### E.4.5.6 Interpretive note (documentary only)

If $\delta\Theta_{ij}^{(s)}$ displays a structured sign pattern, this can be read (within MCP language) as an “elastic load” diagnostic of the RTQ network under different constraints (e.g. colour channel, helicoidal coupling), without promoting $\Theta$ to a sector-dependent free parameter.

### E.4.6 Documentary note: sectoral “breathing” of the effective projection angle

<table><caption>Table 18: Documentary effective angles $\Theta_{\text{eff},ij}^{(s)}$ extracted by inversion (Eqs. (5)–(6)). Values are reported in radians (degrees in parentheses).</caption><thead><tr><th>Storey</th><th>$\Theta_{\text{eff}}^{(\text{L})}$</th><th>$\Theta_{\text{eff}}^{(\text{U})}$</th><th>$\Theta_{\text{eff}}^{(\text{D})}$</th></tr></thead><tbody><tr><td>1 → 2</td><td>0.362819617 rad (20.788033°)</td><td>0.352143550 rad (20.176339°)</td><td>0.363380228 rad (20.820153°)</td></tr><tr><td>2 → 3</td><td>0.352287595 rad (20.184592°)</td><td>0.362825500 rad (20.788370°)</td><td>0.362825500 rad (20.788370°)</td></tr></tbody></table>

#### E.4.6.1 Chiasm (swap) across storeys.
Table 18 exhibits a striking permutation: for the first jump $1 \to 2$, the Up channel sits on the “low” branch while Down and Leptons sit on the “high” branch; for the second jump $2 \to 3$, Leptons move to the “low” branch while Up and Down share the “high” branch. This suggests that the discriminator is not only the presence/absence of the colour axis $d_3$, but also the operator type (linear in $\Theta$ vs quadratic in $\Theta^2$) active at each storey.

#### E.4.6.2 Compact branch summary (documentary)

For the $2 \to 3$ storey in the current minimal skeleton,

$$ \Theta_{\text{low}} := \Theta_{\text{eff},23}^{(\text{L})}, \qquad \Theta_{\text{high}} := \Theta_{\text{eff},23}^{(\text{U})} \approx \Theta_{\text{eff},23}^{(\text{D})}, $$

and we record the midpoint and half-split

$$ \Theta_{\text{mid}} := \frac{\Theta_{\text{low}} + \Theta_{\text{high}}}{2}, \qquad \delta := \frac{\Theta_{\text{high}} - \Theta_{\text{low}}}{2}, $$

together with the sign pattern

$$ \delta\Theta_{23}^{(\text{L})} < 0, \qquad \delta\Theta_{23}^{(\text{U/D})} > 0 \quad (\text{diagnostic only; cf. Eq. (7)}). $$

#### E.4.6.3 A measurable “torsion load” diagnostic (optional)

Once $\Theta_{\text{struct}}$ is fixed by the cosmological construction, $\delta\Theta_{ij}^{(s)}$ in Eq. (7) can be correlated a posteriori with discrete gates ($w_{d3}$, $\text{tilt}_{34}$) and with confinement-related closure defects. This introduces no new PoC parameter: it is a diagnostic observable.

#### E.4.6.4 Diagonal vs off-diagonal (mass vs mixing) — separation of roles

Finally, we reiterate the audit separation: diagonal helix/twist penalties (e.g. $\text{tilt}_{34}$ and helixFactor) belong to the template/sector layer and affect mass-like costs in Sec. E.5, whereas Cabibbo/Wolfenstein-like structures are mixing operators (off-diagonal) and must be treated in a separate layer.

### E.4.7 Documentary addendum: explicit two-regime matching for the $2 \to 3$ jump

#### E.4.7.1 Two regimes for the same $2 \to 3$ jump: geometric anchor vs cosmological deformation

For the $2 \to 3$ stage we record, sector by sector, two complementary readings of the same logarithmic jump

$$ \Delta_{23}^{(s)} := \ln\left(\frac{n_3}{n_2}\right), \qquad s \in \{\text{L, U, D}\}. $$
We use the pivot notation (defined once for all):

$$A_{12} := \frac{3\pi}{2}, \quad A_{23} := \ln\left(\frac{\alpha^{-1}}{\sqrt{\pi}}\right), \quad \alpha \text{ is the fine-structure constant.}$$

Then the documented two-regime dictionary for the $2 \to 3$ stage is:

$$\textbf{Leptons (L):} \quad \Delta_{23}^{(L)}|_{\text{geo}} = \ln(\alpha^{-1}) \ln(\sqrt{\pi}), \qquad \Delta_{23}^{(L)}|_{\text{cosmo}} = A_{23}(1-\Theta),$$

$$\textbf{Up (U):} \quad \Delta_{23}^{(U)}|_{\text{geo}} = \ln(\alpha^{-1}), \qquad \Delta_{23}^{(U)}|_{\text{cosmo}} = A_{23}(1 + \Theta^2),$$

$$\textbf{Down (D):} \quad \Delta_{23}^{(D)}|_{\text{geo}} = \ln(\alpha^{-1}/\pi), \qquad \Delta_{23}^{(D)}|_{\text{cosmo}} = A_{23}(1 - \Theta^2).$$

Imposing $\Delta_{23}|_{\text{geo}} = \Delta_{23}|_{\text{cosmo}}$ yields $\Theta_{\text{eff},23}^{(s)}$ as already defined by the inversion formulas in Eq. (6); the observed degeneracy $\Theta_{\text{eff},23}^{(U)} \simeq \Theta_{\text{eff},23}^{(D)}$ in the minimal skeleton follows algebraically from the choice of pivot $\sqrt{\pi}$ (see the main text discussion).

### E.4.8 Closure bracket for $\Theta_{\text{struct}}$ from the (12) and (23) bridges

Note that the structural closure derived in Step 3 depends only on the $2 \to 3$ operator identity (linear vs quadratic) and on $\alpha$ once the $\sqrt{\pi}$ pivot is fixed; it does not require fixing any particular triplet $\{n_1, n_2, n_3\}$.

This addendum turns the six extracted bridge-angles $\theta_{12}^{(\cdot)}$, $\theta_{23}^{(\cdot)}$ (from Sec. E.4.5.2) into a necessary interval constraint for a single structural angle $\Theta_{\text{struct}}$, and then proposes a minimalist (alpha-only) closure candidate.

#### E.4.8.1 Step 1: explicit formulas for the ordered angles

Using the pivots $A_{12}$ and $A_{23}$ and the constant $\alpha$ defined in Sec. E.4.5.2, the six angles read:

##### (i) (12) bridge.

$$\theta_{12}^{(U)} = \frac{1 + \ln(\pi/2) + \ln(\alpha^{-1})}{A_{12}} - 1 = \frac{2}{3\pi} \left(1 + \ln(\pi/2) + \ln(\alpha^{-1})\right) - 1,$$

$$\theta_{12}^{(L)} = \sqrt{\frac{\ln(207)}{A_{12}} - 1} = \sqrt{\frac{2 \ln(207)}{3\pi} - 1},$$

$$\theta_{12}^{(D)} = 1 - \frac{3}{A_{12}} = 1 - \frac{2}{\pi}.$$

##### (ii) (23) bridge. Let $\ln(\sqrt{\pi}) = \frac{1}{2} \ln \pi$. Then

$$\theta_{23}^{(L)} = 1 - \frac{\ln(\alpha^{-1}) \ln(\sqrt{\pi})}{\ln(\alpha^{-1}/\sqrt{\pi})} = 1 - \frac{\ln(\alpha^{-1}) \frac{1}{2} \ln \pi}{\ln(\alpha^{-1}) - \frac{1}{2} \ln \pi},$$

$$\theta_{23}^{(U)} = \sqrt{\frac{\ln(\alpha^{-1})}{\ln(\alpha^{-1}/\sqrt{\pi})} - 1} = \sqrt{\frac{\ln(\alpha^{-1})}{\ln(\alpha^{-1}) - \frac{1}{2} \ln \pi} - 1},$$

$$\theta_{23}^{(D)} = \sqrt{1 - \frac{\ln(\alpha^{-1}/\pi)}{\ln(\sqrt{\pi}/\alpha^{-1})}}.$$
With $\sqrt{\pi}$ as the canonical (23) pivot, one checks algebraically that

$$ \theta_{23}^{(U)} = \theta_{23}^{(D)}, $$

so the (23) upper and down branches meet (this is the “$\sqrt{\pi}$-closure” already highlighted in Sec. E.4.5.2).

### E.4.8.2 Step 2: the inequality as a necessary compatibility constraint

Assume a single structural angle $\Theta_{struct}$ must be compatible with the whole two-regime bridge system (same $\Theta$ used across sectors, no retuning per family). Then the monotone ordering extracted in practice becomes a hard interval:

$$ \theta_{12}^{(U)} < \theta_{23}^{(L)} < \Theta_{struct} < \theta_{12}^{(L)} < \theta_{23}^{(U)} = \theta_{23}^{(D)} < \theta_{12}^{(D)}. \qquad (8) $$

In fully expanded form (only $\alpha$ and $\pi$, plus the documented 207 pivot):

$$ \frac{2}{3\pi} \left( 1 + \ln(\pi/2) + \ln(\alpha^{-1}) \right) - 1 < 1 - \frac{\ln(\alpha^{-1})}{\ln(\alpha^{-1}) - \frac{1}{2} \ln \pi} < \Theta_{struct} < \sqrt{\frac{2 \ln(207)}{3\pi}} - 1 < \sqrt{\frac{\ln(\alpha^{-1})}{\ln(\alpha^{-1}) - \frac{1}{2} \ln \pi}} - 1 < 1 - \frac{2}{\pi}. $$

Therefore, $\Theta_{struct}$ is not a free dial in Appendix E once the two-regime bridge constraints are enforced: it must lie in the (narrow) intersection interval defined by the (12) and (23) extracted angles.

### E.4.8.3 Step 3: structural closure from the $2 \to 3$ two-regime identity (no ad hoc choice)

A key feature of the $2 \to 3$ bridge is that the $\pi$-dependence cancels when one combines the lepton and quark operators. Write

$$ A := \ln(\alpha^{-1}), \qquad \ell := \ln\left(\frac{\alpha^{-1}}{\sqrt{\pi}}\right) = A - \frac{1}{2}\ln\pi. $$

From the documented $2 \to 3$ forms (Table 17),

$$ 1 - \theta_{23}^{(L)} = \frac{A \cdot \frac{1}{2} \ln \pi}{\ell}, \qquad (\theta_{23}^{(U)})^2 = \frac{\frac{1}{2} \ln \pi}{\ell}, $$

so dividing eliminates $\pi$ and yields the structural identity

$$ \frac{1 - \theta_{23}^{(L)}}{(\theta_{23}^{(U)})^2} = A = \ln(\alpha^{-1}). $$

Imposing this identity as the single-angle self-consistency condition for the structural equilibrium angle $\Theta_{struct}$ gives the closure equation

$$ \ln(\alpha^{-1}) = \frac{1 - \Theta_{struct}}{\Theta_{struct}^2} \iff \ln(\alpha^{-1}) \Theta_{struct}^2 + \Theta_{struct} - 1 = 0. \qquad (9) $$

Equivalently, the closure can be written directly as an $\alpha$-$\Theta$ exponential law:

$$ \alpha^{-1} = \exp\left(\Theta_{struct}^{-2} - \Theta_{struct}^{-1}\right) \qquad (10) $$

The corresponding structural angle is given in closed form by

$$ \Theta_{struct} \sim \Theta_{struct}^{(\alpha)} = \frac{\sqrt{1 + 4 \ln(\alpha^{-1})} - 1}{2 \ln(\alpha^{-1})}, \qquad (11) $$

where the negative root is discarded because the structural domain in MCP is $0 < \Theta < \pi/2$ (and because it would invert the linear $1 \to 2$ down-type operator, leading to the observed blow-up for strange/bottom in the PoC).
### E.4.8.4 Why this is a “closure”

Appendix E separates (i) a deterministic grammar (templates and bridges) from (ii) a mode-selection/closure layer. Here the closure is *not* an extra ansatz chosen “inside a bracket”: it follows directly from the $2 \to 3$ two-regime identities once the $\sqrt{\pi}$ pivot is fixed. Indeed, the (23) leptonic branch is linear in $\Theta$ while the (23) quark branch is quadratic in $\Theta^2$, and dividing the two relations eliminates the explicit $\pi$-dependence, yielding Eq. (9) and hence Eq. (11).

