6.5 Triad Synthesis and Trigger Summary

The Minimum Validation Triad (MVT) is designed so that each pillar can be assessed independently, yet all three fit together into a single falsifiable structure. This section collects the logic in one place and summarizes the concrete “triggers” that turn the Λ-framework from a speculative grammar into an empirically competitive alternative to metric-first models.

A practical note is worth stating explicitly: the laboratory pillar is framed as a high-leverage probe. A positive, cross-lab, coherence-controlled signal would be difficult to accommodate in metric-first models without adding new structure, while a global null outcome would primarily bound or remove the simplest
coherence-to-gravito-inertial coupling schemes, without invalidating the cosmological and constants-geometry pillars by itself.

### 6.5.1 Minimal $\Lambda$-MVT “win condition”

At MVT level, $\Lambda$-Theory does not require that every possible consequence be confirmed. Instead, auditable signature that survives conservative nulls and standard robustness checks:

#### Cosmological pillar (structured Hubble field)

A percent-level low-$\ell$ modulation of the local Hubble field,

$$ | \delta H_0/H_0 | \sim 4-6\% $$

detected with footprint-aware nulls, with a stable axis and persistent amplitude across:

* redshift-depth scans,

* survey jackknifes,

* covariance choices, and showing at least suggestive correlations with environment proxies in the sense “void-like directions → higher local $H_0$”, “clustered directions → lower local $H_0$”.

#### Numerical pillar (angular partition and drifts)

A single structural angle $\Theta_{struct}$ and small leakage $\varepsilon$ reproducing the observed fractions

$$ (\Omega_{\Lambda}, \Omega_{DM}, \Omega_{b}), $$

within uncertainties, and a common drift band $\dot{\Theta}$ inferred from both $\dot{G}/G$ and $\dot{\alpha}/\alpha$, such that:

* late-time probes remain compatible with a common

$$ \Theta_{\infty} \approx 0.33-0.40 \text{ rad,} $$

* current drift bounds are jointly satisfied by a single

$$ | \dot{\Theta} | \lesssim 10^{-17}-10^{-16} \text{ yr}^{-1}. $$

#### Laboratory pillar (coherence-modulated gravito-inertial response)

At least one well-controlled platform (cold atoms, superconductors, high-$Q$ cavities) exhibits a coherence-correlated deviation in $\Delta g/g$, inertial phase, or $\Delta f/f$ consistent with

$$ m_{eff}(K) \approx m_0 [1 - \eta K], $$

with:

* continuous vanishing as $K \to 0$,

* monotonic dependence on $K$,

* a fixed sign consistent with partial $\Lambda$-transparency, and

* a single pair $(\eta, \tau_{\Lambda})$ accounting for both static and dynamic channels and their relaxation behavior.

If all three conditions are met, at least at the “strong hint” level (cosmological + numerical already satisfied within current bounds; laboratory satisfied at the first non-trivial sensitivity), the $\Lambda$-framework becomes empirically non-generic: a single $\Lambda$-grammar organizes structured expansion, dark-sector partition and coherence-dependent gravito-inertial response.
### 6.5.2 Falsification fork and failure modes

The triad is also designed to fail in a controlled way. Each pillar has a clear null-result regime where Λ-Theory loses that piece of its explanatory power:

#### Cosmological failure mode

If future homogeneous SN/BAO/cluster maps drive the low-$z$ Hubble field toward:

* no robust low-$\ell$ anisotropy beyond ΛCDM cosmic variance, and

* no residual environment dependence once peculiar velocities and survey systematics are controlled, then the mapping

$$H_0(\mathbf{x}) \propto \sqrt{\Lambda^P(\mathbf{x})}$$

is effectively ruled out at late times. Λ-Theory can still survive as a network-gravity framework, but it loses the “structured expansion” pillar.

#### Numerical failure mode

If precision cosmology forces ($\Omega_{\Lambda}$, $\Omega_{DM}$, $\Omega_b$) into mutually incompatible $\Theta_{eff}$ bands across independent probes (CMB, BAO, SNe, BBN), or if future measurements of $\dot{G}/G$ and $\dot{\alpha}/\alpha$ cannot be fitted by a single $\dot{\Theta}$ at the preferred $\Theta_{struct}$, the angular partition loses falsifiability. In that case, Λ-Theory cannot claim that the dark sector is a one-angle projection of $\Lambda_0$: the numerical pillar collapses back to a descriptive coincidence.

#### Laboratory failure mode

If the reference laboratory program (MVT-L1-L3) systematically returns null results:

* no $K$-correlated $\Delta g/g$ or inertial induction above the sensitivity implied by cosmology-compatible $\eta$,

* no $K$-correlated $\Delta f/f$ or $\Delta\phi$ in high-$Q$ resonators down to sub-$10^{-15}$ regimes, and no consistent ($\eta, \tau_{\Lambda}$) can be extracted that remains compatible with astrophysical constraints, then the Λ-Lab pillar is falsified at MVT precision. Operationally, this corresponds to

$$\eta \to 0, \tau_{\Lambda} \to 0$$

in the laboratory regime: Λ-Theory becomes effectively indistinguishable from GR in accessible gravito-inertial tests, even if the cosmological and constants-geometry pillars remain viable.

This yields a triad fork:

* either all three pillars survive tightening data, in which case Λ-Theory gains empirical weight;

* or one or more pillars fail, in which case the theory must be restricted to the surviving domains (or abandoned entirely if only a heavily tuned residue remains).
### 6.5.3 Trigger table for the Λ-MVT

The table below closes the MVT loop: it identifies, for each pillar, what must be seen, where it is measured, when it counts as a genuine Λ-trigger, and how the pillar can be decisively ruled out if the pattern fails to appear.

<table><caption>Table 11: Global Synthesis of the Minimal Validation Triad (MVT): Signatures, Observables, and Falsification Logic across Three Scales.</caption><thead><tr><th>Cosmological pillar<br>(environment-dependent Hubble<br>field)</th><th>Numerical pillar (constants<br>graph and dark-sector partition)</th><th>Laboratory pillar<br>(coherence-modulated<br>gravito-inertial response)</th></tr></thead><tbody><tr><td>Local Hubble rate as a response<br>field of Λ-stiffness:<br>H<sub>0</sub>(x) ∝ √Λ<sup>P</sup>(x). Void-like<br>directions (low Λ<sup>P</sup>) yield higher<br>local H<sub>0</sub>; clustered directions (high<br>Λ<sup>P</sup>) yield lower H<sub>0</sub>.</td><td><strong>Core Λ-signature</strong><br>Dark sector and baryons as a<br>one-angle projection of an<br>invariant tension Λ<sub>0</sub>:<br>ρ<sub>DM</sub>/ρ<sub>DE</sub> = tan Θ<sub>struct</sub>,<br>ρ<sub>DM</sub>/ρ<sub>b</sub> ≃ tan Θ<sub>struct</sub>/ε.<br>Correlated drifts of G and α<br>controlled by a single Θ̇.</td><td>Macroscopic coherence K<br>partially re-routes Λ-stiffness,<br>making coherent domains slightly<br>Λ-transparent:<br>m<sub>eff</sub>(K) ≃ m<sub>0</sub>[1 − ηK] with<br>finite relaxation time τ<sub>Λ</sub>. Small,<br>structured deviations in weight,<br>inertia, and metric/frequency<br>proxies track K.</td></tr><tr><td colspan="3"><strong>Primary observables / fit objects</strong></td></tr><tr><td>Low-z SN Hubble residuals Δμ<sub>i</sub>;<br>monopole+dipole fits (Δχ<sub>3</sub><sup>2</sup>);<br>environment proxies E(x), T(x);<br>Cosmicflows bulk-flow templates;<br>LSS reconstructions.</td><td>Audited constants graph<br>(G, A, S); cosmological fractions<br>(Ω<sub>Λ</sub>, Ω<sub>DM</sub>, Ω<sub>b</sub>); derived<br>Θ<sub>eff</sub> = arctan(Ω<sub>c</sub>/Ω<sub>Λ</sub>); drift<br>constraints Ġ/G, α̇/α; inferred<br>Θ<sub>struct</sub>, ε, Θ̇.</td><td>Static gravimetry: ΔW/W<sub>0</sub>,<br>Δg/g near BECs or<br>superconductors; dynamic inertial<br>induction: induced accelerations<br>or inertial phases near<br>accelerated coherent media;<br>high-Q cavities and resonators:<br>coherence-correlated Δf/f, Δφ,<br>Δω<sub>0</sub>/ω<sub>0</sub>. Platforms MVT-L1–L3.</td></tr></tbody></table>

#### Positive trigger (MVT “win” pattern)

Stable, percent-level dipole in the<br>local Hubble field with<br>| δH<sub>0</sub>/H<sub>0</sub> | ≈ 4–6%, axis locked<br>near a fixed direction, robust<br>under redshift cuts, sky cuts,<br>jackknifes and covariance choices;<br>footprint-aware permutation tests<br>yielding global significance ~ 3–4σ.

Environment contrasts and<br>regressors with the expected sign<br>(“void-like → higher H<sub>0</sub>”,<br>“cluster-like → lower H<sub>0</sub>”), and<br>persistence of a low-ℓ mode that<br>cannot be absorbed by a pure 1/z<br>bulk-flow template.

Stable constants graph with<br>representation-invariant (G, A, S)<br>triples; CMB, BAO, SNe and<br>BBN all compatible with a single<br>structural angle Θ<sub>struct</sub> and<br>leakage ε reproducing<br>(Ω<sub>Λ</sub>, Ω<sub>DM</sub>, Ω<sub>b</sub>). Any future<br>non-zero drifts of G and α lie on<br>the same one-parameter curve set<br>by Θ<sub>struct</sub> (a single Θ̇ fits both).

