# Meta-Connective Physics: A Discrete Relational Framework for Spacetime, Matter, and Information

Michaël Vaillant

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# Meta-Connective Physics

## A Discrete Relational Framework for Spacetime, Matter, and Information

Michaël Vaillant

Meta-Connexions

michael.vaillant@meta-connexions.com

November 14, 2025

### Abstract

We introduce the Meta-Connective Physics (MCP), a discrete relational framework in which spacetime geometry, matter, and coherence emerge as phase-coupled projections of a single invariant relational modulus $\Lambda_0$. Physical reality is modeled as a lattice of Relational Tension Quanta governed by a complex tension $\Lambda = \Lambda^P + i\Lambda^I$, where $\Lambda^P$ encodes metric stiffness and inertial response and $\Lambda^I$ encodes phase alignment and coherence, constrained by the monadic conservation law $(\Lambda^P)^2 + (\Lambda^I)^2 = \Lambda_0^2$.

A single structural phase $\Theta$ controls the projection of $\Lambda$ onto observational channels, so that the usual “fundamental constants” ($c, G, \hbar, \alpha, \dots$) are not independent inputs but correlated projections. This yields a hard cross-domain constraint structure: if high-precision datasets require mutually independent drifts (or uncorrelated residual textures) across these channels, the framework is falsified at its core.

In MCP, metric geometry is an effective regime rather than a primitive: it is recovered only in phases where $\Lambda^P$ supports a stable IR mapping, while the underlying description remains relational. Locality is treated as an emergent phase-coherence property, and the conjugate sector $\Lambda^I$ enables phase-conjugate feedback across scales. If confirmed, this implies a revised hierarchy of effective notions in which the metric is derived and cross-domain correlations replace independently adjustable sector parameters.

Gravitation arises as the elastic response of $\Lambda^P$ to coherence variations in $\Lambda^I$, and classical singularities correspond to limiting phase configurations rather than a breakdown of the substrate. The same angular geometry induces a phase-conjugate coupling between a visible sheet $M^+$ and a conjugate sheet $M^-$, shifted by $\pi/2$ in $\Theta$ while originating from the same invariant modulus. Information exchange across this conjugate pair implements multi-scale regulatory loops: inter-sheet backreaction can sustain large-scale negentropic organization while remaining compatible with metric-regime causality. In this picture, curvature sourced in $M^-$ reproduces dark-matter-like phenomenology, while dark-energy-like behavior is associated with slow inter-sheet tension gradients.

A distinctive feature of MCP is a scale-bridging closure in which microphysical bookkeeping (RTQ grain scale and fermionic loop closures) and cosmological anchoring constrain the same $\Theta$-controlled architecture, linking the infinitely small and the infinitely large within a single projection grammar. We formulate conservative negative checks from gravitational-wave propagation, Lorentz invariance, equivalence-principle tests, early-universe consistency, and anisotropies of $H_0$. Finally, the limit $\Theta \to \pi/2$ defines a coherence-dominated, non-metric boundary of validity for effective metric descriptions, providing a controlled setting for future laboratory and cosmological tests.

**Versions:** v1.0. Creation. v2.0. Minor text corrections; supplemental graphics; updated cosmological tests with bulk-flow removal. v2.1. Fixes to Appendices C and D; revised introduction. v3.0. Added causal-depth closure (electron mass scaling law), explicit dimensional selection (d=3) derivation, numerical spinor closure check, and Appendix E: detailed particle modelization and fermion mechanism audit.

**Keywords:** Information-theoretic foundations; Emergent spacetime; Relational networks; Phase geometry; Information coherence; Non-metric regime / coherence boundary; Dark sector; Phase-conjugate cosmology; Correlated variations of constants; Hubble anisotropy (directional H0); Spectral dimension; Micro-macro closure
# 1 Introduction: From Scattered Constants to a $\Lambda$-Grammar

Modern physics is extraordinarily predictive, yet organized as a patchwork of formalisms. Gravitation is encoded geometrically (a metric and curvature), quantum fields are formulated on Hilbert-space operator algebras, thermodynamics via entropic functionals, and information theory through channel capacities. Each regime introduces its own “fundamental” constants $c, \hbar, G, k_B, \alpha, \Lambda_{GR}, \dots$, treated as independent calibrated inputs.

This fragmented approach has been remarkably successful, but it remains primarily *equation-centric*: each regime is organized around its own formalism and set of constants, with unification pursued at the level of field equations rather than underlying structure. Information-based perspectives suggest a different angle, in which physical laws emerge from consistent quantification and inference on relational data rather than being posited as fundamental axioms [Knuth, 2010, 2014]. This viewpoint aligns with the present monograph, which takes an explicitly information-first, structure-centric stance.

This paper starts from a different angle than usual unification attempts. Rather than postulating new fields and then fitting constants, we treat the constants themselves as a structured dataset and ask whether their relations already encode a compact generative scheme. From a data-science and information-theoretic perspective, the guiding question is whether the universe can be regarded as a discrete information-processing network, with the usual laws and constants as emergent summaries of an underlying relational structure constrained by consistent quantification and inference. In this language, a single bounded invariant $\Lambda_0$ and a structural phase angle $\Theta$ organize the constants: the apparent plurality of “fundamental constants” is interpreted as a family of projection channels of one meta-connective network evaluated at a given $\Theta$.