### E.4.8.5 Numerical value and comparison with the cosmological reference

Using $\alpha^{-1} \simeq 137.035999$ (so $\ln(\alpha^{-1}) \simeq 4.920243659$), Eq. (11) yields

$$ \Theta_{\text{struct}}^{(\alpha)} \approx 0.3605140524 \text{ rad } (\approx 20.655934^{\circ}), $$

which sits between the adjacent documentary bounds extracted at storey $2 \to 3$,

$$ \Theta_{\text{eff},23}^{(L)} \approx 0.3522876 < \Theta_{\text{struct}}^{(\alpha)} < \Theta_{\text{eff},23}^{(U)} = \Theta_{\text{eff},23}^{(D)} \approx 0.3628255. $$

For comparison, if the main text uses $\Theta_{\text{struct}}^{(\text{cosmo})} \approx 0.359$ rad, the offset is

$$ \Delta\Theta := \Theta_{\text{struct}}^{(\alpha)} - \Theta_{\text{struct}}^{(\text{cosmo})} \approx 1.51 \times 10^{-3} \text{ rad } (\approx 8.67 \times 10^{-2}^{\circ}), $$

while remaining inside the documentary compatibility bracket (Eq. (8)).

### E.4.8.6 Consistency with the fundamental angular dependence of $\alpha(\Theta)$

In the main text (Sec. 3.4.3.4), the $U(1)$ coupling is controlled by the geometric product of the two tension components, namely

$$ \alpha(\Theta) = \frac{1}{2} \sin(2\Theta) f_{U(1)}(\theta_e), \qquad (12) $$

where $f_{U(1)}(\theta_e)$ is the (sheet-local) electromagnetic efficiency factor calibrated on the reference electron motif/state $\theta_e$.

The structural closure derived here (Eqs. (9)–(11)) should *not* be read as an alternative functional law for $\alpha(\Theta)$ across arbitrary $\Theta$. It is a *mode-selection* (self-consistency) condition that fixes the unique equilibrium angle $\Theta_{\text{struct}}$ from $\alpha$ via the $2 \to 3$ two-regime identities (once the $\sqrt{\pi}$ pivot is fixed). Once $\Theta_{\text{struct}}$ is determined, Eq. (12) becomes an estimator for $f_{U(1)}(\theta_e)$:

$$ f_{U(1)}(\theta_e) = \frac{2\alpha}{\sin(2\Theta_{\text{struct}})}. \qquad (13) $$

Combining Eq. (10) with Eq. (13) yields the explicit single-angle prediction

$$ f_{U(1)}(\theta_e) = \frac{2}{\sin(2\Theta_{\text{struct}})} \exp(-\Theta_{\text{struct}}^{-2} + \Theta_{\text{struct}}^{-1}). \qquad (14) $$

For $\alpha^{-1} = 137.035999$ and $\Theta_{\text{struct}}^{(\alpha)} \approx 0.3605140524$ rad, Eq. (13) gives

$$ f_{U(1)}(\theta_e) \approx 2.210792 \times 10^{-2}, $$

i.e. a few-percent electromagnetic efficiency factor inferred from $\alpha$ at the selected $\Theta_{\text{struct}}$.
### E.4.8.7 Operational note (non-frozen $f_{U(1)}$ as a relaxation diagnostic)

The differential projection relations above are intentionally written in a minimal one-parameter form, with $\Theta$ as the unique control variable. Nevertheless, the $U(1)$ channel in the continuum interface is mediated by a sheet-local, motif-conditioned electromagnetic efficiency factor $f_{U(1)}(\theta_e)$, defined by calibration on the reference electron state $\theta_e$ (Eq. (12)).

Once $\Theta_{struct}$ is fixed by the 2→3 closure identities, Eqs. (13)–(14) do not introduce an additional degree of freedom: they provide an estimator for the implied electron-calibrated value of $f_{U(1)}(\theta_e)$ at the selected equilibrium angle. When the same fermionic grammar is applied across sectors in Appendix E, the corresponding inferred $f_{U(1)}$ values need not coincide exactly at intermediate steps (quark vs lepton motifs), and their departure from the electron reference is treated operationally as a diagnostic of residual sector-dependent relaxation within the shared canonical presets (templates, bridges, and pivots), rather than as a tunable parameter.

In particular, admissible scenarios must keep any motif-to-motif spread (or drift) of $f_{U(1)}$ consistent with the fixed $\alpha$ constraint and with the bridge/template compatibility conditions; large uncorrelated variations would signal a breakdown of the shared-projection assumption.

## E.4.9 Multi-estimators of $\Theta_{struct}$: closure bracket, EM buffer, and the Cabibbo coherence triangle

### E.4.9.1 Why several $\Theta$ values appear

In the framework, $\Theta_{struct}$ is introduced as a global invariant controlling the angular partition of the vacuum tension (structural sector), while $\Theta$ may be read locally with small sector-dependent distortions once a manifested network is created (“$\Theta_{manifest}$”). In practice, $\Theta$ can therefore be inferred through several channels (macro and micro) that need not yield identical central values at the $10^{-3}$ rad level, provided they remain mutually consistent within a tight closure interval.

### E.4.9.2 Channel C1 (cosmological): $\Theta$ from the dark-sector ratio

A first estimate follows from the angular partition of the dark sector, using

$$ \frac{\Omega_{DM}}{\Omega_{DE}} \approx \tan \Theta_{struct}, $$

which yields, for the working PR4-based value adopted throughout this manuscript,

$$ \Theta_{cosmo} = 0.35899 \text{ rad.} $$

The key point is that the discrepancy between $\Theta_{cosmo}$ and any nearby value $\Theta = \Theta_{cosmo} + \delta\Theta$ maps to a very small relative shift on the ratio $\Omega_{DM}/\Omega_{DE}$:

$$ \frac{\delta(\tan \Theta)}{\tan \Theta} \approx \frac{\sec^2 \Theta}{\tan \Theta} \delta\Theta \sim \mathcal{O}(10^{-3}) \times \left( \frac{\delta\Theta}{10^{-3}} \right), $$

so that $\delta\Theta \simeq 1.5 \times 10^{-3}$ rad corresponds to a sub-percent correction on $\Omega_{DM}/\Omega_{DE}$. This is fully compatible with the fact that $\Omega$-parameters are jointly inferred (correlated) in Planck pipelines, and that PR4-level refinements can shift derived ratios at the $10^{-3}-10^{-2}$ level without any conceptual tension.

### E.4.9.3 Channel C2 (micro EM-closure): $\Theta$ from $\alpha$ with an EM buffer

A second estimate is obtained from the $\alpha$-closure derived in Appendix E (electromagnetic (EM) closure channel). In its simplest closed form it yields

$$ \Theta_{\alpha} = 0.360514 \text{ rad}, \quad \Delta\Theta \equiv \Theta_{\alpha} - \Theta_{cosmo} \simeq 1.51 \times 10^{-3} \text{ rad.} $$
At first glance, one might expect $\Theta_\alpha$ and $\Theta_{\text{cosmo}}$ to coincide exactly. However, the EM closure in MCP is not a bare one-variable identity: it explicitly contains a $U(1)$ electromagnetic efficiency factor acting as a physical buffer between the structural angle and the measured fine-structure coupling. The working relation is

$$ \alpha(\Theta) = \frac{1}{2} \sin 2\Theta f_{U(1)}(\theta_e), $$

so that, for fixed $\alpha$, small shifts in $\Theta$ are absorbed by small shifts in $f_{U(1)}$:

$$ f_{U(1)}(\Theta) = \frac{2\alpha}{\sin 2\Theta}. $$

Linearizing,

$$ \frac{\delta f_{U(1)}}{f_{U(1)}} = -2 \cot 2\Theta \delta\Theta. $$

Evaluated around $\Theta \simeq 0.359$ rad, the observed $\Delta\Theta \simeq 1.5 \times 10^{-3}$ rad only requires a few $10^{-3}$ relative change in $f_{U(1)}$ (sub-percent). This provides a clean interpretation: $\Theta_{\text{cosmo}}$ is the best proxy for the global structural value, while $\Theta_\alpha$ is a micro-closure readout slightly renormalized by the local EM channel via $f_{U(1)}$. In other words, the EM coupling plays the expected role of a tampon between two otherwise rigid-looking formulae.

### E.4.9.4 Channel C3 (spectral/mass): $\Theta$ from the quark slope asymmetry $\kappa_{ij}$

A third, purely spectral estimate follows from the symmetric/antisymmetric decomposition of the logarithmic generation costs in the $U$ and $D$ families. Using renormalized indices $n$ with $n_1 = 1$, define

$$ \Delta_{ij}^{(U)} \equiv \ln \left( \frac{n_j^{(U)}}{n_i^{(U)}} \right), \quad \Delta_{ij}^{(D)} \equiv \ln \left( \frac{n_j^{(D)}}{n_i^{(D)}} \right), $$

and the normalized asymmetry

$$ \kappa_{ij} \equiv \frac{\Delta_{ij}^{(U)} - \Delta_{ij}^{(D)}}{\Delta_{ij}^{(U)} + \Delta_{ij}^{(D)}}. $$

With the observed sequences

$$ n^{(U)} = (1, 584, 79408), \quad n^{(D)} = (1, 20, 874), $$

one finds numerically

$$ \kappa_{12} \approx 0.360, \quad \kappa_{23} \approx 0.131, \quad \text{and} \quad \kappa_{23} \approx \kappa_{12}^2, $$

revealing a Wolfenstein-like hierarchical pattern directly at the level of mass-slope costs. The important conceptual point is that $\kappa_{12}$ and $\kappa_{23}$ are sectoral readouts: they probe how each family experiences the structural invariant through its own constraints (tilt, $d_3$-gates, confinement topology). Therefore, the fact that $\kappa_{12}$ sits closer to $\Theta_\alpha$ than to $\Theta_{\text{cosmo}}$ is not paradoxical; it is the expected signature that the spectral channel is slightly more sensitive to the micro-closure pathway (where $\alpha$ and the EM buffer enter explicitly).

### E.4.9.5 The closure bracket: $\Theta_{\text{struct}}$ trapped in a narrow interstice

When the generation formulas are inverted sector-by-sector to extract the “effective angle” $\theta_{ij}^{(s)}$ that would reproduce exactly each observed jump, one obtains a small spread of values bracketing the structural angle:

$$ \theta_{12}^{(U)} < \theta_{23}^{(L)} < \Theta_{\text{struct}} < \theta_{12}^{(L)} \le \theta_{23}^{(D)} < \theta_{12}^{(D)}. $$

Numerically, the bracket width is of order $10^{-2}$ rad, while $\Theta_{\text{cosmo}}$ and $\Theta_\alpha$ lie safely inside it. This is interpreted as a controlled “sectoral breathing”: the global vacuum equilibrium is unique, but each channel reads it with a small, physically interpretable distortion.
### E.4.9.6 The Cabibbo coherence triangle: a micro-macro bridge

Let $\lambda$ denote the Wolfenstein parameter (first-order Cabibbo mixing), conventionally defined from the Cabibbo-Kobayashi-Maskawa (CKM) matrix as $\lambda \simeq \sin \theta_{12}$ (numerically $\lambda \approx 0.225$ at the Particle Data Group (PDG) level). We identify an empirical but highly constraining bridge:

$$ \Theta_{\text{struct}} \approx \sqrt{\frac{8}{\pi}} \lambda \quad \Leftrightarrow \quad \lambda \approx \Theta_{\text{struct}} \sqrt{\frac{\pi}{8}} $$

For $\Theta_{\text{cosmo}} = 0.35899$, this yields

$$ \lambda_{\text{pred}} = 0.35899 \sqrt{\frac{\pi}{8}} \approx 0.224964, $$

remarkably close to PDG-level determinations.

Is $\sqrt{8/\pi}$ numerology? In MCP terms, this factor is precisely the type of coefficient that arises when an underlying structural angle is not read directly, but through an isotropic projection / coherence averaging mechanism. Indeed,

$$ \sqrt{\frac{8}{\pi}} = 2\sqrt{\frac{2}{\pi}}, $$

and $\sqrt{2/\pi}$ is a universal factor appearing in mean-projection problems (e.g., averaging absolute Gaussian projections or isotropic angular readouts). The interpretation is therefore not that $\pi$ is inserted ad hoc, but that weak mixing $\lambda$ behaves as a projected signature of the same control parameter $\Theta_{\text{struct}}$, with $\pi$ encoding the isotropic geometry of the readout.