Reproducible<br>coherence-dependent deviation in<br>weight, inertial phase or high-Q<br>frequency: Δg/g or<br>m<sub>eff</sub>(K) ≃ m<sub>0</sub>[1 − ηK] with<br>fixed sign, monotonic in K and<br>finite relaxation time τ<sub>Λ</sub>. The<br>same (η, τ<sub>Λ</sub>) explains static and<br>dynamic signals on at least one<br>Tier-A platform and is confirmed<br>by independent groups.
<table><caption>Table 11: Global Synthesis of the Minimal Validation Triad (MVT): Signatures, Observables, and Falsification Logic across Three Scales.</caption><thead><tr><th>Cosmological pillar<br>(environment-dependent Hubble<br>field)</th><th>Numerical pillar (constants<br>graph and dark-sector partition)</th><th>Laboratory pillar<br>(coherence-modulated<br>gravito-inertial response)</th></tr></thead><tbody><tr><td colspan="3"><strong>Falsification channel (decisive null pattern)</strong></td></tr><tr><td>Future homogeneous<br>SN/BAO/cluster maps +<br>improved peculiar-velocity fields<br>show: (i) no low-<i>ℓ</i> anisotropy<br>beyond ΛcDM cosmic variance<br>under footprint-aware nulls; (ii) no<br>residual environment dependence<br>once kinematics and systematics<br>are marginalised; and/or (iii) clear<br>1/z scaling compatible with a<br>purely kinematic bulk flow. In that<br>regime, H<sub>0</sub>(x) becomes effectively<br>isotropic and Λ-stiffness<br>heterogeneity is ruled out at late<br>times.</td><td>(i) Future cosmological<br>constraints push different probes<br>(Planck, DESI, SN+BAO, BBN)<br>into non-overlapping Θ<sub>eff</sub> bands,<br>so that no common Θ<sub>∞</sub> and ε fit<br>all; or (ii) measurements of G̃/G<br>and ˙α/α require incompatible ˙Θ<br>at the preferred Θ<sub>struct</sub>; or (iii)<br>the constants graph backbone<br>becomes<br>representation-dependent (no<br>stable (G, A, S) triples). In any<br>of these cases, the numerical<br>pillar is falsified.</td><td>Dedicated K-modulation<br>experiments on cold atoms,<br>superconductors and high-Q<br>resonators find no<br>coherence-correlated Δg/g,<br>inertial phase shift or Δf/f down<br>to sensitivities where any viable<br>Λ-coupling (η, τ<sub>Λ</sub>) would have<br>produced a signal. Combined null<br>results force η → 0 and τ<sub>Λ</sub> → 0,<br>making Λ-dressed inertia<br>indistinguishable from standard<br>GR in the laboratory regime.</td></tr></tbody></table>

**The MCP “barcode”: differential projection laws (one-lever constraint).**

Let Λ₀ be invariant and

$$ \Lambda^P = \Lambda_0 \cos \Theta, \quad \Lambda^I = \Lambda_0 \sin \Theta, \quad \chi(\Theta) = \tan \Theta. $$

A small phase texture δΘ(x, t) induces *correlated* drifts in observables. In compact form:

$$ \delta \ln c = S_c(\Theta) \delta \Theta, \quad \delta \ln G = S_G(\Theta) \delta \Theta, \quad \delta \ln \alpha = S_\alpha(\Theta) \delta \Theta, \quad \delta \ln H_0 = S_{H_0}(\Theta) \delta \Theta, $$

with (as used in the v3 mapping)

$$ S_c(\Theta) = \frac{2}{\sin(2\Theta)}, \quad S_G(\Theta) = \tan \Theta, \quad S_\alpha(\Theta) = 2 \cot(2\Theta) = \cot \Theta - \tan \Theta, \quad S_{H_0}(\Theta) = -\frac{1}{2} \tan \Theta. $$

Hence, for any common δΘ, the model predicts *fixed* inter-channel ratios, e.g.

$$ \frac{\delta H_0}{H_0} = -\frac{1}{2} \frac{\delta G}{G}, \quad \frac{\delta \alpha}{\alpha} = \frac{S_\alpha(\Theta)}{S_G(\Theta)} \frac{\delta G}{G}. $$

This is a hard monadic constraint: if data require independent variations across channels, the one-lever architecture fails.
Ultimate stress-test (monade consistency): Θ-only crash, density survival, and a kill-switch.

Fix $\Theta_{struct}$ from the dark-sector closure and evaluate sensitivities at that angle. Define

$$S_{\alpha} = \frac{d \ln \alpha}{d \Theta}, \quad S_{H_0} = \frac{d \ln H_0}{d \Theta}, \quad R \equiv \frac{S_{\alpha}}{S_{H_0}}.$$

At $\Theta_{struct} \simeq 0.36$ rad, one has numerically $S_{\alpha} \simeq 2.28$, $S_{H_0} \simeq -0.19$, hence $R \simeq -12$.

Case M1a (“Θ-only”): assume an observed Hubble anisotropy is purely phase-driven, $|\delta H_0/H_0| \sim 0.05$. Then

$$\delta\Theta \simeq \frac{0.05}{|S_{H_0}|} \simeq 0.26, \quad \Rightarrow \quad \frac{\delta\alpha}{\alpha} \simeq S_{\alpha}\delta\Theta \sim 0.6,$$

which is catastrophically excluded (chemistry would differ across the sky). Therefore, the Hubble dipole cannot be a large-scale $\Theta$ texture.

Case M1b (MCP v3 survival mode): $\Theta$ is tightly locked by $\alpha$ bounds, and the Hubble dipole must be carried by stiffness/density heterogeneity. Using $|\delta\alpha/\alpha| \lesssim 10^{-6}$ yields

$$|\delta\Theta| \lesssim \frac{10^{-6}}{S_{\alpha}} \sim 4 \times 10^{-7}, \quad \left|\frac{\delta H_0}{H_0}\right|_{\Theta} \lesssim |S_{H_0}| |\delta\Theta| \sim 10^{-7}.$$

Thus the observed percent-level anisotropy must come from the density/stiffness term in

$$H_0(x) \propto \sqrt{n(x) \Lambda_0 \cos \Theta}, \quad \Rightarrow \quad \frac{\delta n}{n} \sim 2 \frac{\delta H_0}{H_0} \sim 10\%.$$

**Kill-switch (cross-map residual slope).** After regressing out the best density-based predictor from $H_0$ maps, any remaining residual $R_{H_0}(\hat{n})$ that is not systematics must be phase-driven and therefore must satisfy a fixed slope:

$$R_{H_0}(\hat{n}) \approx \frac{S_{H_0}}{S_{\alpha}} \left(\frac{\delta\alpha}{\alpha}\right)_{\text{obs}} (\hat{n}) \approx -\frac{1}{12} \left(\frac{\delta\alpha}{\alpha}\right)_{\text{obs}} (\hat{n}).$$

A detected correlation with this slope is a monadic “win”. A robust correlation with a different slope is a decisive falsification of the one-lever architecture.

Decision rule (Model comparison).

M1 (MCP): one field $\Theta(x, t)$ drives correlated drifts: $(\delta \ln \alpha, \delta \ln G, \delta \ln H_0, \dots) \propto \delta\Theta$ with fixed ratios.

M2 (baseline EFT/phenomenology): $\alpha$-variations and $H_0$-variations are independent.

If M2 fits decisively better without paying extra parameters in M1, the monadic core fails. If M1 is competitive (or wins) without extra freedom, the monadic constraint is validated.
# 7 Empirical Consistency and Falsifiability

Λ-Theory posits a single conjugate tension field on an RTQ network, $Λ = Λ^P + iΛ^I$, from which metric, gravitation, “dark sector” and high-coherence regimes emerge. The goal of this section is not to restate the full construction, but to make the proposal empirically sharp:

* by enforcing reduction to known regimes (negative checks),

* by isolating a minimal set of cross-domain signatures that can support or rule out the framework (Minimal Validation Triad),

* and by pointing to concrete, sector-specific tests.

Full derivations of the checks and tests are moved to Supplementary Material:

* **Appendix A** – Constants graph, rewrite rules, negative controls and SI audit (geometry of constants).

* **Appendix C** – $H_0$-dipole pipeline, anisotropy estimators and robustness tests (cosmological pillar).

* **Appendix D** – Detailed derivations of the seven negative checks.

* **Appendix F** – Falsifiability catalog (~125 tests, filtered to standard physical probes).

The main text keeps 7 short and focused on what a referee actually needs: what already constrains Λ-Theory, what would count as genuine evidence for it, and how the framework can be killed.

## 7.1 Empirical tiers and testing philosophy

The empirical program is organized in three tiers:

* **Tier A – Core structural regime**
  Direct consequences of the Λ-formalism at macroscopic scales (GW speed, Lorentz symmetry, Equivalence Principle, CMB/BAO/BBN history, energy closure, large-scale $H_0$ structure). Violation at this tier falsifies Λ-Theory in its current form.

* **Tier B – Structured but model-dependent regime**
  Predictions that depend on the detailed dynamics of the angular tension variable $Θ$ or on specific cosmological histories. Different Λ-models can make different Tier-B predictions while sharing the same core structure.

* **Tier C – Exploratory and analogue regime**
  Quantum time-crystal analogues, analogue-gravity experiments, and other sectorial tests. They probe the same Λ-architecture, but are not required for minimal viability of the physical core.

A second distinction is crucial:

* **Negative checks** = show that the framework is *not already ruled out* by existing high-precision constraints (consistency).

* **Falsifiability conditions** = specify *prospective outcomes* that would kill the theory (or one of its pillars) if realised by future data.

The seven “Checks 1–7” below are intentionally conservative: each is anchored in a standard observable, and each has a detailed derivation in Appendix C. They are not free assumptions; they are where the Λ-formalism is forced to match existing bounds.

A recurring role is played by the Λ-current

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

encoding conserved flows of informational structure across the RTQ network, and by the angular variable $Θ$ that projects the invariant norm

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

onto physical vs informational components.
### 7.1.1 Claims ledger (logical status and falsifiers)

This subsection is a compact reading guide for referees. It does not add assumptions and does not restate the full construction (summarized in Sec. 4.6). Its only purpose is to make explicit, in one place, the logical status of the main statements used in Sec. 7 (Defined / Postulated / Derived), and how each statement is constrained or falsified by (i) the seven negative checks of Sec. 7.2 and (ii) the three pillars of the Minimal Validation Triad (Sec. 7.3).

To avoid ambiguity, the ledger uses the prefix L\# (for Ledger), independent of the fixed sector determinants $C_i$ and of “Check 1–7”. The audit components referenced below are non-generative: no determinants beyond the four fixed anchors are introduced, and no open search over interaction candidates is performed. The audit only enforces SI-traceable canonicalization and invariance across equivalent textbook parametrizations.