We adopt a strict convention throughout, which is described in the following.

## Box — Non-negotiable conventions, for SI-audited reproducibility

We distinguish three related but non-equivalent objects from the outset:

* $\Lambda_{GR}$: the GR cosmological constant (curvature term), with SI units $m^{-2}$.

* $\Lambda_0$: an intrinsic invariant of the underlying relational network (unit-free at the foundational level).

* $(\Lambda^P, \Lambda^I)$: internal projections of $\Lambda_0$, interpreted as physical (curvature-bearing) and informational (coherence-bearing) components. These projections are also unit-free internally; physical units enter only when internal quantities are mapped onto observational channels by coupling them to explicit dimensional carriers (charge scales, mass scales, momentum scales, coarse-grained node densities, etc.).

**SI-only / explicit audit.** All operational carriers used in sector-level expressions are explicitly mapped to SI dimensions. Dimensional factors are not absorbed into alignment coefficients: $A_i$ remains strictly dimensionless, and any dimensional content is carried by $S_i$ (see Appendix A).

**No natural units.** $c$ and $\hbar$ are kept explicit throughout. Any change of convention is declared and remains SI-auditable.

**Emergent time convention.** No ontological time parameter is assumed at the fundamental level. What is commonly called “time” is treated as an emergent, phase-ordered property of the underlying tension/coherence dynamics (developed in Chapter 4). In the IR, however, comparison to GR/QFT requires a coordinate-based representation. When a time-like structure appears below, it should be read as a derived operational construct. Concretely, we encode the emergence of temporal orientation and clock-like behavior through a covariant coherence/causality vector field $T^\mu$, not through a primitive external time. $x^0$ (or any “time coordinate”) is a label used in the continuum IR description. The physically relevant temporal structure is reconstructed from the field $T^\mu$ and its invariants, which encode emergent causal coherence rather than a fundamental time variable.

**Scope discipline.** Statements about non-physical domains are not used as premises for any prediction. Validation relies only on the physics test suite, invariance checks, and audited mappings defined in Appendix A.
## 1.1 Motivation: Physics as a Fragmented Description of a Unified Process

At a conceptual level, general relativity, quantum field theory, statistical mechanics, and information theory all describe aspects of the same physical reality, yet they operate on distinct mathematical substrates and organizing principles:

* GR: a manifold equipped with a metric, where curvature encodes energy-momentum;

* QFT: operator-valued fields defined on a background causal structure, with renormalized couplings and cutoffs;

* Statistical / information theory: ensembles, entropy functionals, and channel capacities;

* Cosmology: effective fluids and parameters ($\Omega_m, \Omega_\Lambda, H_0, \dots$) tuned to observations.

In parallel, a number of information-first and inference-based programmes have shown how significant parts of classical and quantum physics can be reconstructed from entropic or probabilistic principles, rather than postulated as fundamental dynamics [Caticha, 2015, Goyal, 2010]. The $\Lambda$-framework is in the same broad spirit, but focuses on a discrete relational network and a constants-geometry “$\Lambda$-grammar” as its primary organizing tools.

Bridges between these regimes remain partial and often asymmetric. Black-hole thermodynamics and holographic ideas strongly suggest that geometry and information are not independent layers, yet no widely adopted formalism provides an explicit *global* organizing principle that governs both.

From a model-building viewpoint, this landscape resembles heterogeneous effective descriptions of a single underlying process, each with its own preferred primitives and its own constants. A natural question then arises: *if one temporarily brackets historical derivations and examines only the network of relations between constants and equations, does a simpler generative structure become visible?*

The $\Lambda$-framework is an attempt to answer “yes” in a controlled way: not by inventing new numerical relations, but by extracting a small grammar that reorganizes known relations into a bounded, projection-based architecture.

## 1.2 A Data-Scientific Reorganization of Physical Law

The starting point is deliberately empirical and structural. We collect the principal equations in which the standard constants appear—relativistic, quantum, gravitational, thermodynamic, cosmological—and treat them as entries in a relational dataset. Each constant is characterized by:

1. its SI dimensions;

2. the equations in which it appears;

3. its operational role (scale factor for time, length, energy, charge, curvature);

4. its standard combinations with other constants (Planck units, coupling strengths, cosmological densities).

This yields a constants graph: nodes are constants ($c, \hbar, G, k_B, e, \alpha, \Lambda_{GR}, \dots$) and edges are textbook relations (Planck-unit definitions, $\alpha$ rewrites, standard cosmological density identities, etc.). The aim is not pattern-mining or a search over alternative models; it is a canonical re-expression of a fixed set of standard relations under a closed, explicitly declared convention set:

* *No fitted relations.* Only widely used identities and definitions are allowed; no empirically tuned equalities or numerology enter the construction.

* *Dimensional audit.* All manipulations are carried out in a declared unit convention, with *SI used as the audit reference* and explicit conversions whenever sector conventions differ (Appendix A).