### E.4.9.7 Bridge $\lambda \leftrightarrow \kappa_{ij}$

Combining the two empirical identifications

$$ \kappa_{12} \approx \Theta_{\text{struct}}, \quad \Theta_{\text{struct}} \approx \sqrt{\frac{8}{\pi}} \lambda, $$

one obtains the direct spectral-weak bridge

$$ \kappa_{12} \approx \sqrt{\frac{8}{\pi}} \lambda $$

and, using $\kappa_{23} \approx \kappa_{12}^2$,

$$ \kappa_{23} \approx \frac{8}{\pi} \lambda^2, $$

which makes the “Wolfenstein-like” hierarchy visible simultaneously in (i) the mass-slope asymmetry and (ii) weak-flavor mixing. In this reading, $\lambda$ is not an external input but a cross-check channel that must fall inside the same closure interstice.

### E.4.9.8 Summary: three closure channels and their complementarity

The structural angle is therefore constrained by three complementary closure pathways:

1. C1 (cosmological closure): $\Theta_{\text{cosmo}}$ from $\Omega_{\text{DM}}/\Omega_{\text{DE}} \simeq \tan \Theta$, probing the global vacuum partition.

2. C2 (EM micro-closure): $\Theta_{\alpha}$ from the $\alpha$-closure, buffered by $f_{U(1)}$ via $\alpha = \frac{1}{2} \sin 2\Theta f_{U(1)}$, probing a local EM readout that can be weakly renormalized.
3. C3 (spectral/mass closure): $\Theta$ from $\kappa_{ij}$ and the “closure bracket” inferred by inversion of generation jumps, probing the sectoral readout of the invariant through topological constraints (tilt, $d_3$-gates).

The Cabibbo coherence triangle $\Theta \approx \sqrt{8/\pi} \lambda$ provides an additional micro-macro bridge that ties weak mixing to the same structural control parameter, and explains why small ($\sim 10^{-3}$ rad) differences between $\Theta_{\text{cosmo}}$, $\Theta_{\alpha}$, and $\kappa_{12}$ are not a failure of closure but a measurable signature of controlled sectoral renormalization.

## E.5 Numerical Pipeline: From Auto-Sector Calibration to Core Mode

This section documents the forward computation performed by the HTML/JS proof-of-concept (PoC) used in Appendix E, with an explicit separation between (i) the discrete grammar presets, (ii) scenario choices, and (iii) derived outputs. It also introduces a hierarchy of closure modes: a descriptive calibration level (Level 1) and a predictive generative level (Level 2), both sharing the same forward engine and audit pipeline.

**Logical bridge:**

<figure><img src="image_6.png" /></figure>

Appendix E.4 defined the discrete *grammar* and its Auto-Sector classifier, assigning each canonical card to a sector L/U/D with deterministic gates ($w_{d3}$, $\text{tilt}_{34}$). Appendix E.3 fixed the RTQ scale and the resolution parameter $\eta$ either by Compton anchoring or by the causal-depth closure derived from cosmology. We now connect these layers.

### E.5.1 Two closure levels: descriptive calibration vs predictive core generation

The MCP PoC has a dual nature:

* **Level 1 (Auto-Sector calibration, descriptive):** reproduce observed hierarchies given empirical integers $n$ and verify deterministic consistency of the discrete grammar.

* **Level 2 (Core Mode, predictive):** generate $n$ ab initio from $\Theta_{\text{struct}}$ and universal constants, then project (“snap”) to an admissible integer compatible with discrete closure.

Both levels use the same forward kernel (Sec. E.5.4) and the same audit ledger (Sec. E.5.6).

### E.5.2 Level 1 – Auto-Sector calibration mode (descriptive)

Level 1 corresponds to the descriptive (anchored) regime. The integer harmonic indices $n_g$ ($g = 1, 2, 3$ for generations) are treated as empirical inputs obtained by matching to observed masses; the pipeline is used to verify grammar consistency.

#### E.5.2.1 Auto-Sector gates

The Auto-Sector classifier fixes the sectoral gates:

$$L \Rightarrow (w_{d3}, \text{tilt}_{34}) = (0, 0), \quad U \Rightarrow (3, 0), \quad D \Rightarrow (3, 2),$$

so that down-type quarks carry a helicoidal coupling with

$$\text{helixFactor} = \sqrt{1 + \text{tilt}_{34}^2} = \sqrt{5}.$$

In this regime, diagonal (mass-setting) terms are controlled by the sector geometry, while Cabibbo/CKM-like effects are interpreted as off-diagonal overlaps (phase interferences) between neighbouring tilt/sector states rather than as ad-hoc mixing matrices.
### E.5.2.2 External structural anchors

At Level 1, structural parameters are fixed externally (Appendix E.3 and main text):

$$A_{12} = \frac{3\pi}{2}, \quad A_{23} = \ln\left(\frac{\alpha^{-1}}{\sqrt{\pi}}\right), \quad \alpha^{-1} = 137.035999, \quad \Theta_{\text{struct}} \approx 0.36 \text{ rad.}$$

The closure test verifies that reconstructed loop counts and audit ratios remain consistent with anchors, and that the PoC behaves deterministically across sectors.

### E.5.3 Level 2 – Core Mode (generative Target & Snap)

Level 2 replaces empirical $n$ by a structural generation law. Operationally it implements a two-step logic:

$$\text{Target: } n_{\text{theo}}(X) \text{ from } \Theta_{\text{struct}} \quad \Rightarrow \quad \text{Snap: } n_X \in \mathbb{N} \text{ selected by lattice admissibility.}$$

The purpose is to remove ad hoc integer inputs and define, for each state $X$, a *canonical harmonic integer* $n_X$ that is (i) close to the generative target and (ii) compatible with discrete network closure.

#### E.5.3.1 Target generation: cosmological-unified vs pure-geometry models
There are two concep- tually distinct ways to model the $2 \to 3$ generation jump (notably in the Down sector):

##### (a) Pure-geometry model (“$\pi$ rule”)

A striking local relation in the Down sector is

$$\frac{n_b}{n_s} \approx \frac{\alpha^{-1}}{\pi},$$

which is numerically very tight but sector-specific (tied to helicoidal/torsional geometry).

##### (b) Cosmological-unified model (“$\Theta$ rule”)

Core Mode adopts a universal structural control where the same angle $\Theta_{\text{struct}}$ that controls the dark-sector partition in the cosmology chapter also controls the inter-generation deformation in the particle sector. In practice, this yields a unified Up/Down pattern via $\Theta$-dependent operators.

##### Why Core Mode defaults to the $\Theta$ model

Core Mode is a *generation law*, hence it prioritizes: (i) micro–macro unification, (ii) universality across L/U/D, and (iii) hierarchical structure (Wolfenstein-like), whereas the $\pi$ rule is best interpreted as a *local geometric consequence* (or diagnostic) of the Down-sector helicoid. Accordingly, the PoC treats:

Target = $\Theta$-driven (universal), Snap / diagnostics = sectoral closure constraints (where $\pi$-like torsion may appear

#### E.5.3.2 Target generation and scan window

Given $\Theta_{\text{struct}}$ and universal constants (notably $\pi$ and $\alpha$), Core Mode generates a continuous target $n_{\text{theo}}(X) \in \mathbb{R}_{>0}$. Rather than hard-coding a single closed form here, Appendix E refers to the operator tables of Appendix E.4 (two-regime L/U/D operators), which define the mapping

$$(\text{sector}, g; \Theta_{\text{struct}}, \alpha, \pi) \mapsto n_{\text{theo}}(X).$$

For numerical selection, Core Mode defines a deterministic scan window:

$$W_X \equiv [\lfloor n_{\text{theo}}(X)(1-\delta) \rfloor, \lfloor n_{\text{theo}}(X)(1+\delta) \rfloor],$$
where $\delta$ is global (fixed once; not tuned per particle). For robustness diagnostics, the PoC may also scan a full range $n \in [1, n_{max}]$ with $n_{max} = 10^5$.

### E.5.3.3 Snap: lattice closure defects and canonical selection score

*Exact-integer reference scale.* Snap must be evaluated in exact integer arithmetic (BigInt or strict fixed-point) because it is a Diophantine constraint. Core Mode treats the reference loop length as a discrete cardinal:

$$N_{\text{ref}} \in \mathbb{N}.$$

*Grid-closure defect $\varepsilon_N(n)$.* For a candidate integer $n \ge 1$:

$$r_N(n) \equiv N_{\text{ref}} \mod n, \quad 0 \le r_N(n) \le n - 1,$$

and the normalized distance-to-divisibility is

$$\varepsilon_N(n) \equiv \frac{\min(r_N(n), n - r_N(n))}{n} \in [0, \frac{1}{2}].$$

Thus $\varepsilon_N(n) = 0$ iff $n$ divides $N_{\text{ref}}$ (perfect grid closure).

*Spin-bridge defect $\varepsilon_K(n)$.*

The PoC also computes an auxiliary commensurability diagnostic with the weak-sector angular resolution defined in Sec. E.2.1.3.

Using the fixed step count $N_{\text{spin}} \approx 398.932$, we define

$$k(n) = \frac{N_{\text{loop}}^{(n)}}{N_{\text{spin}}}, \quad \varepsilon_K(n) = |k(n) - \text{round}(k(n))|.$$

(The previous placeholder value $N_{\text{spin}} = 2$ corresponded to the topological pass count $N_{\text{pass}}$ and has been removed to avoid conflation.)

*Canonical selection score.* The PoC prioritizes target proximity and grid closure:

$$S(n) \equiv \sqrt{(w_T \Delta_T(n))^2 + (w_N \varepsilon_N(n))^2 + (w_K \varepsilon_K(n))^2}, \quad \Delta_T(n) \equiv |\ln(n/n_{\text{theo}})|.$$

In practice $w_K$ is small (or $\varepsilon_K$ is logged but not used for ranking), while $w_T$ and $w_N$ carry the selection. The canonical integer is then

$$n_X \equiv \underset{n \in W_X}{\operatorname{argmin}} S(n).$$

*Deterministic tie-break.* If multiple integers yield comparable scores:

$$\text{minimize } S(n) \rightarrow \text{minimize } \Delta_T(n) \rightarrow \text{minimize } \varepsilon_N(n) \rightarrow \text{choose smallest } n.$$

This ensures the output is canonical and UI-independent.
### E.5.4 Shared forward kernel (JS PoC): preset vs scenario vs derived

#### E.5.4.1 Three layers: preset card vs scenario knobs vs derived outputs

Throughout Appendix E we separate:

*   **Preset card (discrete grammar input):**

$$C_{\text{preset}} = (D_2, D_{13}, Q; r_1, r_3, w_{d3}, \text{tilt}_{34}; \text{flags}),$$

as defined by the discrete grammar in Appendix E.4.

*   **Scenario knobs (not grammar):** harmonic index $n$, the $\eta$-mode (how resolution is fixed), whether antiParticleLocal is applied, and (optionally) the helix/twist amplitude $\tau$ used only in the down-type channel.

*   **Derived outputs:** holonomy budgets ($\Delta d_1, \Delta d_2, \Delta d_3, \Delta d_{13}$), cycles ($c_1, c_2, c_3$), objective phase $\Phi_{obj}$, RTQ loop count $N_{loop}^{(n)}$, mass map $m_n$, and closure defects/scores $\epsilon_N, \epsilon_K, S(n)$.

#### E.5.4.2 Step A: canonical holonomy budgets from the card

Given $(D_2, D_{13})$ and charge $Q$, the PoC evaluates:

$$\Delta d_1 = 6\pi Q, \quad \Delta d_{13} = -\pi D_{13}, \quad \Delta d_3 = \Delta d_{13} - \Delta d_1, \quad \Delta d_2 = 4\pi,$$

and defines integer cycles

$$c_i := \frac{\Delta d_i}{2\pi}.$$

Local antiparticle bookkeeping (optional scenario) If antiParticleLocal is enabled:

$$Q' = -Q, \quad D'_{13} = D_{13} + 12Q, \quad \tau' = -\tau,$$

which preserves $\Delta d_3$ exactly (sector gates remain consistent under the local map).