<table><caption>Table 12: The Claims Ledger: Logical Status, Core Statements, and Minimal Falsifiers (L1–L12).</caption><thead><tr><th>ID / Status</th><th>Statement</th><th>Constraint channel → minimal falsifier</th></tr></thead><tbody><tr><td>L1. Postulated</td><td>Microscopic substrate: background-free RTQ network with conjugate tension field Λ = Λ<sup>P</sup> + iΛ<sup>I</sup> of invariant modulus Λ<sub>0</sub>. (Sec. 2)</td><td>Tier-A must-pass (indirectly through all checks) → needing additional undeclared primitives to recover the IR interface + Tier-A regimes.</td></tr><tr><td>L2. Derived</td><td>Diffusion on the RTQ substrate exhibits an intermediate-scale plateau d<sub>s</sub> ≃ 3 (effective spatial dimension). (Sec. 4.6.1; Appendix B)</td><td>Tier-A consistency → no stable d<sub>s</sub> ≃ 3 plateau under admissible coarse-grainings of the same substrate class.</td></tr><tr><td>L3. Defined (canonicalization)</td><td>The four fixed interaction determinants admit a canonical three-layer form (G<sub>i</sub>, A<sub>i</sub>, S<sub>i</sub>) under SI-traceable, invariant textbook re-expressions (no additional determinants introduced). (Sec. 3; Appendix A)</td><td>Pillar 2 (constants-geometry) → two equivalent textbook parametrizations of the same fixed determinant yield non-isomorphic canonical triples after declared mapping.</td></tr><tr><td>L4. Derived (sharp consistency)</td><td>Dimensional/SI consistency of audited carriers together with SM-like coupling dimensions selects d = 3 uniquely. (Sec. 4.6.3; Appendix B; Appendix A)</td><td>Tier-A consistency → a consistent alternative d ≠ 3 preserving the same carrier interpretation and SM-like dimensionless couplings under the same scope rules.</td></tr><tr><td>L5. Postulated (explicitly effective)</td><td>A minimal IR effective action is introduced as a continuum interface template (not a full micro-derivation). (Secs. 4.5–4.6.4)</td><td>Checks 1–4,7 → the IR template generically violates hard bounds (GW speed/dispersion, Lorentz/SME limits, EP, CMB/BBN/BAO) without fine tuning.</td></tr><tr><td>L6. Derived (operational)</td><td>In the ordered regime, operational proper time is reconstructed from the coherence/alignment sector T<sup>μ</sup> (not assumed microscopically). (Sec. 4.5.2)</td><td>Checks 2–3 → unavoidable preferred-frame / EP conflict beyond allowed parameter ranges in the simplest ensemble.</td></tr><tr><td>L7. Derived (phenomenology)</td><td>Late-time dark-sector phenomenology is captured by an angular organization of Λ<sub>0</sub> (dual-sheet language; Θ-governed partition). (Secs. 5.1–5.3)</td><td>Pillar 2 + Check 4 → no viable angular regime compatible with early-time bounds and late-time acceleration simultaneously.</td></tr><tr><td>L8. Compatibility (parameter-light)</td><td>Current density ratios are compatible with a small structural angle Θ<sub>struct</sub> plus one efficiency parameter ε (compressed description). (Sec. 5.3; Appendix A)</td><td>Pillar 2 → improved density determinations force disjoint bands (no single Θ<sub>struct</sub>, ε fits across independent data combinations).</td></tr></tbody></table>
<table><caption>Table 12: The Claims Ledger: Logical Status, Core Statements, and Minimal Falsifiers (L1–L12).</caption><thead><tr><th>ID / Status</th><th>Statement</th><th>Constraint channel → minimal falsifier</th></tr></thead><tbody><tr><td>L9. Derived (testable pattern)</td><td>Environment-dependent modulation of H<sub>0</sub>(x) tracks Λ<sup>P</sup>(x), implying LSS-correlated low-<i>ℓ</i> structure in distance residual maps. (Sec. 5.4; Sec. 7.3.1; Appendix C)</td><td>Check 5 + Pillar 1 → future homogeneous maps show statistical isotropy and no robust environment correlation beyond ΛcDM variance/systematics.</td></tr><tr><td>L10. Must-pass (Tier-A)</td><td>In the isotropic limit, GW and EM share a single effective cone (no leading-order dispersion; v<sub>GW</sub> = <i>c</i> within bounds). (Check 1; Appendix D/C)</td><td>Check 1 → robust detection of v<sub>GW</sub> ≠ <i>c</i> or significant frequency-dependent GW dispersion at current sensitivity.</td></tr><tr><td>L11. Must-pass (Tier-A)</td><td>In the simplest ensemble, vacuum birefringence is absent at the fundamental level (single-cone propagation). (Check 7; Appendix D/C)</td><td>Check 7 → robust multi-instrument detection of vacuum birefringence incompatible with residual systematics.</td></tr><tr><td>L12. Program (Tier-C)</td><td>Analogue/high-coherence platforms can probe Λ-like mode structure; this is exploratory and not required for minimal viability. (Sec. 7.3.3; Appendix E)</td><td>Pillar 3 → dedicated campaigns push bounds far below the declared scaling targets across diverse platforms with no consistent anomaly pattern.</td></tr></tbody></table>

**Reading shortcut.** Tier-A “must-pass” constraints are enforced by Checks 1–4 and 7, with Checks 5–6 acting as closure/structure constraints. The Minimal Validation Triad (Sec. 7.3) isolates three cross-domain opportunities: cosmological map structure (Pillar 1), constants-geometry coherence (Pillar 2), and exploratory laboratory analogues (Pillar 3). A decisive failure of any Tier-A must-pass statement rules out the present simplest form of Λ-Theory.

## 7.2 Negative checks against established constraints

The framework can be viewed as an *effective continuum theory* where the tension field $\Lambda^P(x)$ and its phase coherence ($\Theta$-field) generate all known interactions. This subsection summarizes the seven baseline checks that Λ-Theory must satisfy to remain viable.

Each “check” corresponds to a different observational constraint and was verified through a specific physical reasoning. The detailed calculations, effective actions and mapping steps are given in Appendix D (Checks 1–7) and Appendix C (Check 5 cosmological pipeline):

<table><caption>Table 13: Baseline Consistency Checks: Summary of the seven mandatory pass/fail gates imposed by established observational constraints (Checks 1–7).</caption><thead><tr><th>Check</th><th>What is tested</th><th>How it was checked</th><th>Observable bound</th></tr></thead><tbody><tr><td>1</td><td>Gravitational wave speed</td><td>Linear perturbation of Λ<sup>P</sup> → compare v<sub>GW</sub> vs v<sub>γ</sub></td><td>|v<sub>GW</sub> - c|/c &lt; 10<sup>-15</sup></td></tr><tr><td>2</td><td>Lorentz invariance</td><td>Statistical isotropy of connections</td><td>LV coefficients &lt; 10<sup>-18</sup></td></tr><tr><td>3</td><td>Equivalence Principle</td><td>Equality m<sub>g</sub> = m<sub>i</sub> from isotropy</td><td>Δa/a &lt; 10<sup>-13</sup></td></tr><tr><td>4</td><td>Cosmological evolution</td><td>CMB / BAO / BBN consistency</td><td>ΔG/G &lt; 0.02</td></tr><tr><td>5</td><td>H<sub>0</sub> variability</td><td>Local modulation of Λ<sup>P</sup>-density</td><td>ΔH<sub>0</sub>/H<sub>0</sub> ≃ 1–2%</td></tr><tr><td>6</td><td>Tension/energy closure</td><td>No net Λ<sup>P</sup> ↔ Λ<sup>I</sup> transfer</td><td>Δρ/ρ &lt; 10<sup>-5</sup></td></tr><tr><td>7</td><td>Non-birefringence</td><td>Polarization-independent propagation</td><td>Δv<sub>pol</sub>/c &lt; 10<sup>-32</sup></td></tr></tbody></table>
### 7.2.1 Check 1 – Gravitational-wave speed

In the isotropic limit of the $\Lambda$-network, tensor perturbations of $\Lambda^P$ propagate as elastic phonons of the tension field, while EM waves are phase waves in $\Lambda^I$. For an isotropic background one finds a single effective light cone,

$$v_{GW} = v_{\gamma} = c,$$

with no frequency-dependent dispersion term at leading order. GW170817/GRB170817A constrains

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

and GWTC-2 / GWTC-3 catalogs show no statistically significant frequency-dependent deviations from the GR dispersion relation.

Within $\Lambda$-Theory this is automatically satisfied unless large, coherent gradients $\nabla\Lambda^P$ are present on scales $\gtrsim 40$ Mpc (Appendix C.1). Any robust detection of $|v_{GW} - c|$ at the current level, after careful multi-messenger cross-checks, would either impose extreme fine-tuning of $\Lambda$-gradients or falsify the isotropic $\Lambda$-limit.

### 7.2.2 Check 2 – Lorentz invariance

The critical IR requirement is not merely “approximate Lorentz symmetry”, but the emergence of a single relativistic cone governed by a unique second-order hyperbolic operator for all propagating sectors in the ordered regime. In the $\Lambda$-framework this is ensured by statistical isotropy of RTQ connections at the coarse-graining scale: the leading operator takes the universal form

$$D_{IR} \propto \partial_t^2 - c^2 \nabla^2,$$

and any LV contributions enter only through anisotropic/higher-derivative corrections.

Quantitatively, the residual LV coefficients are controlled by fluctuations of local connectivity tensors around isotropy. Coarse-graining over a region containing $N \gg 1$ links with rapidly decorrelating orientations suppresses the anisotropic remainder by a central-limit scaling,

$$|\delta_{LV}| \sim N^{-1/2},$$

so that the effective SME-like coefficients (e.g. $c_{\mu\nu}$ in matter sectors, non-birefringent photon deformations, and æther-like combinations in the tensor sector) are naturally driven far below current bounds for any macroscopic probe volume.

In an SME-like parametrization with a unit timelike vector $u^\mu$ (the macroscopic coherence/clock field direction), the safe-window conditions can be summarized as

$$|c_{\mu\nu}| \ll 10^{-18}, \quad (k_{AF})_{\mu} = 0,$$

together with the tensor-sector single-cone condition already enforced by Check 1 (GW170817): the same underlying isotropy that yields $D_{IR}$ also suppresses any relative cone-splitting between spin-2 and electromagnetic modes.

Any future detection of Lorentz violation incompatible with a single-cone structure—for instance robust anisotropic max-speed shifts that do not scale away under larger averaging volumes, or frequency-dependent GW dispersion beyond the constrained EFT window—would falsify the simplest isotropic-network regime (Appendix D.2).
### 7.2.3 Check 3 – Equivalence Principle

In $\Lambda$-Theory, inertial and gravitational mass correspond to the same projection of $\Lambda^P$ along an internal “mass axis” set by torsion loops vs straight segments. Motion in a $\Lambda^P$-gradient obeys

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

so that $m_g/m_i = 1$ is a structural identity (see Appendix D.3 for the derivation).

Residual anisotropies $\varepsilon$ in the connection tensor induce differential accelerations,

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

with $\alpha$ a dimensionless geometric factor derived in Appendix C.3.1. Eöt-Wash torsion balances and the MICROSCOPE mission constrain

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

implying $\alpha\varepsilon \ll 10^{-13}$. Any confirmed EP violation above this level, with no $\Lambda$-compatible internal source (e.g. an additional torsion sector), would falsify core assumptions on the isotropy of $\Lambda^P$.