* *Minimal structural freedom.* Beyond $\Lambda_0$ and a phase parameter $\Theta$, only a small set of discrete geometric normalizations is introduced; nothing else is adjusted to improve agreement.

To avoid ambiguity: we do not “test candidates” or “reject factorizations” in the sense of exploring an open search space. Rather, we enforce a scope-closure (closed-world) rule: any step that would require (i) non-standard identities, (ii) dimensionally ambiguous carriers, or (iii) representation-dependent choices that fail invariance under equivalent parametrizations (e.g., $\alpha$ vs. $e$) is simply inadmissible in this construction.
The resulting graph, along with the exact identity/definition set and the invariance checks, is provided so the
re-expression can be audited end-to-end.

<figure><img src="image_2.png" /><figcaption>Figure 1: Initial network representation of the constants. Nodes represent constants such as $c, \hbar, G, k_B, e, m_W, \alpha, \alpha_s$ and $\Lambda_{cc}$; edges encode functional or dimensional relations between them (for example the definition of Planck units, the expression of $\alpha$ in terms of $e, \hbar, c$ and $\varepsilon_0$, or the relation between $G, \Lambda_{cc}$ and cosmological densities).</figcaption></figure>

Under these restrictions, a robust regularity appears across the four fundamental interactions: the corresponding
sector determinants admit a common three-layer decomposition. This observation is the empirical trigger for
introducing the $\Lambda$-framework.

For reproducibility, Appendix A provides:

(A.1) the constants graph (nodes/edges) and the exact set of admissible textbook identities/definitions used for canonicalization;

(A.2) the closed-world admissibility criteria (scope closure) excluding non-standard identities, ambiguous carriers, or representation-dependent steps;

(A.3) declared unit conventions and explicit SI audit mappings for sector-specific carriers;

(A.4) invariance checks under equivalent parametrizations (e.g., $\alpha$ vs. $e$) and the resulting stable ($G, A, S$) structure.
# 1.3 The Trigger: A Unified Three-Layer Structure and the Resistivity Table

## 1.3.1 Trigger result: a three-layer factorization

When the constants graph is reorganized under the constraints above, the measured “final determinants” in the electromagnetic, weak, strong and gravitational sectors can be written in the schematic form

$$C_i = \mathcal{G}_i \times \mathcal{A}_i \times \mathcal{S}_i,$$

Where the $(\mathcal{G}, \mathcal{A}, \mathcal{S})$ triplet may be read as ‘Geometry–Alignment–Scale’:

* $\mathcal{G}_i$ is a Geometric normalization fixed by standard symmetry/topology conventions (e.g., $4\pi$, $1/8\pi$, $\pi/\sqrt{2}$);

* $\mathcal{A}_i$ is a dimensionless Alignment / resistivity coefficient, interpreted as the response of the underlying relational network when probed along an internal axis $d_i$;

* $\mathcal{S}_i$ is a dimensional Scale carrier that sets the observational unit scale (charge scale, mass scale, momentum-transfer scale, or coarse-grained node density scale). When a microscopic RTQ grain ($l_0, t_0$) is mentioned, it is a reader-facing micro-scale for “network updates” and is not defined via $G, \hbar, c$. Any occasional Planck-domain numbers are used only as an a posteriori CODATA calibration to give the reader an order of magnitude, and do not enter the audit or parameter extraction.

This decomposition is introduced as a bookkeeping compression of standard relations, not as a redefinition of couplings. Its conceptual impact is that it separates (i) pure geometry, (ii) unit-free network response, and (iii) dimensional conversion—making it plausible that “constants” are shadows of an internal, bounded structure rather than unrelated primitives.

(This “key trigger” interpretation is explicitly stated later in the manuscript in the resistivity synthesis subsection).