#### E.5.4.3 Step B: objective phase functional $\Phi_{obj}$

In the current minimal release:

$$\Phi_{\text{topo}} := \|\Delta \vec{d}\|_2 = \sqrt{\Delta d_1^2 + \Delta d_2^2 + \Delta d_3^2}, \quad \Phi_{\text{obj}} \equiv \Phi_{\text{topo}}.$$

#### E.5.4.4 Step C: fixing the RTQ/rad resolution $\eta$

The PoC supports several mutually-exclusive ways to set $\eta$ (RTQ per radian), treated as scenario choices:

1.  **Theory-internal (Compton-anchored):**

$$\eta_C = \frac{N_{\text{loop}}^{(e)}}{\Phi_{\text{obj}}(e) \cos \Theta_{\text{struct}}}.$$

2.  **Theory-internal (cosmological closure):**

$$\eta_{\text{cosmo}} \equiv N_{\text{cosmo}}^{1/3}.$$

3.  **Validation mode (electron mass calibration):**

$$\eta_{\text{calib}} = \frac{M_P \gamma_{\text{eff}}(e)}{m_e} \frac{1}{\Phi_{\text{obj}}(e) \cos \Theta_{\text{struct}}}.$$
**Locking convention** If lockEta is enabled:

$$ \eta(\text{all templates}) := \eta(e). $$

### E.5.4.5 Step D: RTQ loop length $N_{\text{loop}}^{(n)}$

Let $c_{\Theta} := \cos \Theta_{\text{struct}}$. The PoC computes:

$$ N_{\text{loop}}^{(1)} := \eta \Phi_{\text{obj}} c_{\Theta}, \quad N_{\text{loop}}^{(n)} := \frac{N_{\text{loop}}^{(1)}}{n}, $$

where $n \in \mathbb{N}$ is the harmonic (mode/generation) index.

### E.5.4.6 Step E: mass map and efficiency factors

$$ m_n = \frac{M_P \gamma_{\text{eff}}}{N_{\text{loop}}^{(n)}} = \frac{M_P \gamma_{\text{eff}} n}{\eta \Phi_{\text{obj}} \cos \Theta_{\text{struct}}}. $$

In the current PoC:

$$ \gamma_{\text{eff}} = \gamma_{d2} + w_{d3} \gamma_{d3}^{(\text{phys})} + w_{\text{conf}} E_{\text{conf}}, \quad \gamma_{d3}^{(\text{phys})} = \gamma_{d3} \text{ helixFactor}. $$

### E.5.4.7 Step F: closure defects and SCAN ranking

$$ \varepsilon_N(n) := \frac{\min\{r_n, n - r_n\}}{n} \in [0, \frac{1}{2}]. $$

The spin-bridge defect is implemented via a fixed-point remainder (implementation detail), and logged primarily as an audit quantity (Sec. E.5.3.3). A minimal Euclidean score used in the UI is:

$$ S(n) := \sqrt{\varepsilon_N(n)^2 + \varepsilon_K(n)^2}, $$

while Core Mode (Level 2) uses the weighted Target&Snap score of Sec. E.5.3.3 for canonical selection.

### E.5.5 Practical scan semantics: Scan-n vs Map

#### E.5.5.1 Scan-n (single-state scan).

#### E.5.5.2 Map (batch scan).

### E.5.6 Parameter ledger and implementation crosswalk

To keep Appendix E auditable, we classify every quantity appearing in the HTML PoC into four bins: (A) structural presets, (B) scenario toggles, (C) derived outputs, and (D) UI-only controls.
### E.5.6.1 (A) Structural presets (card inputs).

* Canonical bookkeeping integers: ($D_2, D_{13}$) ∈ $\mathbb{Z}^2$.

* U(1) winding / charge: $Q \in \mathbb{Z}/3$.

* Sector gates: $w_{d3} \in \{0, 3\}$, $\text{tilt}_{34} \in \{0, 2\}$.

* Discrete geometry ratios: ($r_1, r_3$) (visual embedding + visualization-derived $\gamma$ projections).

### E.5.6.2 (B) Scenario toggles (choices per run, not grammar).

* `antiParticleLocal` (boolean).

* `antiCPT` (boolean; global conjugation display-layer flag).

* Harmonic index $n \in \mathbb{N}$ (scanned or generated in Core Mode).

* `etaMode` (enumeration) and `lockEta` (boolean).

* (Core Mode) global scan window parameter $\delta$ and global weights ($w_T, w_N, w_K$) (fixed once; not per particle).

### E.5.6.3 (C) Derived outputs (computed).

* Holonomies $\Delta d_i$, cycles $c_i$, objective proxy $\Phi_{obj}$.

* Helix factor $\text{helixFactor} = \sqrt{1 + \tau^2}$ (down-type channel).

* Loop count $N_{\text{loop}}^{(n)}$, defects $\varepsilon_N, \varepsilon_K$, score $S(n)$.

* Mass $m_n$ (validation overlay; not used for selection unless explicitly stated).

### E.5.6.4 (D) UI-only / rendering controls. Camera/view toggles, shading, 3D mesh resolution, baryon slicing, and other visualization-only knobs.

### E.5.6.5 Implementation crosswalk (HTML/JS names).

<table><thead><tr><th>Appendix symbol</th><th>PoC variable / function (HTML/JS)</th></tr></thead><tbody><tr><td>(D<sub>2</sub>, D<sub>13</sub>, Q)</td><td>spec.D2, spec.D13, spec.Q</td></tr><tr><td>Δd<sub>i</sub></td><td>holonomyFromTemplate(...) → deltaD1, deltaD2, deltaD3</td></tr><tr><td>Local anti-map</td><td>applyLocalAntiMap(spec)</td></tr><tr><td>Φ<sub>topo</sub></td><td>phiTopo deltaD1, deltaD2, deltaD3)</td></tr><tr><td>η modes</td><td>etaFromMode(etaMode, ...)</td></tr><tr><td>Θ<sub>struct</sub></td><td>THETA STRUCT (constant), cosTheta</td></tr><tr><td>N<sub>loop</sub><sup>(n)</sup></td><td>Nloop = eta*Phi*cosTheta/nEff</td></tr><tr><td>Mass map</td><td>computeMass(...) → m = M_PLANCK*gammaEff/Nloop</td></tr><tr><td>helixFactor</td><td>helixFactor = sqrt(1+tilt34*tilt34)</td></tr><tr><td>Scan score S(n)</td><td>scanModes(...) → score = sqrt(epsN^2+epsK^2)</td></tr></tbody></table>

## E.5.7 Summary

* **Level 1** (Auto-Sector) verifies grammar consistency given empirical $n$.

* **Level 2** (Core Mode) generates $n$ via **Target & Snap** and selects a canonical integer in a deterministic way.

* The spin-bridge defect $\varepsilon_K$ is treated as **indicative** at very large scales ($\eta \sim 10^{20}$, $N_{\text{loop}} \sim 10^{21}$ or higher) because modular constraints admit too many near-solutions.
* The predicted mass $m_X$ is a derived output once $n_X$ is selected:

$$m_X = m_{\text{pred}}(n_X; C_X).$$

## E.6 Micro-to-macro bridge: RTQ vectorial update and coarse holonomy budgets

Appendix E intentionally works at two levels: (i) a coarse bookkeeping level (holonomy budgets $\Delta d_i$ and canonical integers $(D_2, D_{13}, Q)$), and (ii) a microscopic level (RTQ ticks and admissible micro-increments). This section states the minimal bridge needed to justify that the coarse budgets used throughout E.3–E.6 are legitimate coarse-grained summaries of an underlying discrete RTQ dynamics.

### E.6.1 RTQ tick and vectorial update

A fermion loop is represented as a closed sequence of RTQ node updates. At RTQ tick $k$, the internal state unit vector $\hat{u}_k$ is updated by a triple of angular micro-increments:

$$\Delta\Omega_k = (\Delta\sigma_{24,k}, \Delta\phi_{1,k}, \Delta\psi_{3,k}),$$

where:

* $\Delta\sigma_{24}$ is the “spine” increment (rotation in the $(d_2, d_4)$ plane),

* $\Delta\phi_1$ is the U(1) phase increment (electric winding on $d_1$),

* $\Delta\psi_3$ is the colour holonomy increment (effective phase on $d_3$).

The spatial displacement in the emergent metric is generated by a (model-dependent) translation operator $\mathcal{T}$:

$$P_{k+1} = P_k + \ell_0 \cdot \mathcal{T}(\hat{u}_k, \Delta\Omega_k),$$

with $\ell_0$ the RTQ grain (Appendix E.3).

### E.6.2 Coarse holonomy budgets as sums of micro-increments

Over one closed loop of $N_{\text{loop}}$ ticks, the coarse budgets are defined as the accumulated micro-updates:

$$\sum_{k=1}^{N_{\text{loop}}} \Delta\sigma_{24,k} = \Delta d_2, \quad \sum_{k=1}^{N_{\text{loop}}} \Delta\phi_{1,k} = \Delta d_1, \quad \sum_{k=1}^{N_{\text{loop}}} \Delta\psi_{3,k} = \Delta d_3.$$

The closure constraints and the canonical bookkeeping map $(Q, D_{13}) \mapsto (\Delta d_1, \Delta d_{13}, \Delta d_3)$ used in the proof-of-concept are stated once in Sec. E.4.1–E.4.1.

### E.6.3 Resolution parameter $\eta$ as a micro-step density

The micro-increments $\Delta\Omega_k$ are drawn from a finite admissible set (a discrete “alphabet”). At coarse audit level we do not need the detailed alphabet: we only need the effective density of admissible steps per radian, encoded by $\eta$ [RTQ/rad]. In the uniform-step idealization, one may think of an effective step size $\Delta\theta_{\min}$ such that

$$\eta \approx \frac{1}{\Delta\theta_{\min}}.$$

Then a coarse objective path-length proxy $\Phi_{\text{obj}}$ (dimensionless) yields the loop tick count

$$N_{\text{loop}}^{(1)} = \eta \Phi_{\text{obj}} \cos(\Theta_{\text{struct}}), \quad N_{\text{loop}}^{(n)} = \frac{N_{\text{loop}}^{(1)}}{n},$$

which is the forward convention used in the numerical pipeline (Sec. E.5) and anchored (in the Compton mode) by Sec. E.3.
### E.6.4 Discrete twist and the helix factor

For quark-type templates, the proof-of-concept distinguishes a “plane colour” preset from a “helicoidal colour” preset through a discrete coupling between $d_3$ and $d_4$ controlled by $\text{tilt}_{34} \in [-2, 2]$. We denote the signed twist parameter by

$$ \tau := s \text{tilt}_{34}, \quad s \in \{\pm 1\}, $$

where the sign $s$ is flipped by the local antiparticle bookkeeping (antiParticleLocal) and may also be flipped by a global conjugation flag (antiCPT) in the display layer. At the coarse level, the only place this enters the forward map is through a multiplicative causal cost factor

$$ \text{helixFactor} = \sqrt{1 + \tau^2}, $$

which preserves masses under $\tau \to -\tau$ and therefore does not split particle/antiparticle masses in this toy model. (Any genuine CP-violating or chirality-sensitive effect would require going beyond this coarse proxy.)

## E.7 Worked particle cards and audit tables

This section gathers auditable tables and a small number of worked “particle cards”. The objective of this section is to make explicit (i) what is entered as a discrete preset (grammar card), (ii) what is chosen as a scenario (e.g. $n$, local antiparticle flag), and (iii) what is computed as a derived quantity (budgets, RTQ counts, closure scores, and optional display outputs).