### 7.2.4 Check 4 – CMB/BAO/BBN history and varying constants

At early times, $\Lambda$-Theory must reduce effectively to $\Lambda$CDM with nearly constant $G$ and standard matter–radiation content. A homogeneous sector can be cast into an effective scalar–tensor action

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

with conformal coupling

$$ A(\Lambda) = \exp[\beta(\Lambda - \Lambda_0)]. $$

BBN, CMB and BAO analyses typically require

$$ | \frac{\Delta G}{G} | \lesssim 0.02 $$

between nucleosynthesis and today, with no rapid drift around recombination. Quasar spectra, atomic clocks and varying-constant cosmography give $|\frac{\Delta\alpha}{\alpha}| \lesssim 10^{-6}-10^{-5}$ over $0 \lesssim z \lesssim 4$. In $\Lambda$-Theory, both $G$ and $\alpha$ are projections of the same angular tension variable $\Theta$. These bounds are enforced by requiring (Appendix C.4) $\dot{\Theta} \simeq 0$, $|\beta(\Lambda - \Lambda_0)| \ll 1$ for $z \gtrsim 2$. Any robust evidence for a large $|\Delta G/G|$ incompatible with slow $\Theta(t)$, or for an uncorrelated drift in $\alpha$ with no counterpart in the gravitational sector, would falsify single-angle $\Lambda$-cosmologies.

### 7.2.5 Check 5 – $H_0$ variability as constraint

$\Lambda$-Theory links the local Hubble rate to the stiffness of the tension field:

$$ H_0(x) \propto \sqrt{\Lambda^P(x)}. $$

Void-like regions (lower $\Lambda^P$) should yield systematically higher inferred $H_0$, while overdense regions (clusters, superclusters; higher $\Lambda^P$) should yield lower $H_0$. This implies non-random spatial structure in $H_0$ maps: correlations with large-scale structure and a possible low-$\ell$ anisotropy component.
Section 6 and Appendix B present an internal reduction test on a Pantheon+/SH0ES-style SN compilation, using residuals

$$ \Delta\mu_i = \mu_i - \mu_{th}(z_i; H_0), $$

and comparing a monopole-only model to a monopole+dipole model. The dipole amplitude $A_\mu = \| \mathbf{D} \|$ maps to an effective modulation

$$ | \frac{\delta H_0}{H_0} | \approx \frac{\ln 10}{5} A_\mu. $$

Permutation tests that preserve survey footprints yield an empirical p-value $p_{emp} \lesssim 10^{-5}$ for the observed low-$\ell$ structure, with caveats about peculiar velocities and systematics (Appendix B).

As a negative check, Λ-Theory remains compatible with current data provided future homogeneous SN/BAO/-cluster maps and improved peculiar-velocity fields either:

* confirm a few-percent environment-dependent $H_0$ structure consistent with $\Lambda^P$-heterogeneity, or

* drive the residual modulation below the level where $\Lambda^P$-heterogeneity plays a significant role.

If forthcoming datasets converge toward an effectively isotropic $H_0$ field with no environment correlation beyond ΛCDM cosmic variance, strong $\Lambda^P$-heterogeneity interpretations are disfavoured (Tier A/B falsifier; see 7.3.1).

### 7.2.6 Check 6 – Tension/energy closure

The conjugate Λ-cycle is norm-conserving at the coarse-grained level:

$$ \partial_t((\Lambda^P)^2 + (\Lambda^I)^2) = 0. $$

Exchanges between metric sheet $M^+$, conjugate sheet $M^-$ and high-coherence sectors are oscillatory rather than secular.

Any sustained leakage would appear as an anomalous contribution to cosmological closure at the $\mathcal{O}(10^{-5})$ level (as estimated in Appendix D.6) or larger, in tension with tight CMB+BAO constraints on near-flatness. Persistent evidence for missing or extra energy not attributable to standard components or Λ-oscillations would rule out simple $M^+/M^-$ coupling schemes.

### 7.2.7 Check 7 – Absence of birefringence

Λ-Theory predicts that all propagating modes share a single effective cone at the fundamental level. Photons see

$$ \tilde{g}_{\mu\nu} = g_{\mu\nu} + D u_{\mu} u_{\nu}, $$

with $D \ll 1$, and parity-odd SME coefficients are set to zero at the fundamental level in the minimal ensemble. Vacuum birefringence therefore vanishes in the core framework Appendix D.7.

Astrophysical polarimetry and SME analyses constrain the relative speed of polarizations to

$$ \frac{\Delta v_{pol}}{c} \lesssim 10^{-32}. $$

Hints of a small cosmic birefringence angle in Planck polarization data can be interpreted either as residual systematics or as a sign that explicit parity-violating $\Lambda^I$ couplings or large-scale anisotropies in $\Lambda^P$ must be added. A robust detection of birefringence at current sensitivities, confirmed across instruments, would kill the simplest non-birefringent RTQ ensembles.
## 7.3 Minimal Validation Triad and falsifiability map

The seven checks above ensure that Λ-Theory is not already excluded. To make the framework properly falsifiable, we isolate a *Minimal Validation Triad* of cross-domain pillars, all rooted in the Tier-A structural regime:

1. *Cosmological pillar* – environment-dependent $H_0$ and large-scale structure.

2. *Constants-geometry pillar* – angular organization of fundamental constants.

3. *Laboratory pillar* – analogue and high-coherence tests of Λ-dynamics.

Each pillar has (i) a structural prediction, (ii) a positive trigger, and (iii) a null pattern that would falsify it.

### 7.3.1 Cosmological pillar: $H_0$ variability and Λ-stiffness

#### Structural prediction

Late-time expansion tracks Λ-stiffness,

$$H_0(\mathbf{x}) \propto \sqrt{\Lambda^P(\mathbf{x})},$$

with voids and overdensities imprinting a non-random pattern on local $H_0$ estimates. The framework predicts:

* correlations between $H_0$ maps and the cosmic web (voids/filaments/clusters), and

* a low-$\ell$ anisotropy component (at least a dipole) in $H_0$ or related distance indicators at low redshift.

#### Positive trigger

A convergent, multi-probe picture in which:

* percent-level $H_0$ modulations are confirmed by homogeneous SN/BAO/cluster catalogs and standard sirens,

* the pattern tracks large-scale structure in a way that cannot be reabsorbed into peculiar-velocity fields or known systematics, and

* the preferred axes inferred from independent tracers (SNe, clusters, radio/quasar dipoles) agree within uncertainties,

would strongly support the Λ-stiffness mapping rather than kill it.

#### Null falsifier (redshift-scaling discriminant)

If future surveys (Euclid, LSST, Roman, DESI, eROSITA, etc.) show that:

* $H_0$ becomes statistically isotropic at the few-percent level or better after marginalising over peculiar velocities and known systematics, and

* no robust correlation remains between $H_0$-residual maps and large-scale environment,

then the cosmological pillar in its current form is falsified or demoted to a descriptive picture, and $\Lambda^P$-heterogeneity cannot play a leading role in the $H_0$ tension.

If the inferred dipole amplitude follows the kinematic template $A_\mu(z) \propto 1/z$ (with a stable direction consistent with independent peculiar-velocity reconstructions) and no statistically significant residual remains once this component is removed, then the $\Lambda^P$-texture interpretation is strongly disfavoured in its current form.

### 7.3.2 Constants-geometry pillar: angular structure in the constants

#### Structural prediction

The dark-sector partition and baryon fraction are controlled by a single structural angle $\Theta_{struct}$ and a condensation efficiency $\varepsilon$ for torsion loops:

$$\frac{\rho_{DM}}{\rho_{DE}} \approx \tan \Theta_{struct}, \qquad \frac{\rho_{DM}}{\rho_b} \approx \frac{\tan \Theta_{struct}}{\varepsilon}.$$
All “constants”

$$ \{c, G, \alpha, \hbar, \dots\} $$

are angular projections of the same invariant $\Lambda_0$ through $\Theta$, and may co-vary only in a sector-coherent way (no independent drifts).

Appendix A formalizes this as a constants graph with rewrite rules, negative controls and SI audit, and shows that a small-angle $\Theta_{struct}$ and $\varepsilon \sim \mathcal{O}(10^{-1})$ jointly reproduce current density ratios.

### Positive trigger

As measurements sharpen (Planck re-analyses, DESI/Euclid/BBN constraints, improved $\Omega_m$ and $\Omega_l$), a stable small-angle solution emerges:

* $\Theta_{struct}$ and $\varepsilon$ remain statistically consistent across independent data combinations and analysis pipelines,

* the constants graph backbone (set of nodes, edges and rewrite rules) remains invariant under equivalent parametrizations (e.g. $\alpha$ vs $\varepsilon$, different $G$ conventions), and

* mild but coherent co-variation of constants (from quasar spectra, atomic clocks, standard sirens) can be fitted by a single $\Theta(t)$ within the tight bounds of Check 4.

This would support the idea that “constants geometry” carries non-trivial explanatory content rather than being numerology.

### Null falsifier

The constants-geometry pillar is falsified if:

* improved determinations of density ratios force $\Theta_{struct}$ and $\varepsilon$ into disjoint best-fit bands (no single pair fits all sectors), or

* robust and uncorrelated drifts are found between constants (e.g. $\alpha$ varies while $G$ and $H(z)$ remain fixed within errors), or

* an alternative factorization of the constants, of comparable or lower complexity, explains the same phenomenology without invoking a single $\Lambda$-angle or the RTQ structure.

In any of these cases the numerical pillar loses its explanatory advantage and must be abandoned or drastically revised.

### 7.3.3 Laboratory pillar: analogue and high-coherence tests

#### Structural prediction

In the laboratory, $\Lambda$-Theory predicts that suitable analogue media (fluids, metamaterials, BECs, Josephson arrays) can approximate an effective Lagrangian of the form

$$ \ell_{eff} \sim \frac{1}{2} Z(\Lambda)(\partial\Lambda)^2 - V(\Lambda), $$

with emergent phonon-like excitations mimicking tensor or phase modes of $\Lambda$. It also predicts tiny couplings between coherence and local “time structure”: in highly coherent regimes, residual $\Lambda$-dependent contributions to phase noise or clock drift could appear beyond standard decoherence models.

#### Positive trigger

A positive trigger would be:

* reproducible dispersion relations in analogue systems that match the $\Lambda$-derived phonon spectrum (Appendix D) and are not explained by standard hydrodynamics or condensed-matter models, and/or

* statistically significant, coherence-correlated anomalies in ultra-stable clocks or interferometers, with a scaling law matching a $\Lambda$-based prediction (e.g. dependence on coherence fraction, geometry of the effective tension field) and surviving cross-checks across devices and laboratories.
## Null falsifier

If dedicated Λ-laboratory campaigns:

* push upper bounds on such analogue and coherence effects orders of magnitude below the natural Λ-scaling predicted in Appendix D, and

* do so across diverse platforms (fluids, BECs, Josephson arrays, optical cavities) without revealing any consistent anomaly pattern,

then the laboratory pillar, at least in its simplest coupling schemes, is falsified. The core large-scale framework could still survive, but it would lose its main near-term laboratory handles.