## 1.3.2 Formal resistivity table (interpretation, conventions, and audit contract)

<table><caption>Table 1: Formal resistivity table for the four interaction channels. Each interaction “strength” is written schematically as $C_i = \mathcal{G}_i \mathcal{A}_i \mathcal{S}_i$, where $\mathcal{G}_i$ is a pure geometric factor, $\mathcal{A}_i$ a dimensionless resistivity / alignment coefficient, and $\mathcal{S}_i$ the dimensional carrier that restores the correct SI scaling.</caption><thead><tr><th>Interaction</th><th>Geom. factor <i>G</i><sub>i</sub></th><th>Resist. <i>A</i><sub>i</sub></th><th>Carrier <i>S</i><sub>i</sub></th><th>Final determinant <b>C</b><sub>i</sub></th><th>Axis <b>d</b><sub>i</sub></th><th>Common name</th></tr></thead><tbody><tr><td>EM</td><td>1</td><td>&alpha;</td><td>&hbar;c</td><td><i>C</i><sub>EM</sub> = &alpha;&hbar;c &equiv; <i>e</i><sub>*</sub><sup>2</sup> &equiv; <i>k</i><sub>e</sub> <i>e</i><sup>2</sup></td><td><i>d</i><sub>1</sub> (U(1))</td><td>Coulomb strength (<i>e</i><sup>2</sup>/(4&pi;&epsilon;<sub>0</sub>))</td></tr><tr><td>Weak</td><td><sup>&pi;</sup>&frasl;<sub>&radic;2</sub></td><td>&alpha; sin<sup>-2</sup> &theta;<sub>W</sub></td><td><i>M</i><sub>W</sub><sup>-2</sup></td><td><i>G</i><sub>F</sub> = <sup>&pi;</sup>&frasl;<sub>&radic;2</sub> &alpha; sin<sup>-2</sup> &theta;<sub>W</sub> <i>M</i><sub>W</sub><sup>-2</sup></td><td><i>d</i><sub>2</sub> (SU(2))</td><td>Fermi constant</td></tr><tr><td>Strong</td><td>4&pi;</td><td>&alpha;<sub>s</sub>(<i>Q</i><sup>2</sup>)</td><td>&hbar;c</td><td><i>C</i><sub>S</sub>(<i>Q</i><sup>2</sup>) &equiv; <i>g</i><sub>s,*</sub><sup>2</sup>(<i>Q</i><sup>2</sup>) = 4&pi; &alpha;<sub>s</sub>(<i>Q</i><sup>2</sup>) &hbar;c</td><td><i>d</i><sub>3</sub> (SU(3))</td><td>Strong coupling</td></tr><tr><td>Gravitational</td><td><sup>1</sup>&frasl;<sub>8&pi;</sub></td><td>&alpha;<sub>G</sub></td><td><i>c</i><sup>3</sup>&hbar;<sup>-1</sup>(&rho;)<sup>-1</sup></td><td><i>G</i><sub>P</sub> = <sup>1</sup>&frasl;<sub>8&pi;</sub> &alpha;<sub>G</sub> <i>c</i><sup>3</sup>&hbar;<sup>-1</sup>(&rho;)<sup>-1</sup></td><td><i>d</i><sub>4</sub></td><td>Newton constant</td></tr></tbody></table>

Table 1 implements the factorization introduced in Sec. 1.3.1. The closed-world admissibility rules and rewrite identities used to obtain this canonical form are detailed in Appendix A.
Representation notes (to prevent ambiguity).

EM sector. We define the determinant

$$C_{EM} = \alpha \hbar c \equiv e_*^2,$$

with

$$e_*^2 \equiv k_e e^2, k_e = \frac{1}{4\pi\varepsilon_0},$$

so the familiar Coulomb strength $e^2/(4\pi\varepsilon_0)$ is just an equivalent representation of the same determinant.

**Weak sector.** The same $\alpha$ appears in the alignment slot, while the sector-specific geometry and carrier are encoded by $\pi/\sqrt{2}$ and $M_W^{-2}$, respectively. In high-energy convention $M_W$ is often treated as an energy scale; in SI audit this is made explicit (no absorption of powers of $c$ or $\hbar$ into $A_i$). Sector conversion conventions and audit mappings are declared in Appendix A.

**Gravitational sector.** In the gravitational row, $\Lambda^P$ denotes the *internal, unit-free physical-tension component* of the $\Lambda$-formalism. The symbol $\Lambda^P$ used in Table 1 denotes the *effective dimensionless gravitational alignment coefficient*, i.e.

$$A_G \equiv \alpha_G \in \mathbb{R},$$

associated with axis $d_4$. In the simplest mapping used throughout the paper, $\alpha_G$ is the sector projection of the internal phase (introduced below), but the key point here is operational: $\Lambda^P$ is dimensionless, and the SI dimensions of Newton's constant arise entirely through the carrier

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

We use $\langle \rho \rangle$ as a coarse-grained *a real* node-density carrier with SI dimension

$$[\langle \rho \rangle] = m^{-2}.$$

Angle brackets here *do not* denote a quantum expectation value; they denote coarse-graining at the deployment scale used for the gravitational projection channel.

Sector unit conventions are declared explicitly; SI serves as the audit reference. Closed-world admissibility (scope closure), rewrite identities, and invariance checks are documented in Appendix A.
Referee reader map / falsification contract

This manuscript separates two complementary layers: (i) an *auditable verification layer* (Claims Ledger, Checks, and Minimal Validation Triad), and (ii) a *generative architecture* in which a single invariant modulus $\Lambda_0$ and its angular projections constrain multiple domains (cosmology, constants, particle spectra, and an informational/coherence sector). The verification layer provides an explicit testing path; the architectural layer defines the core claim.