### E.7.1 Canonical convention (why reduced templates are not admissible inside bookkeeping)

All computations in the proof-of-concept (holonomy budgets, local antiparticle update, and derived cycles) use the canonical integer card ($D_2, D_{13} \in \mathbb{Z}^2$ through:

$$ \Delta d_{13} = -\pi D_{13}, \quad \Delta d_1 = 6\pi Q, \quad \Delta d_3 = \Delta d_{13} - \Delta d_1. $$

A reduced/gcd-normalized template ($d_2, d_{13}$) = ($D_2/3, D_{13}/6$) is useful for classification only, but it must not be used inside the bookkeeping because the local integer update law $D_{13} \mapsto D_{13} + 12Q$ (needed to preserve $\Delta d_3$ under $Q \to -Q$) is canonical-integer specific.
<table><caption>Table 19: Legend for the transposed summary tables (construction vs scenario vs derived quantities). This table is the audit key: every symbol appearing in the transposed summaries is classified as preset / scenario / derived.</caption><thead><tr><th>Row / symbol</th><th>Meaning</th><th>Status</th></tr></thead><tbody><tr><td>r<sub>1</sub> = A<sub>1</sub>/A<sub>2</sub>, r<sub>3</sub> = A<sub>3</sub>/A<sub>2</sub></td><td>Discrete geometry ratios used by sector rules / template classing (and by the 3D embedding)</td><td>preset</td></tr><tr><td>w<sub>d3</sub></td><td>SU(3) gate: 0 singlet, 3 fundamental channel (quark strands)</td><td>preset</td></tr><tr><td>tilt<sub>34</sub></td><td>Discrete d<sub>3</sub> ↔ d<sub>4</sub> coupling preset (plane vs helicoidal)</td><td>preset</td></tr><tr><td>Q</td><td>Charge quantum number (q = Qe), with Q ∈ Z/3</td><td>preset</td></tr><tr><td>D<sub>2</sub>, D<sub>13</sub></td><td>Canonical bookkeeping integers in Δd<sub>13</sub> = −πD<sub>13</sub></td><td>preset</td></tr><tr><td>antiParticleLocal</td><td>Local antiparticle bookkeeping flag (same-sheet)</td><td>scenario</td></tr><tr><td>antiCPT</td><td>Global conjugation/display flag (kept distinct from local bookkeeping)</td><td>scenario</td></tr><tr><td>n</td><td>Harmonic (mode) index</td><td>scenario</td></tr><tr><td>Q<sub>eff</sub>, D<sub>13,eff</sub></td><td>Values after scenario flags are applied</td><td>derived</td></tr><tr><td>Δd<sub>1</sub>, Δd<sub>2</sub>, Δd<sub>13</sub>, Δd<sub>3</sub></td><td>Coarse holonomy budgets (radians)</td><td>derived</td></tr><tr><td>Cycles (c<sub>1</sub>, c<sub>2</sub>, c<sub>13</sub>, c<sub>3</sub>)</td><td>Integer cycles: c<sub>i</sub> = Δd<sub>i</sub>/(2π)</td><td>derived</td></tr><tr><td>Φ<sub>topo</sub></td><td>Proxy objective: Φ<sub>topo</sub> = √Δd<sub>1</sub><sup>2</sup> + Δd<sub>2</sub><sup>2</sup> + Δd<sub>3</sub><sup>2</sup></td><td>derived</td></tr><tr><td>η</td><td>RTQ/rad resolution (set by Sec. E.3 in Compton mode)</td><td>derived (closure)</td></tr><tr><td>N<sub>loop</sub></td><td>Loop RTQ count (coarse): N<sub>loop</sub><sup>(n)</sup> = N<sub>loop</sub><sup>(1)</sup>/n</td><td>derived</td></tr><tr><td>helixFactor</td><td>Causal cost factor: √1 + τ<sup>2</sup> with τ = s tilt<sub>34</sub></td><td>derived</td></tr><tr><td>γ<sub>d2</sub>, γ<sub>d3</sub>, γ<sub>eff</sub></td><td>Visualization-dependent projection/efficiency proxies</td><td>derived (display)</td></tr><tr><td>N<sub>spin</sub>, k</td><td>Spin-bridge step count and ratio k = N<sub>loop</sub>/N<sub>spin</sub></td><td>derived</td></tr></tbody></table>

### E.7.2 Minimal fermion grammar cards (presets only)

<table><caption>Table 20: Minimal fermion grammar cards (canonical presets) used by the proof-of-concept. The table lists *presets only* (Appendix E.4): the canonical integers ($D_2, D_{13}$), charge $Q$, and the discrete gates ($w_{d3}, \tilde{\text{tilt}}_{34}$) as determined by the discrete grammar (Sec. E.4). The harmonic mode index $n$ and antiparticle flags are *scenario* (not presets) and therefore do not appear here. Quark masses are scheme/scale dependent; PDG conventions apply.[Navas et al., 2024]</caption><thead><tr><th>State</th><th>Sector</th><th>r<sub>1</sub></th><th>r<sub>3</sub></th><th>w<sub>d3</sub></th><th>tilt<sub>34</sub></th><th>Q</th><th>D<sub>2</sub></th><th>D<sub>13</sub></th><th>PDG mass</th></tr></thead><tbody><tr><td>Electron e<sup>-</sup></td><td>L</td><td>2</td><td>0</td><td>0</td><td>0</td><td>-1</td><td>3</td><td>6</td><td>0.51099895000(15) MeV</td></tr><tr><td>Muon μ<sup>-</sup></td><td>L</td><td>2</td><td>0</td><td>0</td><td>0</td><td>-1</td><td>3</td><td>6</td><td>105.6583755(23) MeV</td></tr><tr><td>Tau τ<sup>-</sup></td><td>L</td><td>2</td><td>0</td><td>0</td><td>0</td><td>-1</td><td>3</td><td>6</td><td>1776.93(9) MeV</td></tr><tr><td>Up u</td><td>U</td><td>2</td><td>1</td><td>3</td><td>0</td><td>+2/3</td><td>6</td><td>12</td><td>2.16 ± 0.07 MeV</td></tr><tr><td>Charm c</td><td>U</td><td>2</td><td>1</td><td>3</td><td>0</td><td>+2/3</td><td>6</td><td>12</td><td>1.27(2) GeV</td></tr><tr><td>Top t</td><td>U</td><td>2</td><td>1</td><td>3</td><td>0</td><td>+2/3</td><td>6</td><td>12</td><td>172.7(3) GeV</td></tr><tr><td>Down d</td><td>D</td><td>2</td><td>1</td><td>3</td><td>2</td><td>-1/3</td><td>9</td><td>18</td><td>4.7 ± 0.07 MeV</td></tr><tr><td>Strange s</td><td>D</td><td>2</td><td>1</td><td>3</td><td>2</td><td>-1/3</td><td>9</td><td>18</td><td>93.5<sup>+11</sup><sub>-5</sub> MeV</td></tr><tr><td>Bottom b</td><td>D</td><td>2</td><td>1</td><td>3</td><td>2</td><td>-1/3</td><td>9</td><td>18</td><td>4.18(3) GeV</td></tr></tbody></table>
### E.7.3 Particles summaries (scenario + derived outputs)

<table><caption>Table 21: Lepton-family summary (proof-of-concept; $η$ mode = Compton). Transposed view. Presets are fixed by Table 20. This table reports: (i) scenario choices, (ii) derived outputs computed by the forward map, and (iii) resultant masses and precision errors relative to PDG targets. $N_{\text{spin}} = 398.932$ for all columns.</caption><thead><tr><th></th><th>e<sup>-</sup></th><th>μ<sup>-</sup></th><th>τ<sup>-</sup></th><th>e<sup>+</sup> (local anti)</th></tr></thead><tbody><tr><td colspan="5"><em>Scenario choices (not grammar presets)</em></td></tr><tr><td>n (mode index)</td><td>1</td><td>207</td><td>3477</td><td>1</td></tr><tr><td>Card (D<sub>2</sub>, D<sub>13</sub>, Q; g)</td><td>(3, 6, -1; 1)</td><td>(3, 6, -1; 2)</td><td>(3, 6, -1; 3)</td><td>(3, 6, -1; 1)</td></tr><tr><td colspan="5"><em>Computed (derived outputs)</em></td></tr><tr><td>Q<sub>eff</sub></td><td>-1</td><td>-1</td><td>-1</td><td>+1</td></tr><tr><td>D<sub>13,eff</sub></td><td>6</td><td>6</td><td>6</td><td>-6</td></tr><tr><td>Cycles (c<sub>1</sub>, c<sub>2</sub>, c<sub>13</sub>, c<sub>3</sub>)</td><td>(-3, 2, -3, 0)</td><td>(-3, 2, -3, 0)</td><td>(-3, 2, -3, 0)</td><td>(3, 2, 3, 0)</td></tr><tr><td>Δd<sub>1</sub></td><td>-6π</td><td>-6π</td><td>-6π</td><td>+6π</td></tr><tr><td>Δd<sub>2</sub></td><td>+4π</td><td>+4π</td><td>+4π</td><td>+4π</td></tr><tr><td>Δd<sub>13</sub></td><td>-3π</td><td>-3π</td><td>-3π</td><td>+3π</td></tr><tr><td>Δd<sub>3</sub></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Φ<sub>topo</sub> (rad)</td><td>22.654347</td><td>22.654347</td><td>22.654347</td><td>22.654347</td></tr><tr><td>η (RTQ/rad)</td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td></tr><tr><td>N<sub>loop</sub></td><td>4.86868 × 10<sup>21</sup></td><td>2.35202 × 10<sup>19</sup></td><td>1.40025 × 10<sup>18</sup></td><td>4.86868 × 10<sup>21</sup></td></tr><tr><td>helixFactor</td><td>1.000000</td><td>1.000000</td><td>1.000000</td><td>1.000000</td></tr><tr><td>γ<sub>eff</sub></td><td>0.203777</td><td>0.203777</td><td>0.203777</td><td>0.203777</td></tr><tr><td>k = N<sub>loop</sub>/N<sub>spin</sub></td><td>1.2228 × 10<sup>19</sup></td><td>5.9071 × 10<sup>16</sup></td><td>3.5167 × 10<sup>15</sup></td><td>1.2228 × 10<sup>19</sup></td></tr><tr><td><strong>Calculated Mass (MeV)</strong></td><td><strong>0.510999</strong></td><td><strong>105.777</strong></td><td><strong>1776.743</strong></td><td><strong>0.510999</strong></td></tr><tr><td><strong>Precision Error (%)</strong></td><td><strong>0.000%</strong></td><td><strong>0.112%</strong></td><td><strong>0.011%</strong></td><td><strong>0.000%</strong></td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Figure 4: Geometric reconstruction of the lepton family. The visualization displays the spinor closure $Δd_2 = 4π$ characteristic of the Lepton sector, generated from the canonical card (D<sub>2</sub>, D<sub>13</sub>) = (3, 6). See the table above for the corresponding holonomy budget.</figcaption></figure>
<table><caption>Table 22: Up-type quark family summary (proof-of-concept; $\eta$ mode = Compton). Transposed view. Presets are fixed by Table 20. This table reports: (i) scenario choices (antiParticleLocal, n), (ii) derived outputs computed by the forward map, and (iii) resultant masses and precision errors relative to PDG targets. $N_{\text{spin}}$ = 398.932 for all columns.</caption><thead><tr><th></th><th>u</th><th>c</th><th>t</th><th>ū (local anti)</th></tr></thead><tbody><tr><td colspan="5"><em>Scenario choices (not grammar presets)</em></td></tr><tr><td>n (mode index)</td><td>1</td><td>584</td><td>79408</td><td>1</td></tr><tr><td>Card (D<sub>2</sub>, D<sub>13</sub>, Q; g)</td><td>(6, 12, +2/3; 1)</td><td>(6, 12, +2/3; 2)</td><td>(6, 12, +2/3; 3)</td><td>(6, 12, +2/3; 1)</td></tr><tr><td colspan="5"><em>Computed (derived outputs)</em></td></tr><tr><td>Q<sub>eff</sub></td><td>+2/3</td><td>+2/3</td><td>+2/3</td><td>-2/3</td></tr><tr><td>D<sub>13,eff</sub></td><td>12</td><td>12</td><td>12</td><td>20</td></tr><tr><td>Cycles (c<sub>1</sub>, c<sub>2</sub>, c<sub>13</sub>, c<sub>3</sub>)</td><td>(2, 2, -6, -8)</td><td>(2, 2, -6, -8)</td><td>(2, 2, -6, -8)</td><td>(-2, 2, -10, -8)</td></tr><tr><td>Δd<sub>1</sub></td><td>+4π</td><td>+4π</td><td>+4π</td><td>-4π</td></tr><tr><td>Δd<sub>2</sub></td><td>+4π</td><td>+4π</td><td>+4π</td><td>+4π</td></tr><tr><td>Δd<sub>13</sub></td><td>-6π</td><td>-6π</td><td>-6π</td><td>-10π</td></tr><tr><td>Δd<sub>3</sub></td><td>-16π</td><td>-16π</td><td>-16π</td><td>-16π</td></tr><tr><td>Φ<sub>topo</sub> (rad)</td><td>53.314595</td><td>53.314595</td><td>53.314595</td><td>53.314595</td></tr><tr><td>η (RTQ/rad)</td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td></tr><tr><td>N<sub>loop</sub></td><td>1.14579 × 10<sup>22</sup></td><td>1.96197 × 10<sup>19</sup></td><td>1.44292 × 10<sup>17</sup></td><td>1.14579 × 10<sup>22</sup></td></tr><tr><td>helixFactor</td><td>1.000000</td><td>1.000000</td><td>1.000000</td><td>1.000000</td></tr><tr><td>γ<sub>eff</sub></td><td>2.041062</td><td>2.041062</td><td>2.041062</td><td>2.041050</td></tr><tr><td>k = N<sub>loop</sub>/N<sub>spin</sub></td><td>2.8776 × 10<sup>19</sup></td><td>4.9275 × 10<sup>16</sup></td><td>3.6239 × 10<sup>14</sup></td><td>2.8776 × 10<sup>19</sup></td></tr><tr><td><strong>Calculated Mass (MeV)</strong></td><td><strong>2.175</strong></td><td><strong>1270.106</strong></td><td><strong>172699.6</strong></td><td><strong>2.175</strong></td></tr><tr><td><strong>Precision Error (%)</strong></td><td><strong>0.687%</strong></td><td><strong>0.008%</strong></td><td><strong>0.000%</strong></td><td><strong>0.687%</strong></td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Figure 5: Geometric reconstruction of the Up-type quark family (Sector U). The visualization renders the canonical preset ($D_2, D_{13}$) = (6, 12) with charge $Q = +2/3$. Unlike the down-sector, this topology is characterized by a *plane colour coupling* ($\text{tilt}_{34} = 0$), resulting in a simpler projection onto the causal axes. See Table 22.</figcaption></figure>
<table><caption>Table 23: Down-type quark family summary (proof-of-concept; $\eta$ mode = Compton). Transposed view. Presets are fixed by Table 20. This table reports: (i) scenario choices (antiParticleLocal, n), (ii) derived outputs computed by the forward map, and (iii) resultant masses and precision errors relative to PDG targets. $N_{\text{spin}} = 398.932$ for all columns.</caption><thead><tr><th></th><th>d</th><th>s</th><th>b</th><th>d&#773; (local anti)</th></tr></thead><tbody><tr><td colspan="5"><em>Scenario choices (not grammar presets)</em></td></tr><tr><td>n (mode index)</td><td>1</td><td>20</td><td>874</td><td>1</td></tr><tr><td>Card (D<sub>2</sub>, D<sub>13</sub>, Q; g)</td><td>(9, 18, -1/3; 1)</td><td>(9, 18, -1/3; 2)</td><td>(9, 18, -1/3; 3)</td><td>(9, 18, -1/3; 1)</td></tr><tr><td colspan="5"><em>Computed (derived outputs)</em></td></tr><tr><td>Q<sub>eff</sub></td><td>-1/3</td><td>-1/3</td><td>-1/3</td><td>+1/3</td></tr><tr><td>D<sub>13,eff</sub></td><td>18</td><td>18</td><td>18</td><td>14</td></tr><tr><td>Cycles (c<sub>1</sub>, c<sub>2</sub>, c<sub>13</sub>, c<sub>3</sub>)</td><td>(-1, 2, -9, -8)</td><td>(-1, 2, -9, -8)</td><td>(-1, 2, -9, -8)</td><td>(1, 2, -7, -8)</td></tr><tr><td>Δd<sub>1</sub></td><td>-2π</td><td>-2π</td><td>-2π</td><td>+2π</td></tr><tr><td>Δd<sub>2</sub></td><td>+4π</td><td>+4π</td><td>+4π</td><td>+4π</td></tr><tr><td>Δd<sub>13</sub></td><td>-9π</td><td>-9π</td><td>-9π</td><td>-7π</td></tr><tr><td>Δd<sub>3</sub></td><td>-16π</td><td>-16π</td><td>-16π</td><td>-16π</td></tr><tr><td>Φ<sub>topo</sub> (rad)</td><td>52.192057</td><td>52.192057</td><td>52.192057</td><td>52.192057</td></tr><tr><td>η (RTQ/rad)</td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td><td>2.2954 × 10<sup>20</sup></td></tr><tr><td>N<sub>loop</sub></td><td>1.12167 × 10<sup>22</sup></td><td>5.60833 × 10<sup>20</sup></td><td>1.28337 × 10<sup>19</sup></td><td>1.12167 × 10<sup>22</sup></td></tr><tr><td>helixFactor</td><td>2.236068</td><td>2.236068</td><td>2.236068</td><td>2.236068</td></tr><tr><td>γ<sub>eff</sub></td><td>4.391828</td><td>4.391828</td><td>4.391828</td><td>4.391828</td></tr><tr><td>k = N<sub>loop</sub>/N<sub>spin</sub></td><td>2.8170 × 10<sup>19</sup></td><td>1.4085 × 10<sup>18</sup></td><td>3.2232 × 10<sup>16</sup></td><td>2.8170 × 10<sup>19</sup></td></tr><tr><td><strong>Calculated Mass (MeV)</strong></td><td><strong>4.780</strong></td><td><strong>95.607</strong></td><td><strong>4178.009</strong></td><td><strong>4.780</strong></td></tr><tr><td><strong>Precision Error (%)</strong></td><td><strong>1.709%</strong></td><td><strong>2.253%</strong></td><td><strong>0.048%</strong></td><td><strong>1.709%</strong></td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Figure 6: Geometric reconstruction of the Down-type quark family (Sector D). The visualization renders the canonical preset (D<sub>2</sub>, D<sub>13</sub>) = (9, 18) with charge $Q = -1/3$. Crucially, the helicoidal coupling (tilt<sub>34</sub> = 2) acts along the causal dimension $d_4$. Since this figure displays the spatial projection onto ($d_1, d_2, d_3$), the $d_4$ excursions are masked. The loop therefore accumulates extra path length (mass) in the hidden dimension while maintaining the smooth spatial topology shown here.</figcaption></figure>
### E.7.4 Worked composite cards (baryons)