## 7.4 Outlook: sharpening tests and data needs

Taken together, Checks 1–7 and the three pillars above define how Λ-Theory can be pressured empirically:

* *Checks 1–7* enforce reduction to known physics (GW speed, Lorentz symmetry, EP bounds, CM-B/BAO/BBN history, $H_0$ constraints, energy closure and birefringence).

* *Cosmological pillar:* increasingly homogeneous and deep 3D maps of $H_0$ and large-scale structure (SN/BAO, clusters, standard sirens, peculiar-velocity reconstructions) test the Λ-stiffness picture.

* *Constants-geometry pillar:* improved determinations of density ratios and multi-probe constraints on $\Delta G/G$ and $\Delta\alpha/\alpha$ sharpen the angular partition and constants graph.

* *Laboratory pillar:* dedicated analogue and high-coherence experiments with clearly specified Λ-scaling targets probe the near-field regime.

Appendices A–E collect the technical material required for replication: graph structures and rewrite rules, $H_0$-anisotropy pipelines, detailed check derivations, and the broader falsifiability catalog. The main text stays deliberately conservative: it only commits to tests that a physics readership can, in principle, implement and audit with existing or near-future tools.

In this sense, Λ-Theory is not presented as an unfalsifiable meta-framework, but as a concrete, over-constrained proposal: it must survive simultaneous pressure from GW propagation, cosmology, constants geometry and laboratory analogues. Any decisive failure along the lines identified above would either kill the framework or force a non-trivial revision of its core assumptions.
# 8 Discussion and Outlook: Λ-Based Physics of Information

Λ-Theory was introduced as a discrete, relational framework where a single conjugate tension field $\Lambda = \Lambda^P + i\Lambda^I = \Lambda_0(\cos\Theta + i\sin\Theta)$ on an RTQ network underlies spacetime, matter, interactions, and coherence. Internal axes $d_1-d_4$ encode U(1), SU(2), SU(3), and CPT/causal orientation; a global phase $d_5 = \Theta$ sets the balance between curvature $\Lambda^P$ and coherence $\Lambda^I$; emergent spacetime corresponds to correlated orientations in higher, statistical dimensions $d_6-d_9$; and a composite $d_{10}$ couples Λ-sheets to high-coherence regimes. This final section interprets this architecture as a physics of information, situates it with respect to existing frameworks, and outlines open problems and future work.

## 8.1 From Laws as External Rules to Λ as Internal Grammar

Conventional fundamental physics treats laws as external rules imposed on a pre-given manifold: GR postulates a metric and field equations; QFT postulates fields on spacetime with specified symmetries and couplings; thermodynamics and information theory introduce entropic or coding functionals on top of this structure. In such a picture, constants $(c, \hbar, G, \alpha, ...)$ enter as independent parameters to be measured and inserted into the laws.

In Λ-Theory, by contrast, “laws” are emergent regularities of how $\Lambda^P$ and $\Lambda^I$ organize themselves. RTQs are not embedded points but relational tension quanta whose connections encode internal angles $d_1-d_4$ and a global phase $\Theta$. Gauge “fields” arise as collective patterns of U(1)/SU(2)/SU(3) misalignment on these axes [Overduin and Wesson, 1997a, Bohm, 1980, Healey, 2007]; mass and inertia correspond to torsion loops on the SU(2) axis; confinement is encoded by SU(3) loop closure; causal order follows from synchronization on the CPT/temporal axis; and effective constants are angular projections of $\Lambda_0$.

In this sense, Λ plays the role of an internal grammar rather than an external code:

* the alphabet consists of internal axes and phases ($d_1, \dots, d_5$);

* the syntax is given by allowed angular couplings and topological motifs (loops, braids, flux tubes);

* the semantics corresponds to emergent spacetime, fields, particles, and thermodynamic/informational quantities.

What we usually call “physical laws” (Einstein equations, Yang-Mills, Schrödinger/Dirac, thermodynamic relations) appear as effective rules governing certain coarse-grained Λ-regimes, much like grammar rules emerge from deeper statistical properties of text corpora. This is close in spirit to information-theoretic reconstructions of physics [Shannon, 1948, Jaynes, 1957, Landauer, 1961], but Λ-Theory insists that information is not just a description of physical states: information and curvature are two conjugate manifestations of a single tension invariant.

## 8.2 Monist Interpretation: Matter, Information, and Cognitive Structure as Phase Regimes

The resulting ontological picture is monist but phase-structured. There is a single relational substrate—the Λ-field on an RTQ network—with two orthogonal components:

* $\Lambda^P$: curvature, rigidity, energetic “cost” of changing internal phases;

* $\Lambda^I$: coherence, order, informational memory and constraint.

The invariant $\Lambda^{P2} + \Lambda^{I2} = \Lambda_0^2$ is not interpreted as a statement about two “substances”, but as a bookkeeping identity: curvature-like and coherence-like behaviors are complementary projections of a single invariant tension. Varying the phase parameter $\Theta$ interpolates between regimes where $\Lambda_0$ is expressed predominantly as metric stiffness $\Lambda^P$ or as coherence $\Lambda^I$. Within this single Λ-fabric, the following regimes are distinguished:

### Physical matter

(small $\Theta$, $\Lambda^P \gg \Lambda^I$): curvature-dominated phase in which spacetime, inertia, and local interactions admit standard metric-effective descriptions (GR/QFT).

### Far-from-equilibrium structured matter
(Θ locally near $\pi/4$, $\Lambda^P \approx \Lambda^I$): sustained curvature-coherence exchange supporting persistent organization in driven, dissipative conditions. Here we only claim a structural regime of non-equilibrium physics, in line with classical work on self-organization [Prigogine and Stengers, 1984, England, 2013], without committing to any specific biological model.

## Coherence boundary / informational limit

(Θ → $\pi/2$, $\Lambda^P \to 0$, $\Lambda^I \to \Lambda_0$): coherence-dominated, non-metric regime where adjacency is defined by similarity of phase configurations rather than spatial distance. In this limit the usual metric notion of locality ceases to be the appropriate organizing principle; it functions instead as a boundary condition for metric-effective physics.

The coherence-dominated limit is formulated entirely inside the Λ-framework and does not introduce extra ontological entities (no separate “mental” or “observer” sector). The manuscript makes no claim about biological or cognitive mechanisms, and does not propose any dedicated “observer physics”. It only formulates a phase-structured physical language in which curvature-like and coherence-like components exchange under the invariant constraint $\Lambda_0^2$. Where a metric description exists, standard relativistic consistency conditions apply. In particular, absence of superluminal signalling in metric-effective regimes; Einstein [1911].

From an information-theoretic viewpoint, the “informational limit” is meant in a minimal, operational sense: information is treated as a measure of distinguishability/ordering compatible with standard definitions [Shannon, 1948, Brillouin, 1956]. Any extension of these notions to biological, cognitive or social domains would require domain-specific modelling and independent empirical criteria, and is therefore explicitly left outside the scope of this paper.

## 8.3 Relation to Existing Frameworks

### 8.3.1 Links and Differences with Relational and Emergent Approaches

Λ-Theory naturally interfaces with several relational or emergent programs.

* *Causal set / causal dynamical triangulations.* These treat spacetime as a discrete causal order or path integral over combinatorial geometries [Bombelli et al., 1987, Ambjørn et al., 2012]. Λ-Theory is also discrete and relational, but its fundamental objects are not events with partial order, but RTQs with internal angular data. Causality emerges from synchronization on the CPT/temporal axis $d_4$ (Kuramoto-type dynamics; [Acebrón et al., 2005, Rodrigues et al., 2016], rather than being imposed as a primitive partial order).

* *Loop quantum gravity and spin networks.* LQG encodes quantum geometry in spin networks and spin foams [Rovelli, 2004, Thiemann, 2007]. Λ-Theory also uses graph-like structures, but adds a separate layer for coherence $\Lambda^I$ and internal gauge axes $d_1-d_3$ in a way closer to fibre-bundle internal spaces [Overduin and Wesson, 1997a, Healey, 2007]. The metric and its quantisation become derived from $\Lambda^P$-gradients rather than being quantised from the start.

* *Emergent/entropic gravity.* Approaches by [Sakharov, 1967, Jacobson, 1995, Verlinde, 2011] and others relate gravity to thermodynamic or entropic principles. Λ-Theory is compatible with this spirit: curvature is an elastic response of a tension field $\Lambda^P$ whose degrees of freedom carry entropic/informational content. However, it adds a microscopic angular grammar: “entropic” effects are not generic but arise from the way Λ redistributes between $\Lambda^P$ and $\Lambda^I$ on internal axes.

* *Holography and information geometry.* Holographic dualities suggest that bulk geometry is encoded by boundary quantum information [Susskind, 1995, Ryu and Takayanagi, 2006]. Information geometry interprets statistical manifolds as curved spaces of probability distributions [Amari and Nagaoka, 2000]. Λ-Theory resonates with both: $\Lambda^I$ plays a holographic “memory” role for $\Lambda^P$ dynamics, and effective metrics arise from informational tensions. The difference is that holography is usually formulated on fixed manifolds, whereas Λ-Theory attempts to derive both metric and information spaces from the same discrete tension field.

* *Relational quantum mechanics & QBism.* Interpretations emphasising relations between systems or observer-centred probabilities [Rovelli, 1996, Fuchs and Schack, 2013] conceptually align with Λ's relational stance, but Λ-Theory is not primarily interpretational: it proposes a specific micro-structure from which those relational features might arise.
In short, Λ-Theory sits at the intersection of causal set ideas, loop-like/topological matter models [Skyrme, 1961, Faddeev and Niemi, 1996, Volovik, 2003], entropic gravity, and holographic/information-geometric approaches, but adds a concrete angular decomposition of a single tension invariant that simultaneously accounts for interactions, constants, and coherence.

### 8.3.2 Complementarity with Quantum Information and Complexity-Based Cosmology

Quantum information theory and complexity-based cosmology increasingly view spacetime and dynamics as emergent from entanglement, computation, or algorithmic complexity. Λ-Theory is compatible with—and potentially complementary to—these views:

* **Quantum information.** In Λ-language, entanglement and coherence can be represented as patterns of Λ¹-phase alignment across RTQs; decoherence and measurement correspond to redistribution of Λ from Λ¹ to ΛP (and vice versa) under environmental coupling, with a minimal action scale set by RTQ reorientation [Wheeler, 1990, Landauer, 1961, Lloyd, 2006]. This echoes “it-from-bit” and “information is physical” perspectives, but grounds them in a specific relational micro-mechanism. We do not claim here a complete reformulation of quantum theory in Λ-variables; only that the framework is compatible with such attempts.