<table><thead><tr><th>Goal / question</th><th>Where to look (sections / appendices)</th></tr></thead><tbody><tr><td>Minimal reading path (fast verification + core architecture)</td><td>Architecture: Secs. 2–4<br>Must-pass gates: Sec. 7.2 (Checks 1–7)<br>Cross-domain opportunity test (MVT): Sec. 7.3 + synthesis in Sec. 6.5<br>Concrete implementation: Appendix E (Proof-of-Concept; autoSector / Core Mode)</td></tr><tr><td>Layer (ii): generative architecture — what is the single invariant and how does it project?</td><td>Invariant + projections: Secs. 2–4 (definitions, RTQ substrate, emergence route)<br>Angle-driven partition / dual-sheet picture: Sec. 5 (Angular cosmology, M<sup>±</sup>, information limit N)</td></tr><tr><td>Layer (ii): monadic constraint — one-parameter differential projection laws</td><td>Sec. 7.3.2 (Numerical pillar: differential projection laws / one-parameter ratios)<br>(optional supporting derivations) Appendix D</td></tr><tr><td>Layer (ii): stringent consistency test — phase-driven vs density-driven H<sub>0</sub> texture</td><td>Sec. 7.3.2 (phase vs density test; residual slope prediction)<br>Appendix C (cosmology pipeline, estimators, robustness / null tests)</td></tr><tr><td>Layer (ii): EM closure nuance — f<sub>U(1)</sub> not strictly frozen (relaxation tracker)</td><td>Appendix E (near the definition/estimation of f<sub>U(1)</sub>;<br>quark/lepton relaxation discussion)</td></tr><tr><td>Layer (i): logical status of each main claim + minimal falsifier</td><td>Sec. 7.1.1 (Claims Ledger L#; Defined / Postulated / Derived + falsifier)</td></tr><tr><td>Layer (i): empirical kill-switches (must-pass gates)</td><td>Sec. 7.2 (Checks 1–7)<br>Appendix D (derivations and operational definitions used in the checks)</td></tr><tr><td>Layer (i): audit the constants-geometry machinery (no cherry-picking)</td><td>Appendix A (constants graph, admissible rewrites, negative controls, SI traceability, invariance checks)</td></tr><tr><td>Layer (i): concrete cosmology pipeline (anisotropy / H<sub>0</sub> dipole tests)</td><td>Appendix C (pipeline, estimators, robustness tests, permutations / null patterns)</td></tr><tr><td>Layer (i): fully enumerated tests map (complete falsifiability catalog)</td><td>Appendix F</td></tr></tbody></table>
### 1.3.3 Dimensional audit of the sector carriers (SI-first, no hidden conventions)

For later use we treat the carriers $S_i$ as sector-defining deployment scales with explicit SI dimensions. The minimal audit set is:

$$ 
\begin{aligned}
S_{EM} &= \hbar c, \quad [\hbar c] = J m = kg m^3 s^{-2}, \\
S_S &= \hbar c, \quad [\hbar c] = J m = kg m^3 s^{-2}, \\
S_W^{(audit)} &= G_F^{1/2}, \quad [S_W^{(audit)}] = kg^{-1} m^{-2} s^2, \\
S_G &= c^3 \hbar^{-1} \langle \rho \rangle^{-1}, \quad [\langle \rho \rangle] = m^{-2}, \quad [S_G] = [G].
\end{aligned}
$$

Two remarks matter for interpretation:

1. Sharing $\hbar c$ as carrier in EM and strong sectors is consistent with a genuinely volumic propagation channel at the deployment scale.

2. Gravity's carrier is curvature-like: $c^3/\hbar$ multiplied by an areal density inverse. This is why gravity naturally appears as a codimension-2 tension-density channel in the $\Lambda$-decomposition.

### 1.3.3 Dimensional selection: why $d=3$ is derived (not assumed)

For clarity, the $(3+1)$-dimensional continuum is not postulated as a starting stage; it emerges as a compatibility regime selected by independent constraints. In fact, the effective dimensionality is constrained twice: (i) by the SI-audited carrier structure of the constants graph, and (ii) by the requirement that the IR effective theory admits the standard renormalizable coupling pattern (dimensionless gauge/Yukawa/quartic couplings). Together with the diffusion-based spectral plateau $d_s \simeq 3$, this provides a robust, overdetermined selection of $d=3$.

(i) Carrier audit constraint. In $d$ spatial dimensions, the Newton constant has SI dimension

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

But the resistivity table enforces the gravitational carrier

$$ S_G = c^3 \hbar^{-1} \langle \rho \rangle^{-1}, \quad [\langle \rho \rangle] = m^{-2}, $$

so that

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

Therefore, the identification $[S_G] = [G_d]$ forces

$$ d = 3. $$

This is not an aesthetic choice: it is a direct consequence of the closed-world SI audit (in particular the fact that the gravitational carrier is an inverse-area object).

(ii) Renormalizable IR coupling constraint. Independently of gravity, the effective IR description is assumed to be a standard local QFT in $D=d+1$ spacetime dimensions. In mass units, the engineering dimensions of the usual couplings are

$$ [g] = M^{(4-D)/2}, \quad [y] = M^{(4-D)/2}, \quad [\lambda] = M^{4-D}, $$

for gauge ($g$), Yukawa ($y$), and quartic scalar ($\lambda$) couplings. The Standard Model pattern that these couplings are dimensionless is recovered only when $D=4$, i.e.

$$ D=4 \iff d=3. $$
Hence the constants-graph carrier audit and the IR renormalizability requirement select the same spatial dimensionality.