**E.7.4.1 Additivity at the canonical level** For coarse descriptor auditing, composites add linearly at the canonical level:

$$ (D_2, D_{13})_{\text{comp}} = \sum_{i} (D_2, D_{13})_i, $$

consistent with the HTML proof-of-concept, which stores baryons as explicit constituent lists (e.g. comp: ["u", "u", "d"]).

### E.7.5 Particle card: Proton ($p = uud$)

**Particle card: Proton**

$p = uud$

**Template grammar (canonical additivity).** Using Table 20 for quark presets:

$$ u : (D_2, D_{13}) = (6, 12), \quad d : (D_2, D_{13}) = (9, 18), $$

hence

$$ (D_2, D_{13})_p = (6, 12) + (6, 12) + (9, 18) = (21, 42). $$

**U(1) winding closure (charge).**

$$ Q_p = \frac{2}{3} + \frac{2}{3} - \frac{1}{3} = 1 \quad \Rightarrow \quad q_p = +e, $$

equivalently $n_{d_1}(p) = 3Q_p = 3$.

**SU(3) triadic closure (confinement).** Each quark strand carries a non-trivial centre phase $\omega = \exp(i2\pi/3)$ on $d_3$. A baryon closes because the three strands satisfy

$$ \omega \cdot \omega \cdot \omega = \exp(i2\pi) = 1, $$

so the composite is a colour singlet (no net $Z_3$ obstruction).

**Loop-length scale (Compton audit).** The RTQ loop length is anchored (order-of-magnitude) by the reduced Compton wavelength:

$$ N_{\text{loop}}(p) \approx \frac{\lambda_{C,p}}{\ell_0}, \quad N_{\text{bulk}}(p) \sim \left( \frac{\lambda_{C,p}}{\ell_0} \right)^3, $$

using inputs in Sec. E.3.

**Interpretation.** A colour-singlet composite knot whose long-lived existence is tied to triadic $Z_3$ closure on $d_3$, with net electromagnetic winding $Q_p = +1$.

### E.7.6 Particle card: Neutron ($n = udd$)

**Particle card: Neutron**

$n = udd$

**Template grammar (canonical additivity).**

$$ (D_2, D_{13})_n = (6, 12) + (9, 18) + (9, 18) = (24, 48). $$

**U(1) winding closure (neutrality).**

$$ Q_n = \frac{2}{3} - \frac{1}{3} - \frac{1}{3} = 0 \quad \Rightarrow \quad q_n = 0, $$

equivalently $n_{d_1}(n) = 3Q_n = 0$.

**SU(3) triadic closure (confinement).**

$$ \omega \cdot \omega \cdot \omega = 1, $$

so the neutron is also a colour singlet despite being composed of non-singlet quark strands.

**Loop-length scale (Compton audit).** Because $m_n \simeq m_p$ (sub-percent difference), the RTQ loop scale is the same order of magnitude:

$$ N_{\text{loop}}(n) \sim \frac{\lambda_{C,p}}{\ell_0}, \quad N_{\text{bulk}}(n) \sim \left( \frac{\lambda_{C,p}}{\ell_0} \right)^3. $$

**Interpretation.** A colour-singlet composite knot with zero net electromagnetic winding, whose stability again relies on triadic $Z_3$ closure on $d_3$.
### E.7.7 Audit checklist (how to verify Appendix E quickly)

To audit Appendix E efficiently:

1. Verify presets: check Table 20 (these are the only template inputs).

2. Verify bookkeeping: recompute $(\Delta d_1, \Delta d_{13}, \Delta d_3)$ from $(Q, D_{13})$ using Sec. E.4.1–E.4.1.

3. Verify antiparticle bookkeeping: apply `antiParticleLocal` and confirm $\Delta d_3$ is preserved by construction.

4. Verify scale mode: in Compton mode, check $\eta$ and $N_{loop}$ consistency with Sec. E.3.

5. Verify closure scores: compute $k = N_{loop}/N_{spin}$ using $N_{spin}$ fixed in Sec. E.2.1.3.
# F Falsifiability catalog (Tests map)

This appendix collects, in a compact form, the main qualitative and semi-quantitative predictions of Λ-Theory across several physical domains. Its purpose is not to introduce new assumptions, but to provide a “map” of how the framework connects to existing and future experimental tests.

The catalog is organized in five blocks:

• **F1 – Discrete substrate and pre-metric structure** (finite-state network, spacetime granularity, pre-metric noise).

• **F2 – Internal axes $d_1–d_5$ and interaction sectors** (U(1), SU(2), SU(3), time/causality, Θ-phase).

• **F3 – Tension layers, particles and fundamental constants** ($\Lambda^P$ layers, particle topology, bosons/fermions, emergent {$c, \hbar, G, \alpha$}).

• **F4 – Time, entropy and emergent gravity** (onset of time, arrow of time, curvature, decoherence, gravity as $\nabla\Lambda^P$).

• **F5 – Angular phase, dark sector and cosmology** (Θ-phase, multi-sheet structure, dark matter/energy, $H_0$ tension).

Each table entry contains a *Category*, a *Prediction* (with a cross-reference to the main text), and a *Signature/Test* suggesting how the prediction could be confronted with data.

## Tier labels (A / B / C)

Each prediction is assigned a *Tier* indicating how directly it connects to empirical tests:

• **Tier A** – High-priority or strongly constrained predictions. These are either:

– directly involved in the negative Checks 1–7 (Appendix D), or

– structural “pillars” of the framework that admit clear falsification channels with existing or near-term data.

• **Tier B** – Intermediate predictions. These are testable in principle with foreseeable extensions of current experiments/surveys, or require modest additional modelling before becoming falsifiable.

• **Tier C** – Long-term or speculative predictions. These typically involve cross-domain or emergent phenomena (e.g. fine “foam-like” noise, subtle cosmological fossils) and are included to document the full logical scope of the framework, even if their experimental access is distant.

Tier labels are descriptive rather than normative: they reflect the present status of testability and may evolve as new data or methods become available.

## Symbols ★ and ✓

Two additional symbols are used:

• ★ (star) marks *structural pillars* of Λ-Theory. These entries encode the core architectural claims of the framework (e.g. “constants as projections of $\Lambda_0$”, “particles as Λ-loops”, “gravity from $\nabla\Lambda^P$”, “Θ as angular phase linking sectors”). They are central to the internal coherence of the theory, independently of current data.