* **Complexity and tensor-network cosmology.** Tensor-network descriptions of AdS/CFT and proposals linking wormhole volume to computational complexity treat spacetime as a “geometry of quantum circuits” [Swingle, 2012, Brown et al., 2016]. Λ-Theory can be read, at an effective level, as a “geometry of reconfiguration costs”: ΛP measures how hard it is to rewire internal phases; ΛI measures coherence resources; and Θ determines the regime in which complexity (as circuit depth) and geometry (as curvature) coincide. This is a heuristic, not a derivation of specific complexity conjectures.

* **Information-theoretic reconstructions of constants and laws.** Work linking thermodynamics, information and gravitation [Jaynes, 1957, Landauer, 1961, Jacobson, 1995, Dvali, 2018, Bhattacharya and Potvin, 2025] shows that many “laws” can be recast as extremization of information-related functionals. Λ-Theory extends this formally by showing how effective constants may be encoded as angular projections—e.g. $\chi(\Theta) \propto \tan \Theta$, $G(\Theta) \propto 1/(\langle\rho\rangle \Lambda_{\text{struct}}^P(\Theta))$—subject to the Λ-grammar and carrier structure. These relations are not assumed to hold exactly in nature; they are proposed as testable parametrizations within the empirical program of Sec. 6.

Thus, Λ-Theory does not compete with quantum information and complexity-based approaches; it offers a concrete relational substrate on which such information structures could live, potentially providing a candidate “micro-ontology” for emergent space, time, and law, without pre-empting more detailed constructions.

## 8.4 Open Problems and Future Work

The present work only sketches the Λ-framework and its empirical architecture. Substantial development is required before it can be seriously confronted with data and compared to established formalisms.

### 8.4.1 Mathematical Completion of the Λ-Field Equations

Several mathematical tasks remain open:

1. *RTQ dynamics and continuum limit.* A precise update rule for RTQ states and connections must be specified (e.g. as a stochastic process or cellular automaton on a graph), with a controlled continuum limit reproducing effective field theories on emergent manifolds. This includes rigorous derivations of:

* the effective Einstein-like equations from $\nabla\Lambda^P$;

* gauge-field dynamics from internal axes $d_1-d_3$;

* the Θ-dependence of effective constants.

2. *Well-posed Λ-field equations.* A continuous description of Λ(x) requires a Lagrangian or Hamiltonian formulation where the decomposition

$$ \Lambda^P = \Lambda_{\text{struct}}^P + \Lambda_{\text{manif}}^P, \Lambda_{\text{confined}}^P \subset \Lambda_{\text{manif}}^P $$
can be derived, not postulated, and where the conservation law $\Lambda_P^2 + \Lambda_I^2 = \Lambda_0^2$ is preserved under
dynamics.

3. Renormalization and coarse-graining. The behavior of $\Lambda$ under coarse-graining must be studied (e.g. using renormalization-group ideas on graphs), clarifying how low-energy GR + SM arise as fixed points and how $\Lambda$-specific signatures survive at observable scales.

4. $\Theta$-dynamics and stability. The evolution of $\Theta$, both globally and locally, must be tied to RTQ dynamics and constrained by stability and observational bounds (e.g. on variations of constants, $H_0$ tension, DM/DE ratios).

5. Explicit network models and simulations. A concrete RTQ-network model implementing the internal axes ($d_1, \dots, d_5$) and emergent spacetime degrees of freedom ($d_6, \dots, d_9$) should be constructed and explored numerically. Even in reduced toy settings (e.g. 1+1 dimensions, single gauge axis), simulations could test whether (a) a stable light cone and effective metric emerge from $\Lambda^{(P)}$ gradients, (b) topological motifs behave as quasi-particles with mass/charge properties, and (c) $\Theta$-controlled variations of effective parameters reproduce qualitative features of the phenomenology discussed here.

Without these mathematical developments, $\Lambda$-Theory remains at the level of a *structured hypothesis* rather than a complete theory.

## 8.4.2 Detailed Sectorial Papers (Particles, Cosmology, High-Coherence Systems)

Given the scope, detailed treatment must be split into sectorial studies:

* Particle and interaction sector. A dedicated paper should map the topological-motif picture of fermions and bosons (loops, braids, flux tubes) onto the full Standard Model, including flavour, CKM/PMNS mixing, anomalies, and possible beyond-SM structures [Skyrme, 1961, Faddeev and Niemi, 1996, Ma and Yang, 2016, Volovik, 2003].

* Cosmology and dark sector. Another paper should develop $\Lambda$-cosmology in detail: $\Lambda^P$-density driven $H_0$ variation, DM as curvature in the mirror sheet $M^-$ [Foot, 2004, 2014, Ciarcelluti, 2009, Yin et al., 2020], DE as slow $\Theta$-drift ("cosmic osmosis"), and the consistency of the tan $\Theta$ explanation of dark-sector ratios with current and future datasets (Planck; Euclid, DESI, LSST, Roman).

* High-coherence systems and laboratory tests. A third line of work should focus on the laboratory pillar: coherence-tunable gravito-inertial response and cavity shifts in superconductors, BECs, photonic condensates, and engineered metamaterials. This includes both dedicated experiments and reinterpretation of existing high-precision data.

Each sectorial paper should translate the qualitative picture given here into explicit models and falsifiable predictions, compatible with the empirical framework of Sec. 6.

## 8.4.3 Toward an Ecology of $\Lambda$-Coherence

Finally, the broader informational and ecological implications of $\Lambda$-Theory are explicitly speculative and lie outside the core validation program developed in Secs. 5–7. In the conservative reading adopted so far, biological, cognitive, cultural, and planetary processes are only treated as possible $\Lambda$-bridges stabilizing $\Theta$ in narrow viability bands, potentially forming a multi-scale “ecology of $\Lambda$-coherence” [Prigogine and Stengers, 1984, Lovelock, 1979, Varela et al., 2001, West, 2017].

Within this cautious framing, the framework simply offers a language to parameterize cross-scale coherence without committing to any specific “noospheric” mechanism. The default null hypothesis is that, at macroscopic scales, there is no additional $\Lambda^P-\Lambda^I$ coupling specifically associated with cognitive or social variables beyond standard electromagnetic, gravitational, and quantum interactions. Future work could, in principle:

* model how $\Lambda^P-\Lambda^I$ exchanges constrain the long-term stability of biospheres and technospheres;

* characterize collective coherence order parameters (e.g. $|\psi|$ for neural or social systems) and their possible coupling to $\Lambda^I$ at planetary scale;
* use controversial claims of global coherence effects (RNG networks, Schumann resonances) as simple *null tests*, i.e. as empirical constraints on any hypothesized cross-scale Λ-coupling [Nelson and Bancel, 2011, Nelson and collaborators, 2021, Nickolaenko and Hayakawa, 2014].

In all such cases, Λ-Theory does not assume that non-local coherence effects exist beyond standard physics: it merely provides a unified parametrization in which such claims can be bounded or falsified. The consistency checks, minimal validation triad, and cosmological signatures discussed in Secs. 6-7 remain fully intact even if all such effects turn out to be null within observational precision; the question of whether information-rich structures (life, mind, culture) feed back on the physical domain can then be addressed, if at all, strictly as an optional extension subject to the same fundamental constraints.
# 9 Conclusion

For centuries, physics has progressed by isolating fragments of reality. We learned to quantify matter, then energy, then space and time, each captured by its own formalism and set of “fundamental” constants. This strategy has been remarkably successful, yet it leaves us with a patchwork: general relativity, quantum field theory, thermodynamics, information theory, and neuroscience coexist without a common structural substrate.

The Λ-framework explored here proposes such a substrate. Rather than taking objects, fields, or observers as primitive, it starts from a discrete relational network and a single invariant tension Λ₀, decomposed into two orthogonal components:

Λ<sub>P</sub>(curvature, energetic cost, metric structure), Λ<sub>I</sub>(coherence, informational order, memory), subject to the conservation law

$$ \Lambda_P^2 + \Lambda_I^2 = \Lambda_0^2. $$

The phase Θ selects how much of Λ₀ is expressed as curvature versus coherence in any given regime.

In this picture:

* spacetime and gravitation emerge in regimes where Λ<sup>P</sup> dominates (small Θ);

* non-equilibrium matter and life arise near a balance Λ<sup>P</sup> ≈ Λ<sup>I</sup>, where curvature-coherence exchange can sustain low-entropy structures;

* cognitive organization corresponds to configurations where this exchange becomes reflexive and self-referential;

* coherence-dominated phases (Θ → π/2) act as informational boundaries, storing and redistributing order without introducing new substances.

The core proposal is conceptually modest but far-reaching: physics, information, and cognition are different phase regimes of a single relational tension field, rather than separate ontological categories. Constants become geometric parameters of this field; “laws” become grammars of allowed phase transformations.

Empirically, Λ-Theory is not free to describe anything. It is constrained by:

* the fixed invariant Λ₀ and the angular structure encoded by Θ;

* the architecture of the RTQ network (discrete substrate, internal axes d₁–d₅, bi-sheet structure);

* standard precision tests on gravitational-wave speed, Lorentz invariance, the equivalence principle, and early-universe cosmology (CMB/BAO/BBN);

* dark-sector observables (dark-matter/energy ratios, H₀ variability and anisotropy);

* possible signatures of coherence-tunable gravitation or curvature-coherence feedback in laboratory systems.

A single clear violation of any Tier A prediction, or of any of the Checks 1–7, would falsify the framework. Partial confirmation would still leave it in competition with other relational or emergent approaches.

Conceptually, however, the Λ-formalism offers something that current fragmented models lack: a quantitative way to talk about how geometric structure and informational order co-vary across scales, from microscopic interactions to biological organization and collective dynamics. Questions usually phrased in terms of “meaning”, “information”, or “global coherence” need not be banished from physics, provided they are reformulated as constraints on Λ<sup>P</sup>–Λ<sup>I</sup> exchange and on the allowed flows between metric domains and their coherence boundaries.

Whether Λ-Theory ultimately proves correct is an empirical matter. Its value, even if it fails, is to sharpen a testable version of a long-standing intuition: that the apparent convergence of gravitation, quantum theory, thermodynamics, and information is not accidental, but the shadow of a deeper unity in which curvature and coherence are two faces of the same underlying fabric.
# A Reproducibility / Audit Manifest

## A.1 Constants graph and identity/definition set used for canonicalization

The network diagrams shown in the main text were developed as exploratory, reader-facing visualizations (initially programmatically, then refined manually). They are not claimed to “discover” the ($G$, $A$, $S$) structure. Instead, this appendix specifies a closed-scope audit protocol:

Given (i) the four sector determinants explicitly anchored in Table 1, (ii) a restricted library of exact textbook identities and representation changes, and (iii) the explicitly introduced model-level alignment object $\lambda^P$ in the gravitational sector, the audit verifies that the factorization

$$C_i = G_i \times A_i \times S_i$$

is canonical, SI-traceable, and invariant under equivalent parametrizations.