*Synthesis.* The framework thus exhibits a three-way consistency: (1) diffusion on the RTQ substrate yields a spectral plateau $d_s \simeq 3$; (2) the gravitational carrier audit implies $d = 3$; (3) the existence of a renormalizable SM-like IR sector implies $d = 3$. This makes the emergent $(3+1)$ continuum a *derived compatibility regime* of the network, not a postulated background stage.

A systematic dimensional analysis with a free spatial dimension $d$, and its link to RTQ ensemble spectral dimension, is given in Appendix B.5.

### 1.3.4 From Resistivity Table to the $\Lambda$-pivot (what is inferred, and what is postulated)

Table 1 suggests a compact unification principle: across sectors, most differences are carried by three elements:

1. $G_i$: fixed symmetry/topology normalization.

2. $d_i$: which internal axis is probed.

3. $S_i$: which dimensional carrier performs the projection to observables.

This leaves $A_i$ as the natural locus for a common, unit-free organizing invariant. The $\Lambda$-framework therefore introduces a bounded internal invariant $\Lambda_0$ and an internal phase $\Theta$ such that the internal tension components are written as

$$ \Lambda^P = \Lambda_0 \cos(\Theta), \quad \Lambda^I = \Lambda_0 \sin(\Theta), \quad \Lambda_0^2 = (\Lambda^P)^2 + (\Lambda^I)^2. $$

Here $\Lambda^P$ is interpreted as curvature-bearing stiffness and $\Lambda^I$ as coherence-bearing flexibility **internally**. Both are unit-free at the foundational level; **units enter only at projection**, through the carriers $(e, M_W, Q, \langle\rho\rangle, \dots)$. The paper’s falsifiability architecture is built around this separation.

## 1.4 Aim and Scope of This Paper: a compact, testable $\Lambda$-grammar

The full network dynamics (RTQs, internal dimensions, particle phenomenology, cosmology, and high-coherence regimes) is developed elsewhere. The goal of the present article is narrower: to extract a $\Lambda$-grammar—the minimal axiomatic and rewrite structure suggested by the trigger factorization—and present it in a compact form designed to be *auditable and falsifiable*.

By “$\Lambda$-grammar” we mean:

* a bounded invariant $\Lambda_0$ carried by discrete relational nodes (RTQs);

* an internal phase $\Theta$ and a finite set of internal axes $\{d_i\}$ defining admissible projections;

* an explicit rewrite layer mapping $(\Lambda_0, \Theta, d_i)$ projections to standard constants and effective couplings, with geometric factors and SI-audited carriers made explicit.

To prevent predictable misunderstandings, we state what the paper *does not* claim:

* it does *not* redefine measured couplings; it reorganizes them as $G_i \times A_i \times S_i$ under closed-world rules;

* it does *not* numerically fit constants or introduce coincidence equalities;

* it does *not* add ad hoc fields solely to reproduce known values.

### 1.4.1 Internal Propagation Tilt $\chi(\Theta)$ vs. Operational Light Speed: Avoiding the VSL Pitfall

The $\Lambda$-formalism naturally yields the internal dimensionless ratio

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

which parametrizes the relative ease of coherence updates versus physical stiffness.
In metric-defined regimes ($\Lambda^P > 0$), $\chi(\Theta)$ maps onto an operational propagation bound in the IR description. In the limit $\Lambda^P \to 0$, the metric sector no longer constrains propagation and the formal divergence indicates a transition to a non-metric regime; it is not an assertion of operational superluminal signaling inside a metric domain.

Accordingly, this paper treats $\chi(\Theta)$ as the primary internal quantity; the symbol $c$ is reserved for its stable mapping in metric regimes.

*Negative check (must-pass): any demonstrable operational superluminal signaling in metric-defined regimes ($\Lambda^P > 0$) would falsify the mapping.*

### 1.4.2 Quantization of $\Theta$ and falsifiability (finite-state hypothesis)

Because the substrate is discrete and finite-state, the framework takes $\Theta$ to admit a quantized spectrum

$$ \Theta_n = n \Delta \Theta_{\min}. $$

This is introduced as a capacity constraint: if $\Theta$ were continuously tunable with arbitrary precision, the internal state per RTQ would be unbounded.

Quantized $\Theta$ implies discrete admissible partitions and therefore testable restrictions, notably correlated (non-independent) drifts among $\{c, G, \hbar, \alpha, \dots\}$ under the rewrite layer. The falsification logic is direct: if high-precision datasets show statistically robust, mutually independent drifts incompatible with the predicted correlation structure, the $\Lambda$-grammar collapses.

### 1.4.3 What the paper will deliver

Concretely, the article aims to:

* formulate the $\Lambda$-framework axiomatically in network terms (RTQs, internal axes, complex tension $\Lambda = \Lambda^P + i\Lambda^I$);

* derive a compact rewrite layer expressing standard constants as constrained projections of $\Lambda_0$ (explicit geometric normalizations and SI-audited carriers; no free numerical fitting);

* sketch the cosmological anchoring (including the relationship between $\Lambda_{GR}$ and projected/averaged internal balance);

* provide an explicit falsifiability architecture, separating negative consistency checks from positive tests and identifying a minimal validation core.