• ✓ (check mark) marks predictions that enter *explicitly* into at least one of the *negative Checks 1–7* detailed in Appendix D:

1. GW speed (GW170817, GWTC)

2. Lorentz invariance

3. Equivalence principle

4. CMB/BAO/BBN history and slow evolution of couplings
5. $H_0$ variability (environment + anisotropy)

6. $\Lambda$-cycle energy closure

7. Absence of vacuum birefringence

A single entry may carry both ★ and ✓ when it is both a structural pillar and already confronted with precise observational bounds. In the tables, the symbols can be placed next to the *Category* or *Prediction* label (e.g. “Gravity – Waves ★✓”), as you already do for ★.

## F.1 Discrete Substrate and Pre-Metric Structure

### F.1.1 Discretization & Pre-Metric Structure

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Discretization ★</td><td>The physical world is a finite-state RTQ network</td><td>Discrete, non-Gaussian metric noise at the Planck scale (LIGO/Virgo, atom interferometers)</td><td>A</td></tr><tr><td>Pre-metric structure</td><td>Observable spacetime granularity</td><td>Gravitational phase noise (“holographic-like” structured correlations)</td><td>B</td></tr><tr><td>Primitive causality</td><td>Structural nonlocal correlations</td><td>Distance-independent long-range entanglement tests</td><td>B</td></tr><tr><td>Quantized Θ</td><td>Discrete multiverse (quantized Θ)</td><td>Stepwise variations in fundamental constants over cosmological time</td><td>B</td></tr></tbody></table>

### F.1.2 Propagation and Local Physics

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>χ(Θ)★✓</td><td>Across Θ-sectors, the effective light speed scales with the angular ratio</td><td>Tiny anisotropies in photon/neutrino propagation across strong curvature regions</td><td>A</td></tr><tr><td>Projected constants★</td><td>{ħ, c, G} are angular projections of Λ<sub>0</sub></td><td>Correlated, never independent, drifts of constants (atomic clocks + quasar spectra)</td><td>A</td></tr><tr><td>Statistical geometry</td><td>Metric has fluctuating non-Gaussian statistics</td><td>Weak-lensing non-Gaussianities; excess kurtosis in shear maps</td><td>C</td></tr><tr><td>No singularities</td><td>Regular black holes (Λ<sup>I</sup>→0 cores)</td><td>Modified QNM ringdown; anomalous precession near BHs</td><td>B</td></tr></tbody></table>

### F.1.3 Topology, Rewiring, and Local Phase

<table><thead><tr><th>Category</th><th>Prediction (Λ-Theory)</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Topological rewiring</td><td>Discrete network rewiring</td><td>Neutrino-like oscillatory patterns in unrelated sectors</td><td>C</td></tr><tr><td>Local variability</td><td>Local laws depend on Θ(x)</td><td>Regional drifts of c, G, α correlated with cosmic structure</td><td>B</td></tr><tr><td>Coherence-dominant regime</td><td>Quasi-instantaneous coherence</td><td>Anomalously long-range phase retention in quantum systems</td><td>C</td></tr></tbody></table>
<table><thead><tr><td>Category</td><td>Prediction (Λ-Theory)</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Crystallization</td><td>Local Λ<sup>I</sup>→Λ<sup>P</sup> conversion</td><td>Local inertial/mass anomalies in highly coherent media</td><td>C</td></tr></tbody></table>

## F.2 Internal Axes $d_1-d_5$ and Interaction Sectors

### F.2.1 Predictions Associated with $d_1$ – U(1) Axis (Electromagnetism)

<table><thead><tr><td>Category (d<sub>1</sub>)</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>U(1) Topology ★</td><td>Electric charge is a winding number</td><td>Topological defects behaving as trapped charges</td><td>A</td></tr><tr><td>Photon nature ★✓</td><td>Photon = U(1) realignment wave (not pointlike)</td><td>Micro-discontinuities or quantized-phase steps in propagation</td><td>A</td></tr><tr><td>EM nonlinearity</td><td>EM energy ∝ Λ<sup>P</sup> (∂φ)<sup>2</sup> ⇒ weak nonlinearities at high intensity</td><td>Nonlinear optical deviations in extreme cavities</td><td>B</td></tr><tr><td>Field-line coherence</td><td>E/B field lines = coherence paths</td><td>Synchronization-pattern signatures in EM fields</td><td>C</td></tr></tbody></table>

### F.2.2 Predictions Associated with $d_2$ – SU(2), Mass, Higgs, Inertia

<table><thead><tr><td>Category (d<sub>2</sub>)</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Geometric mass ★</td><td>Mass = angular projection<br>m ∝ Λ<sup>P</sup> cos(θ<sub>2</sub>)</td><td>Cosmological drift of particle masses if θ<sub>2</sub> varies</td><td>A</td></tr><tr><td>Higgs orientation</td><td>Higgs = average orientation of d<sub>2</sub></td><td>Collective (order-parameter-like) signatures in Higgs behavior</td><td>B</td></tr><tr><td>Chirality</td><td>Chirality = geometric property</td><td>Weak, topology-dependent chirality asymmetries</td><td>C</td></tr><tr><td>Mass hierarchy</td><td>Mass hierarchy = stationary angles in d<sub>2</sub></td><td>Predictable mass ratios from geometric spectrum</td><td>B</td></tr><tr><td>Higgs mode</td><td>Higgs = gapped vibration (phonon-like) of d<sub>2</sub></td><td>Secondary collective modes in collider data</td><td>C</td></tr></tbody></table>

### F.2.3 Predictions Associated with $d_3$ – SU(3), Confinement, Matter

<table><thead><tr><td>Category (d<sub>3</sub>)</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>QCD confinement</td><td>Confinement = angular stabilization → fixed string tension</td><td>Universal QCD string tension, not a free parameter</td><td>B</td></tr><tr><td>Gluon propagation</td><td>Gluons = δφ(d<sub>3</sub>) excitations</td><td>Micro-dispersions or phase steps in strong field propagation</td><td>C</td></tr><tr><td>Baryons as loops</td><td>Baryons = closed d<sub>3</sub> loops</td><td>Stable spectra arising naturally from loop structures</td><td>C</td></tr><tr><td>Exotic hadrons</td><td>Existence of exotic d<sub>3</sub> loops</td><td>Prediction of multi-branch, non-standard hadrons</td><td>C</td></tr></tbody></table>
<table><thead><tr><th>Category (d<sub>3</sub>)</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Hadronic mass</td><td>Proton mass = curvature energy in d<sub>3</sub></td><td>Small drift in proton mass if &theta;<sub>3</sub> or &Theta; evolves</td><td>C</td></tr></tbody></table>

### F.2.4 Predictions Associated with $d_4$ – Causality, CPT, Time

<table><thead><tr><th>Category (d<sub>4</sub>)</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Arrow of time</td><td>Arrow = synchronization transition (Kuramoto)</td><td>Reversible or neutral “pre-temporal pockets”</td><td>B</td></tr><tr><td>Decoherence</td><td>Decoherence = resynchronization of d<sub>4</sub></td><td>Decoherence rates depending on local &Lambda;<sup>P</sup></td><td>B</td></tr><tr><td>Antimatter in d<sub>4</sub></td><td>Antimatter = inverted synchronization</td><td>Matter/antimatter asymmetries = sync asymmetries</td><td>C</td></tr><tr><td>Causal front</td><td>Existence of a propagating causal-sync front</td><td>v<sub>sync</sub> &le; c, measurable in cosmology</td><td>B</td></tr></tbody></table>

### F.2.5 Predictions Associated with $d_5$ – $\Theta$ (Physical / Informational Phase)

<table><thead><tr><th>Category (Θ)</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Correlated constants ★</td><td>Fundamental constants are functions of Θ</td><td>Coordinated drifts of {c, G, ħ, α} in cosmic history</td><td>A</td></tr><tr><td>No singularities</td><td>Θ bounded ⇒ singularities impossible</td><td>Regular black-hole interiors</td><td>B</td></tr><tr><td>Physical regimes</td><td>Distinct regimes: curvature / balance / coherence</td><td>EM propagation anomalies in ultra-coherent media</td><td>C</td></tr><tr><td>CPT-mirror</td><td>M<sup>+</sup> and M<sup>-</sup> = Θ and Θ + π</td><td>Cosmological CPT-mirror signatures</td><td>C</td></tr><tr><td>Present Θ ≈ π/4</td><td>Present constants reflect near-equilibrium state</td><td>Astrophysical consistency of present constant ratios</td><td>B</td></tr><tr><td>Local Θ variations</td><td>Local Θ shifts near strong gravity</td><td>Slight modifications of EM propagation</td><td>C</td></tr></tbody></table>

### F.2.6 Predictions from the Full Dimensional Architecture (Section 2.4)

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Emergent spacetime</td><td>(x,y,z,t) are emergent from internal axes</td><td>Detectable spacetime foam as synchronization noise</td><td>B</td></tr><tr><td>Long-range correlations</td><td>Geometry inherits long-range sync correlations</td><td>CMB alignments, vacuum-structure correlations</td><td>C</td></tr><tr><td>Hidden dimension</td><td>Existence of d<sub>10</sub> (Λ<sup>P</sup> ↔ Λ<sup>I</sup> coupling)</td><td>informational anomalies</td><td>C</td></tr></tbody></table>

## F.3 Tension Layers, Particles and Fundamental Constants

### F.3.1 Three Layers of Physical Tension (Λ<sup>P</sup>)
<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Λ<sup>P</sup><sub>struct</sub> – Background stiffness ★</td><td>c varies inversely with Λ<sup>P</sup><sub>struct</sub></td><td>Effective c slightly lower in high-curvature / high-Λ<sup>P</sup> regions</td><td>A</td></tr><tr><td>Λ<sup>P</sup><sub>struct</sub> – Time dilation</td><td>Gravitational time dilation = variation in Λ<sup>P</sup><sub>struct</sub></td><td>Deviations from pure GR metric if Λ<sup>P</sup> fluctuations measurable</td><td>B</td></tr><tr><td>Λ<sup>P</sup><sub>manif</sub> – Energy transport</td><td>EM/weak/strong waves = redistributions of Λ<sup>P</sup><sub>manif</sub></td><td>Wave energy transport must correlate with Λ-gradient fluctuations</td><td>B</td></tr><tr><td>Λ<sup>P</sup><sub>confined</sub> – Mass from torsion</td><td>Mass ∝ torsion stored in closed Λ-loops</td><td>Mass spectrum tracks loop geometry; identical topology → identical mass</td><td>B</td></tr></tbody></table>

### F.3.2 Particle Topology (What is a Particle?)

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Loop invariants ★</td><td>Particles = stable topological motifs</td><td>Winding number must remain invariant (no spontaneous change)</td><td>A</td></tr><tr><td>Photon nature (d<sub>1</sub>)</td><td>Photon = minimal coherence loop, strictly massless</td><td>Photon mass > 10<sup>-18</sup> eV falsifies</td><td>A</td></tr><tr><td>Open vs closed Λ<sup>P</sup></td><td>Massive states must correspond to closed Λ-loops</td><td>Observation of mass without loop topology falsifies</td><td>B</td></tr></tbody></table>

### F.3.3 Bosons (U(1), SU(2), SU(3), Gravity)

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Bosons – Angular modes</td><td>Interaction type = axis of oscillation (d<sub>1</sub>/d<sub>2</sub>/d<sub>3</sub>)</td><td>Each boson must couple along the predicted symmetry axis</td><td>B</td></tr><tr><td>Photon (U(1))</td><td>Photon mass strictly zero (no d<sub>2</sub> projection)</td><td>Photon mass constraints must always trend toward zero</td><td>A</td></tr><tr><td>Weak SU(2)</td><td>W/Z mass from d<sub>2</sub> projection geometry</td><td>Weinberg angle must match Λ-angle relationships</td><td>B</td></tr><tr><td>Gluons SU(3)</td><td>Gluons massless but confined</td><td>No free gluons ever observed</td><td>A</td></tr><tr><td>Confinement SU(3)</td><td>Linear potential from triadic loop closure</td><td>Confinement must hold at all accessible energies</td><td>B</td></tr><tr><td>Gravity – Emergence ★</td><td>G emerges from Λ<sup>P</sup> gradients</td><td>Slight variation of G with network density (not perfectly rigid)</td><td>A</td></tr><tr><td>Gravity – Waves ✓</td><td>Gravitational waves = Λ-phonons (∂<sub>t</sub>∇Λ<sup>P</sup>)</td><td>Dispersion relation must match Λ-phonon predictions</td><td>B</td></tr></tbody></table>