Accordingly, the audit addresses reproducibility and non-arbitrariness of the factorization, not historical provenance. It neither searches for interaction “candidates” nor generates alternative determinants. Its sole function is to guaranty that, for each fixed sector determinant $C_i$ (the four anchors of Table 1), every admissible textbook representation reduces to the same canonical triple ($G_i$, $A_i$, $S_i$).

### A.1.1 Graph definition (nodes / edges)

We represent the constants dataset as a directed multigraph

$$G = (V, E).$$

Nodes. $V$ contains the canonical symbols needed to express the Table 1 determinants and their allowed representation changes, including at least

$$\{c, \hbar, G, k_B, e, \varepsilon_0, \mu_0, k_e, \alpha, \alpha_s, \alpha_G, Q, M_W, \theta_W, G_F, \Lambda_{GR}, \langle\rho\rangle, \dots\},$$

plus any explicitly declared derived objects used as determinants or carriers in the manuscript (e.g. $e_*^2$, $g_{s,*}^2(Q^2)$, declared reference scales for $Q$, etc.). Each node carries:

1. a canonicalized symbol (A.1.3);

2. an audited SI dimension vector (A.3);

3. a role tag in {geom, alignment, carrier, determinant}.

Optional internal-axis notation. When referring to internal axes $d_1$ (U(1)), $d_2$ (SU(2)), $d_3$ (SU(3)), one may introduce auxiliary axis-specific parameters for interpretation. In the audit itself, however, the only canonical alignment objects are those appearing in Table 1:

$$\alpha \text{ (EM), } \alpha \sin^{-2} \theta_W \text{ (weak), } \alpha_s(Q^2) \text{ (strong), } \alpha_G \text{ (gravity).}$$

Any $\Theta$-based interpretation of these objects is treated as a model-level reading and is not used as an audit rewrite rule.

Edges. $E$ contains only exact relations (no fitted relations), typed into three categories:

* Definitional edges (exact definitions), e.g.

$$k_e = (4\pi\varepsilon_0)^{-1}, \quad \alpha = \frac{k_e e^2}{\hbar c},$$

* Unit/representation edges (meaning-preserving changes), e.g.
$$h = 2\pi\hbar, \quad c = (\varepsilon_0\mu_0)^{-1/2},$$

* Determinant edges mapping each sector determinant to its $(\mathcal{G}, \mathcal{A}, \mathcal{S})$ factorization (Table 1).

The draw.io diagrams in the main text remain illustrative triggers. The audit graph $\mathcal{G}$ is formal and canonical: it contains a single literal $\pi$, and any “$\pi$-injection points” are recorded only as edge annotations (metadata), not as distinct nodes (cf. Appendix A.4.4).

### A.1.2 Canonical “target determinants” (Table 1 anchors)

The factorization is performed only for the determinants explicitly anchored in Table 1:

* Electromagnetism

$$C_{EM} \equiv \alpha \hbar c \equiv e_*^2 \equiv k_e e^2 \equiv \frac{e^2}{4\pi\varepsilon_0}.$$

* Weak

$$C_W \equiv G_F.$$

* Strong

$$C_S(Q^2) \equiv g_{s,*}^2(Q^2) \equiv 4\pi \alpha_s(Q^2) \hbar c, \quad \alpha_s(Q^2) \equiv \frac{g_s^2(Q^2)}{4\pi}, \quad g_{s,*}^2(Q^2) \equiv g_s^2(Q^2) \hbar c.$$

Here $g_s(Q^2)$ and $\alpha_s(Q^2)$ are dimensionless, while $g_{s,*}^2(Q^2)$ carries the dimensions of $\hbar c$, consistently with Table 1.

* Gravitation

$$C_G \equiv G_P.$$

In the audit, $G_P$ is the gravitational sector determinant whose calibrated SI value is identified with the measured Newton constant $G$ (numerical equality under the declared SI mapping in A.3), while keeping the internal $(\mathcal{G}, \mathcal{A}, \mathcal{S})$ decomposition explicit.

Scope closure. No determinants beyond the four fixed anchors are introduced by the audit. It documents inadmissible operations and representation-breaking manipulations—i.e., steps that fall outside the declared admissible language and that would otherwise create the illusion of extra freedom.

### A.1.3 Rewrite language and canonicalization

#### Rewrite language and canonicalization

We use a finite, terminating rewrite convention $R = \{r_k\}$ whose sole purpose is to *canonicalize* representations of the already-fixed sector determinants $C_i$ (Table 1 anchors) and to extract a unique $(\mathcal{G}_i, \mathcal{A}_i, \mathcal{S}_i)$ factorization. The audit is *non-generative*: it does not discover, rank, or optimize over alternative interaction candidates.

Before applying any rule, each expression is normalized to remove purely representational degrees of freedom (equivalent parametrizations, trivial algebraic rearrangements, and unit-convention rearrangements when explicitly declared), so that only physically meaningful—and auditable—content remains.

#### Geometric prefactors are fixed by declared conventions

Factors such as $4\pi$, $8\pi$, $2\sqrt{\pi}$, etc., are not “tuned” during the audit. They enter only through standard textbook definitions or coupling normalizations. For documentation purposes, locations where such factors enter may be recorded as an edge annotation

$$\text{geoHint} \in \{\pi, 2\pi, 4\pi, 8\pi, 2\sqrt{\pi}, \dots\},$$
but these annotations are non-generative and do not introduce additional algebraic freedom.

### Allowed rule sources (closed world)

The rewrite convention is restricted to:

(a) exact definitional equalities;

(b) representation changes that preserve meaning;

(c) sector-standard relations used in mainstream formulations of the same determinant.

No approximations promoted to equalities, no fitted integers/rationals, and no cross-sector “coincidence” identities are admitted (A.2.1). The rewrite convention is applied only to reduce admissible textbook representations of a fixed $C_i$ to its canonical normal form.

### A.1.4 Core rewrite rules used to recover Table 1

#### What is (and is not) allowed to vary

The audit does not permit free tuning of numerical prefactors. The only admissible geometric prefactors are those that enter through (i) standard sector conventions (e.g. $4\pi$ in the definition of $\alpha$ or $\alpha_s$), or (ii) explicit normalization choices already fixed by the mainstream form of the corresponding action or field equations (e.g. $1/8\pi$ in the Einstein-Hilbert normalization). Any non-standard integer/rational inserted solely to improve “fit” is rejected by the numerology filters (A.2.1).

Below is the minimal rule set sufficient to derive the Table 1 factorizations.

#### (EM) Electromagnetism

Rules (definitional / representation-preserving):

$$r_{EM1}: e_*^2 \Rightarrow k_e e^2, \quad r_{EM2}: k_e \Rightarrow (4\pi\epsilon_0)^{-1}, \quad r_{EM3}: \alpha \Rightarrow \frac{k_e e^2}{\hbar c}.$$

Then the admissible forms

$$k_e e^2, \quad \frac{e^2}{4\pi\epsilon_0}, \quad \alpha \hbar c$$

are reduced by $R$ to the canonical determinant form $C_{EM} = \alpha \hbar c$, hence

$$(\mathcal{G}_{EM}, \mathcal{A}_{EM}, \mathcal{S}_{EM}) = (1, \alpha, \hbar c).$$

Alternative EM representations (e.g. directly in terms of $\epsilon_0, \mu_0$, or via other equivalent textbook parametrization) are treated as invariance tests (A.4), not as distinct determinants.

#### (Weak) Weak sector

We enforce the Table 1 carrier choice

$$S_W \equiv M_W^{-2},$$

with $M_W$ interpreted as an energy scale in SI (see A.3.2). We also enforce the Table 1 alignment object

$$\mathcal{A}_W \equiv \alpha \sin^{-2}\theta_W$$

and the geometric factor $\mathcal{G}_W = \pi/\sqrt{2}$.

A minimal textbook rewrite path is:

$$r_{W1}: G_F \Rightarrow \frac{g^2}{4\sqrt{2}} M_W^{-2}, \quad r_{W2}: g^2 \Rightarrow \frac{4\pi\alpha}{\sin^2\theta_W}.$$
Combining gives the canonical Table 1 form

$$C_W = G_F \Rightarrow \frac{\pi}{\sqrt{2}} \alpha \sin^{-2}\theta_W M_W^{-2},$$

so that

$$(G_W, A_W, S_W) = \left(\frac{\pi}{\sqrt{2}}, \alpha \sin^{-2}\theta_W, M_W^{-2}\right).$$

Any equivalent Standard Model parametrization (e.g. via $g$ and $M_W$, or via the Higgs vacuum expectation value $v$) is treated as an invariance test (A.4), not as an alternative definition.

### (Strong) Strong sector

We enforce the Table 1 carrier choice

$$S_S \equiv \hbar c, \quad A_S \equiv \alpha_S(Q^2), \quad G_S \equiv 4\pi.$$

Rules:

$$r_{S1}: g_{s,*}^2(Q^2) \Rightarrow 4\pi \alpha_s(Q^2) \hbar c, \quad r_{S2}: \alpha_s(Q^2) \Rightarrow \frac{g_s^2(Q^2)}{4\pi}, \quad r_{S3}: g_{s,*}^2(Q^2) \Rightarrow g_s^2(Q^2) \hbar c.$$

Then

$$C_S(Q^2) \equiv g_{s,*}^2(Q^2) \Rightarrow 4\pi \alpha_s(Q^2) \hbar c,$$

hence

$$(G_S, A_S, S_S) = (4\pi, \alpha_S(Q^2), \hbar c).$$

If alternative QCD conventions are used (different normalizations of $g_s$ or $\alpha_s$), the audit requires an explicit mapping in A.3 and an invariance check in A.4.

### (Gravitation) Gravitational sector

We enforce the Table 1 alignment and carrier:

$$A_G \equiv \alpha_G, \quad S_G \equiv c^3 \hbar^{-1} (\langle \rho \rangle)^{-1}.$$

The gravitational determinant is rewritten as:

$$r_{G1}: G \Rightarrow \frac{1}{8\pi} \alpha_G \frac{c^3}{\hbar} (\langle \rho \rangle)^{-1}.$$

Hence:

$$(G_G, A_G, S_G) = \left(\frac{1}{8\pi}, \alpha_G, c^3 \hbar^{-1} (\langle \rho \rangle)^{-1}\right).$$

We denote by $\langle \rho \rangle$ the coarse-grained, areal-density carrier used in the gravitational sector; angled brackets here do not denote a quantum expectation value. $\langle \rho \rangle$ sI audited dimensions are declared explicitly (A.3).