Read as intended, the paper presents a top-down structural grammar: a minimal internal set

$$ (\Lambda_0, \Lambda^P, \Lambda^I, \Theta, d_1, \dots) $$

from which the usual constants and interaction strengths arise as constrained projections. The emphasis is not to claim complete dynamical closure at this stage, but to specify a generative structure precise enough to be audited, constrained, or refuted.
## 2 Axiomatic $\Lambda$-Framework: Relational Tension and Internal Dimensions (Theory I)

This section formulates the $\Lambda$-framework at the axiomatic level. The basic ingredients are (i) a background-free network of Relational Tension Quanta (RTQs), (ii) a complex tension field $\Lambda = \Lambda^P + i\Lambda^I$ of fixed modulus $\Lambda_0$, and (iii) a finite set of internal axes $d_1 - d_5$ encoding interaction channels and a global phase $\Theta$. Spacetime, fields, masses, and constants will later be treated as emergent projections of these internal structures.

The effective number of spatial dimensions will not be assumed a priori. Instead, it will be treated as a spectral dimension $d_s$, inferred from diffusion on the RTQ hypergraph (Appendix B), and later shown to select three spatial dimensions in the low-energy regime.

### 2.1 Relational Tension Quanta (RTQs) as Fundamental Nodes

#### 2.1.1 Discrete Network Without Background Metric

##### Postulate 1 (Ontology)

The fundamental substrate is a discrete, background-free network of Relational Tension Quanta (RTQs). There is no pre-given manifold, no intrinsic coordinates, and no prior metric—only RTQs and the relations between them. This postulate follows the spirit of background independence in quantum gravity and aligns with causal/discrete approaches where spacetime structure is not assumed but reconstructed from order and connectivity [Brightwell and Gregory, 1991, Smolin, 2006, Sorkin, 2003, Knuth and Bahreyni, 2014].

More recently, Knuth and Bahreyni [2014] have shown how a Minkowski-like spacetime geometry and Lorentz transformations can emerge from a consistent quantification of a causal network of events as seen by an embedded observer, which is conceptually close to the RTQ ontology adopted here.

This concept echoes the historical vision of quantum geometrodynamics, where space and time emerge from the relationships between fundamental elements rather than being pre-given structures [Wheeler, 1957]. This approach challenges the classical notion of a continuous spacetime background and instead suggests that geometry itself is dynamically generated through the interactions and connections of RTQs.

An RTQ is defined purely by its adjacency pattern to other RTQs. The substrate can be represented as an oriented hypergraph:

* Nodes: RTQs.

* Hyperedges: multi-body relations encoding higher-order interactions [Carletti et al., 2020].

* Orientation: incidence signs encode directed influence / information flow [Vu, 2003].

Hyperedges allow non-pairwise structures (e.g., SU(3) baryonic combinatorics) without introducing a continuous background or embedding fields. Topology itself is dynamical: edges may switch on/off or rewire, echoing models where geometry emerges from graph/hypergraph updates and “graphity”-like phase structure [Konopka et al., 2008, Oriti, 2018, Wolfram, 2020].

No edge carries an intrinsic length. Locality, distance, dimension, and curvature are emergent statistical descriptors of adjacency patterns rather than primitive inputs:

* dense, coherent connectivity → effectively positive curvature;

* sparse, noisy connectivity → nearly flat or negative curvature.

“Space” is thus a large-scale shadow of hypergraph adjacency, not a primitive container.

##### Calibration note (Planck symbols are reader-facing references, not RTQ primitives)

At the RTQ level, we denote the fundamental update scales by $(\ell_0, t_0)$. These are defined independently of $\{G, c, \hbar\}$ inside the ontology above. When we quote the “Planck domain” values from [NIST CODATA, 2022a],

$$ \ell_P \approx 1.616 \times 10^{-35} m, t_P \approx 5.391 \times 10^{-44} s, $$
we use them only as CODATA-calibrated reference units to communicate typical magnitudes to the reader. Importantly, $(\ell_P, t_P)$ are conventionally derived from $\{G, c, \hbar\}$; therefore, in this framework they must not be treated as fundamental inputs. Any identification $(\ell_0, t_0) \mapsto (\ell_P, t_P)$ is understood as an a posteriori calibration step once the effective constants are recovered from the audited carrier/projection relations. Motivationally, this is compatible with “it from bit” and informational readings of fundamental discreteness, while keeping the non-circular status of $(\ell_0, t_0)$ explicit [Rovelli, 1996, Wheeler, 1989, Zizzi, 2010].

## 2.1.2 Local Update Rules and absence of Global Time

### Postulate 2 (No global time)

There is no external time parameter and no universal clock. RTQs undergo local state transitions with a minimal update duration $t_0$. These transitions are network update quanta (a counting/ordering structure), not slices of an underlying background time. The notion of macroscopic time is recovered only after coarse-graining, when a stable, timelike coherence flow exists [Smolin, 2006, Barbour, 2012, Caticha, 2015, 2018].