### F.3.4 Fermions (Spin 1/2)

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Loop structure</td><td>Fermions = closed torsional loops</td><td>720° spatial holonomy must hold</td><td>A</td></tr><tr><td>Charge winding</td><td>Charge = winding around d<sub>1</sub></td><td>Charge quantisation must be absolute</td><td>A</td></tr></tbody></table>
<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Colour winding</td><td>Charge/colour = winding numbers on d<sub>1</sub>/d<sub>3</sub></td><td>No isolated colour allowed</td><td>A</td></tr><tr><td>Spinor holonomy</td><td>Spinor behavior from topology of loops (double cover)</td><td>Fermion sign reversal exactly at 720° only</td><td>A</td></tr><tr><td>Pauli topology</td><td>Pauli exclusion = topological self-avoidance</td><td>No two identical fermionic loops can overlap</td><td>A</td></tr><tr><td>Mass torsion (d<sub>2</sub>)</td><td>Mass ∝ cos(θ_d2)</td><td>Mass ratios must follow angular geometry</td><td>B</td></tr><tr><td>Neutrino torsion/coherence</td><td>Neutrino oscillations = torsion ↔ coherence switching</td><td>Oscillation lengths tied to Λ<sup>P</sup>–Λ<sup>I</sup> exchange</td><td>B</td></tr></tbody></table>

### F.3.5 Fundamental Constants as Emergent Quantities

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>χ(Θ) variation</td><td>χ(Θ) ∝ tan(Θ)</td><td>Cross-sector / cross-Θ comparison of χ(Θ)</td><td>A</td></tr><tr><td>c near curvature</td><td>c decreases in strong curvature</td><td>Detectable near extreme gravitational sources</td><td>B</td></tr><tr><td>ħ emergence</td><td>ħ emerges from discrete action quanta in Λ-network</td><td>ħ variation must track fundamental Λ-geometry</td><td>B</td></tr><tr><td>α (Θ)</td><td>α (Θ) = projection of Λ<sub>0</sub> on d<sub>1</sub></td><td>Slow drift of α in cosmic history</td><td>A</td></tr><tr><td>G(ρ) ✓</td><td>G(Θ) determined by Λ<sup>P</sup> and ⟨ρ⟩ (network density)</td><td>Slight early-universe variations of G</td><td>B</td></tr></tbody></table>

### F.3.6 Synthesis (Cross-Domain Constraints)

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Constant co-variation ★</td><td>All constants co-vary with Θ and Λ<sub>0</sub></td><td>c, α, ħ cannot drift independently</td><td>A</td></tr><tr><td>Matter emergence</td><td>Fermions differ only by loop topology and internal winding</td><td>Detection of mass without loop topology falsifies</td><td>A</td></tr><tr><td>Boson/Fermion split</td><td>Bosons/fermions distinction is purely topological</td><td>Possible torsion-collapse transitions</td><td>C</td></tr></tbody></table>

## F.4 Time, Entropy and Emergent Gravity

### F.4.1 Emergence of Time

<table><thead><tr><th>Category</th><th>Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Time – Coherence field ★</td><td>Time = measurable order parameter of T_local</td><td>Temporal flickering or causal instability in high-coherence systems (BECs, Josephson arrays)</td><td>A</td></tr><tr><td>Time – Threshold</td><td>Critical Λ<sup>P</sup> threshold for emergence of time</td><td>Reversible or near-reversible dynamics in ultra-coherent media</td><td>B</td></tr></tbody></table>
<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Time – Synchronization</td><td>Finite-speed propagation of temporal coherence</td><td>Measurable front of synchronization in quantum networks</td><td>A</td></tr></tbody></table>

### F.4.2 Arrow of Time

<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Entropy – Orientation disorder</td><td>Entropy = geometric disorder in T_local</td><td>Correlation entropy ↔ orientation noise in quantum information systems</td><td>A</td></tr><tr><td>Reversibility – Microscopic</td><td>Microscopic reversibility = fluctuations in d<sub>4</sub></td><td>Variation in internal orientation field during Loschmidt echo experiments</td><td>B</td></tr><tr><td>Entropy–Curvature link</td><td>Entropy ↑ → Λ<sup>I</sup> ↓ → Λ<sup>P</sup> ↑ → curvature ↑</td><td>Cosmological correlation: entropy-rich regions slightly more curved</td><td>A</td></tr></tbody></table>

### F.4.3 Emergence of Gravity

<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Gravity – Curvature field</td><td>Curvature = ∇Λ<sup>P</sup>, not Einstein equation</td><td>Non-standard perturbations near strong info/Λ<sup>P</sup> gradients</td><td>A</td></tr><tr><td>Gravity – Variation of G</td><td>Slight G-variation in cosmic voids</td><td>Ultra-weak gravitational anomalies in BAO mapping</td><td>B</td></tr><tr><td>Equivalence principle ★✓</td><td>Inertial = torsional equivalence</td><td>No violation of EP at 10<sup>-14</sup> (MICROSCOPE-2 / STEP)</td><td>A</td></tr><tr><td>Black holes – Interior structure</td><td>Λ<sup>I</sup>→0 inside BH instead of singularity</td><td>Specific internal signatures in QNM ringdowns</td><td>B</td></tr></tbody></table>

### F.4.4 Curvature ↔ Temporal Disorder

<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Curvature–Entropy</td><td>Curvature proportional to temporal decoherence</td><td>Anomalies near highly dissipative astrophysical systems</td><td>B</td></tr><tr><td>Metric Transparency</td><td>Λ<sup>P</sup>→0 ⇒ soft antigravity regime</td><td>Micro-weight reduction in maximally coherent systems</td><td>B</td></tr><tr><td>Informational Flux ✓</td><td>Conservation of J<sub>μ</sub> = Λ<sup>P</sup> ∂<sub>μ</sub>Λ<sup>I</sup></td><td>ΛCDM-modified simulations must show conserved J<sub>μ</sub> patterns</td><td>B</td></tr></tbody></table>

### F.4.5 Time Crystals

<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Time-crystal regime</td><td>Λ<sup>P</sup> ≈ Λ<sup>I</sup> yields natural phase oscillations</td><td>Subharmonic oscillations without external drive</td><td>B</td></tr><tr><td>Gravitational modulation</td><td>ω decreases when Λ<sup>P</sup> increases</td><td>Oscillation shifts in gravitational gradients</td><td>C</td></tr></tbody></table>
<table><thead><tr><td>Category</td><td>Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Cosmological fossil oscillations</td><td>Tiny primordial oscillations</td><td>Fine modulations in CMB power spectrum</td><td>C</td></tr></tbody></table>

## F.5 Angular Phase, Dark Sector and Cosmology

### F.5.1 Multi-Sheet Angular Continuum (M$^+$ / M$^-$)

<table><thead><tr><td>Category</td><td>Chapter – Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Constants – Angular dependence ⚫</td><td>Angular drift of fundamental constants (Θ-dependent)</td><td>Correlated variation of α, G, ℏ across cosmic epochs; atomic clocks + quasar spectra</td><td>A</td></tr><tr><td>Dark Sector – Ratio ⚫</td><td>Fixed DM/DE ratio = tan Θ</td><td>Predicts Ω DM = 27%, Ω DE = 68% exactly for Θ≈22°</td><td>A</td></tr><tr><td>Mirror Universe – Nature of DM ⚫</td><td>Dark Matter = curvature field of M<sup>-</sup> (mirror sheet)</td><td>Smooth halos, core-cusp resolution, gravitational-only DM</td><td>A</td></tr><tr><td>Multi-Universe – Pairing</td><td>Universes occur in conjugate Θ, Θ+π/2 pairs</td><td>Single physical twin universe (not a multiverse)</td><td>B</td></tr></tbody></table>

### F.5.2 Phase Granularity (ΔΘ_min)

<table><thead><tr><td>Category</td><td>Chapter – Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Phase Quantization</td><td>Universal phase quantum ΔΘ_min</td><td>Weak oscillations in GWs, CMB microstructure, condensed-matter coherence noise</td><td>B</td></tr></tbody></table>

### F.5.3 Local Phase Dynamics Θ(x,t)

<table><thead><tr><td>Category</td><td>Chapter – Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Solitons – Particle ontology ⚫</td><td>Particles = Sine-Gordon solitons of Θ(x,t)</td><td>Internal oscillations, topological stability, quantized spectrum</td><td>A</td></tr><tr><td>Phase Topology – Folds</td><td>“Spacetime folds” induced by Θ fluctuations</td><td>Micro-lensing anomalies, BH trajectory oscillations</td><td>B</td></tr></tbody></table>

### F.5.4 M$^+$ / M$^-$ Mass Polarity & Curvature

<table><thead><tr><td>Category</td><td>Chapter – Prediction</td><td>Signature / Test</td><td>Tier</td></tr></thead><tbody><tr><td>Dark Lensing</td><td>M<sup>-</sup> mass produces lensing without baryons</td><td>Dark-lensing clusters; lensing maps with no visible mass</td><td>A</td></tr><tr><td>MOND Regime</td><td>MOND-like behavior from Λ<sup>P</sup>–Λ<sup>I</sup> resonance</td><td>Acceleration anomaly at a<sub>0</sub>-scale in outer galaxies</td><td>B</td></tr></tbody></table>

### F.5.5 Phase-Conjugate Coupling & Stability
<table><thead><tr><th>Category</th><th>Chapter – Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Phase-Conjugate Stability</td><td>No runaway instabilities (system constrained by invariant modulus Λ<sub>0</sub>)</td><td>Absence of catastrophic runaway events</td><td>A</td></tr><tr><td>Weak-field Scaling</td><td>g<sup>-</sup> = e<sup>2Φ</sup>g<sup>+</sup> where e<sup>2Φ</sup> ≈ tan<sup>2</sup> Θ</td><td>Weak-field deviations: alignment of polarizations, modified gravity at very low field</td><td>C</td></tr></tbody></table>

### F.5.6 Continuum of Θ-Universes

<table><thead><tr><th>Category</th><th>Chapter – Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Attractor Structure</td><td>Only 3 stable universe families (Θ=0, π/4, π/2)</td><td>Tri-modal structure in numerical cosmology</td><td>C</td></tr><tr><td>Long-lived complexity window</td><td>Long-lived bound structures and stable non-equilibrium organization are expected for Θ ∈ (0, π/4)</td><td>Anthropic constraints: habitability tied to angular phase</td><td>C</td></tr></tbody></table>

### F.5.7 Dark Energy as Cosmic Osmosis

<table><thead><tr><th>Category</th><th>Chapter – Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>DE Evolution</td><td>Λ<sub>eff</sub> must slowly decrease (dΛ/dt<0)</td><td>Non-exponential acceleration; long-term drift detectable by Euclid/JWST</td><td>B</td></tr><tr><td>DE Equation of State</td><td>w cannot be < -1 (phantom energy impossible)</td><td>No Big Rip; observational constraints w ≥ -1</td><td>A</td></tr></tbody></table>

### F.5.8 Dark Matter as Mirror Curvature

<table><thead><tr><th>Category</th><th>Chapter – Prediction (Λ-Theory)</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Non-Detection</td><td>No WIMP/axion possible (DM is not a particle)</td><td>Persistent null results in all DM detectors</td><td>A</td></tr><tr><td>Smoother Halos</td><td>DM distribution smoother than visible matter</td><td>Halo profiles from Gaia/Euclid</td><td>A</td></tr><tr><td>Angular Ratio</td><td>DM/DE ratio = tan Θ</td><td>Match with Planck cosmology</td><td>A</td></tr></tbody></table>

### F.5.9 H<sub>0</sub> Tension and Angular Drift

<table><thead><tr><th>Category</th><th>Chapter – Prediction</th><th>Signature / Test</th><th>Tier</th></tr></thead><tbody><tr><td>Hubble Tension ★✓</td><td>H<sub>0</sub> tension = local δΘ (texture / drift channel)</td><td>Local H<sub>0</sub> > global H<sub>0</sub>; tension traced to δΘ(x)</td><td>A</td></tr><tr><td>Directional Anisotropy</td><td>H<sub>0</sub> anisotropies aligned with the cosmic web</td><td>Cross-correlation SN Ia ↔ density maps</td><td>A</td></tr><tr><td>Temporal Scale</td><td>H<sub>0</sub> ≈ 7 × 10<sup>-11</sup> yr<sup>-1</sup> (i.e. ˙a/a today)</td><td>Long-stride cosmology: SKA/LSST/JWST</td><td>B</td></tr></tbody></table>
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