## A.2 Admissible operation language (scope closure; anti-post-hoc freedom)

The audit is intentionally non-generative: it does not enumerate alternative “candidate determinants”, and it does not introduce any determinants beyond the four fixed anchors $C_i$. Its role is purely canonicalization: within a declared scope, it ensures that equivalent textbook representations of a fixed $C_i$ reduce to the same canonical triple $(G_i, A_i, S_i)$.
Accordingly, we explicitly define the admissible operation language used by the audit. Any manipulation outside this language is simply out of scope and is not part of the method, even if it can be arranged to reproduce a numerical value. This prevents post-hoc normalization freedom by construction: the only permitted steps are those that preserve meaning, remain SI-auditable, and are invariant under equivalent parametrizations.

### A.2.1 Admissible identities (standard-definition closure)

Only standard identities and definitions are admissible, in the following strict sense:

* No approximate relations are used as equalities (e.g., no use of “≈” or asymptotic substitutions to engineer exact cancellations or to manufacture $\pi$-factors).

* No empirically tuned integers/rationals are inserted unless they belong to explicit, mainstream conventions (e.g., $4\pi$ in the Coulomb-law normalization, $1/8\pi$ in the Einstein-equation convention, etc.).

* No cross-sector “numerical match” relations are introduced: sectors are linked only through textbook relations that are already standard in the literature.

With this closure, $G_i$ factors can only originate from declared conventions, never from numerological tuning.

### A.2.2 SI-auditable carriers (dimensional traceability)

The dimensional carrier $S_i$ must be operationally pinned and auditable in pure SI. In particular:

* Conversions between mass and mass-energy must carry explicit $c^2$ bookkeeping; toggling between “$M$” and “$E$” without declaring the mapping is out of scope.

* A scale symbol must keep a declared meaning (e.g., $Q$ cannot silently change role between energy, momentum, or inverse length).

* By convention, $A_i$ remains strictly dimensionless; all SI dimensions are carried by $S_i$. Any dimensional reshuffling into $A_i$ is out of scope.

This block is the SI traceability guardrail: the audit is performed in $(kg, m, s)$ units, with explicit mappings whenever a sector uses alternate conventions.

### A.2.3 Invariance under equivalent parametrizations (representation closure)

Canonicalization must be invariant under meaning-preserving rewrites. Concretely, if the same fixed determinant $C_i$ is written in equivalent textbook forms, the extracted triple $(G_i, A_i, S_i)$ must be the same up to declared mappings. Examples of equivalent parametrizations include:

* $\alpha$ vs. $e$ vs. $\epsilon_0$-based forms in electromagnetism;

* alternate but equivalent symbolizations (aliases) that do not change meaning;

* sector-specific convention changes (e.g., QCD normalizations of $g_s$ vs. $\alpha_s$) provided the mapping is declared and the canonical triple is shown invariant once mapped.

This is not a “selection” step; it is the defining requirement of canonicalization: representation choices are allowed only insofar as they preserve the same canonical output.

### A.2.4 Audit record (what is documented)

For transparency, the supplementary material records:

* the declared admissible identity/definition set used for canonicalization;

* the SI mapping conventions used to pin $S_i$;

* the invariance checks showing that equivalent textbook representations of each $C_i$ collapse to the same $(G_i, A_i, S_i)$.

This record is not a catalog of “alternatives”. It is the minimal documentation needed to audit that the procedure is closed, SI-traceable, and invariant.
## A.3 Declared unit conventions and explicit SI audit mappings

The manuscript enforces the “units appear at projection” rule. However, the audit still requires explicit SI traceability for all carrier objects $S_i$ and for any sector-specific conventions.

### A.3.1 SI audit basis

We use SI base units ($m, kg, s, A$), with $C = As$ and $J = kg m^2 s^{-2}$.

No natural-units convention ($\hbar = c = 1$) is used anywhere in the audit.

### A.3.2 Sector carrier mappings (Table 1 and audit carriers)

Carriers are audited as follows.

#### Electromagnetism

Table 1 uses the carrier

$$S_{EM} = \hbar c, \quad [\hbar c] = J \cdot m = kg m^3 s^{-2}.$$

The Coulomb form $k_e e^2 = e^2/(4\pi\epsilon_0)$ is used as a cross-check (A.4).

**Weak sector** Table 1 uses the carrier

$$S_W = M_W^{-2}$$

with $M_W$ interpreted as a mass-energy scale:

$$[M_W] = J, \quad [M_W^{-2}] = J^{-2} = kg^{-2} m^{-4} s^4.$$

This matches the declared dimensions of the Fermi constant $G_F$, so that the determinant $C_W = G_F$ can be written in the Table 1 factorized form with

$$(G_W, A_W, S_W) = (\frac{\pi}{\sqrt{2}}, \alpha \sin^{-2} \theta_W, M_W^{-2}).$$

in audited SI.

For dimensional arguments in Appendix B it is convenient to also introduce an *audit carrier* for the weak sector,

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

whose SI dimension is

$$[S_W^{(audit)}] = [G_F]^{1/2} = kg^{-1} m^{-2} s^2.$$

This repackaging leaves the canonical triple $(G_W, A_W, S_W)$ of Table 1 unchanged; it is used only to express the weak sector as a structural carrier in the dimensional analysis with a free spatial dimension $d$ (Appendix B.5).

#### Strong interaction

Table 1 uses

$$S_S = \hbar c, \quad A_S = \alpha_s(Q^2),$$

so that

$$[C_S(Q^2)] = [g_{s,*}^2(Q^2)] = [\hbar c] = kg m^3 s^{-2}.$$

If one chooses to explicitly represent the underlying coupling $g_s$, then

$$[g_s] = 1, \quad [\alpha_s] = 1, \quad g_{s,*}^2 = g_s^2 \hbar c.$$
Any running interpretation of $\alpha_s(Q^2)$ requires an explicit declaration of what $Q$ denotes (energy, momentum transfer, or inverse length), the unit convention used, and the mapping between conventions (e.g. $E = pc$, $k = p/\hbar$).

## Gravitation

Table 1 uses

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

As

$$G = \frac{1}{8\pi} \alpha_G S_G, \quad [\lambda^P] = 1,$$

the audit enforces

$$[S_G] = [G].$$

Therefore,

$$[c^3 \hbar^{-1} \langle \rho \rangle^{-1}] = [G] \Rightarrow [\langle \rho \rangle^{-1}] = [G] [\hbar] c^{-3}.$$

Consequently,

$$[\langle \rho \rangle] = ([G] [\hbar] c^{-3})^{-1} = m^{-2},$$

consistent with a coarse-grained areal-density (inverse-area) carrier. Any alternative label used in exploratory figures is non-binding; the SI audit declaration above is the authoritative dimensional mapping.

## Structural reading

In the $\Lambda$-framework these carriers are interpreted as minimal structural requirements for consistent RTQ propagation in each sector, rather than arbitrary bookkeeping choices. Electromagnetism and the strong interaction share a *volumic* carrier

$$[S_{EM}] = [S_S] = kg m^3 s^{-2},$$

compatible with a genuinely three-dimensional propagation medium. The weak sector is anchored on a *short-range*, *quasi-surfacic* carrier,

$$[S_W^{(audit)}] = kg^{-1} m^{-2} s^2,$$

while gravity is tied to a *curvature-like*, *codimension-2* tension carrier $S_G$ with $[\langle \rho \rangle] = m^{-2}$. In Appendix B.5 these carriers are promoted to functions of a free spatial dimension $d$ and combined with the RTQ spectral analysis to constrain the effective dimensionality of the emergent 4D $\Lambda$-framework.

### A.3.3 Sector-specific “carrier hygiene” rules

* A carrier is not allowed to “silently” include $\hbar$ or $c$. If a mass is used instead of a mass-energy, the explicit $c^2$ factor is required.

* A carrier definition may be augmented by a declared reference scale (e.g. $Q_0$) to keep carriers dimensionless, but that augmentation must be explicit and listed.

## A.4 Invariance checks under equivalent parametrizations and stable ($G, A, S$) recovery

This subsection specifies the invariance tests used to ensure that the extracted structure is not an artifact of representation choices.
### A.4.1 Canonical triple extraction

For each sector $i$, apply the rewrite system $R$ to reduce any admissible representation of the determinant $C_i$ into the canonical normal form:

$$C_i \Rightarrow \mathcal{G}_i A_i S_i,$$

with $\mathcal{G}_i \in \{1, \pi/\sqrt{2}, 1/(8\pi), \dots\}$ drawn from the declared geometric library, $A_i$ dimensionless, and $S_i$ matching the Table 1 carrier.

### A.4.2 Example invariance: $\alpha$ vs. $e$ (electromagnetism)

Starting from any of the admissible EM representations:

$$k_e e^2, \quad \frac{e^2}{4\pi\varepsilon_0}, \quad \alpha \hbar c,$$

the audit must recover the same canonical triple:

$$(\mathcal{G}_{EM}, \mathcal{A}_{EM}, \mathcal{S}_{EM}) = (1, \alpha, \hbar c).$$

Failure to recover the same triple is logged as an invariance break.

### A.4.3 Representation invariance: $\varepsilon_0$ vs. $\mu_0$, $h$ vs. $\hbar$

Equivalent rewrites such as

$$c = (\varepsilon_0 \mu_0)^{-1/2}, \quad h = 2\pi\hbar$$

must not change the extracted $(\mathcal{G}, \mathcal{A}, \mathcal{S})$ beyond the explicitly declared regrouping (e.g. pushing $2\pi$ factors into $\mathcal{G}$ vs $\mathcal{A}$ according to the canonical geometric library). Any “migration” of dimensional content into $\mathcal{A}$ is forbidden and counted as a failed audit.

### A.4.4 “$\pi$-injection markers” are metadata, not algebra

Exploratory diagrams may display multiple “$\pi$” icons to keep the visual map readable (injection markers). The audit graph does not instantiate multiple $\pi$ nodes. Instead:

* $\pi$ is a single node (or literal constant);

* the locations where $\pi$, $4\pi$, $8\pi$, $2\sqrt{\pi}$, ... enter are stored as edge annotations *geoHint*;

* invariance requires that collapsing visual markers into a single $\pi$ symbol leaves all canonical triples unchanged.

### A.4.5 Output of invariance testing

The invariance suite outputs, for each sector $i$:

* the set of admissible representations tested;

* the canonical triple $(\mathcal{G}_i, A_i, S_i)$ recovered;

* pass/fail flags;

* if failed, the minimal counterexample rewrite sequence.

The stable structure reported in the main text is the subset of factorizations that pass all invariance checks and match Table 1. Dimensional signatures of the determinants in the $(\hbar, c)$ sublattice (when expressed in audited SI) are treated as diagnostic invariants during canonicalization, not as additional assumptions.