This view is closely aligned with entropic-dynamics approaches, in which time is interpreted as an ordering parameter for inferential updates rather than as a fundamental background variable [Caticha, 2015].

Each local update can involve:

* a change in the internal orientation $(\Lambda^P, \Lambda^I, \Theta, d_i)$,

* adjacency rewiring (creation/destruction of hyperedges),

* local redistribution of tension between $\Lambda^P$ and $\Lambda^I$.

neighboring RTQs constrain one another via relations that propagate along links at a finite rate controlled by the local tension configuration:

* high $\Lambda^P$: strong coupling, fast propagation of constraints;

* low $\Lambda^P$: weak coupling, slow/irregular propagation.

The emergent propagation bound in metric regimes will later be denoted $c_{phys}$; internally, we track the dimensionless ratio $\chi(\Theta) = \Lambda^I/\Lambda^P$ as the primary parameter (cf. 2.2.3 and 1.4.1).

Large-scale temporal order and causal structure appear only after coarse-graining many such local updates. Macroscopic “time” corresponds to a synchronized phase of the causal axis $d_4$ kuramoto-type dynamics [Acebrón et al., 2005, Rodrigues et al., 2016] (cf. Sec. 2.3.4), and not to a fundamental variable.

Quantum non-locality is encoded as persistent topological configurations (e.g. long-range correlations in adjacency), not as superluminal updates.

## 2.2 Complex Tension as Fundamental Quantity

The network is endowed with a bipolar tension field $\Lambda$ that quantifies both curvature-bearing stiffness and coherence-bearing flexibility.

### 2.2.1 Definition of $\Lambda=\Lambda^P+i\Lambda^I$

### Postulate 3 (Complex tension)

Each link (and, by extension, each RTQ) carries a complex tension

$$ \Lambda = \Lambda^P + i\Lambda^I, \quad |\Lambda| = \Lambda_0 = \text{const}, $$

where:

* $\Lambda^P$: physical tension — the crystallized, curvature-generating part of the field; it controls inertia, metric rigidity, and the capacity to perform mechanical work.

* $\Lambda^I$: informational tension — the coherent, order-bearing part; it measures negentropic structure and phase alignment.
Internally, $\Lambda_0$, $\Lambda^P$, and $\Lambda^I$ are taken as dimensionless. Physical units emerge only when these internal quantities are projected onto observational channels via dimensional carriers (charges, masses, momentum scales, node densities; cf. Appendix A).

This split is not a dualism between “matter” and “information”, but a decomposition of a single connective resource into crystallized vs stored components [Schrödinger, 1944, Shannon, 1948, Landauer, 1961].

The choice of a complex tension $\Lambda = \Lambda^P + i\Lambda^I$ mirrors the role of complex amplitudes in quantum theory and is consistent with information-geometric reconstructions in which the complex Hilbert-space structure is uniquely selected by probabilistic and operational consistency requirements [Goyal, 2010].

### 2.2.2 Polar Decomposition and the Invariant $\Lambda_0$

**Postulate 4 (Polar form and invariant)** $\Lambda$ is constrained to a fixed modulus $\Lambda_0$. We write

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

with $\Theta \in [0, \pi/2]$ an internal phase angle in tension space.

* $\Lambda_0$ is a bounded invariant of the network: a generalized “tension Hamiltonian” conserved under internal dynamics.

* $\Theta$ controls the balance between curvature-bearing stiffness ($\Lambda^P$) and coherence-bearing flexibility ($\Lambda^I$).

Qualitatively:

* $\Theta \simeq 0$: $\Lambda^P$ dominates $\to$ metric, inertia-dominated, “matter-like” regime.

* $\Theta \simeq \pi/2$: $\Lambda^I$ dominates $\to$ highly coherent, non-metric, “information-like” regime.

* Intermediate $\Theta$: mixed regimes where spacetime and coherence co-exist.

In the cosmological channel, projection of $\Lambda^P$ on an emergent metric yields a curvature density with units [$m^{-2}$], allowing $\Lambda^P$ to be related to an effective cosmological constant. In microscopic channels, combinations such as

$$ \hbar \sim \Lambda_0 \ell_P t_P $$

can be interpreted as action quanta, with $\Lambda_0$ playing the role of a fundamental tension scale.

### 2.2.3 Angle $\Theta$ as Phase Regulator Between Curvature and Coherence

The internal phase $\Theta$ is a dynamical degree of freedom, not a fixed label. It is constrained only by the invariant radius $\Lambda_0$; the orientation in the $(\Lambda^P, \Lambda^I)$ plane is free to evolve.

Local evolution of $\Theta$ is driven by:

* reduction of local tension imbalances,

* diffusion of angular gradients across the network,

* constraints imposed by interaction axes $d_1 - d_4$.

A useful dimensionless control parameter is the coherence-to-stiffness ratio

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

Internally, $\chi(\Theta)$ measures how easily coherence updates percolate through the network relative to metric stiffness. In metric-defined regimes ($\Lambda^P > 0$), $\chi(\Theta)$ will later be mapped to an operational propagation bound $c_{phys}$ (Section 3). In the limit $\Lambda^P \to 0$, the divergence of $\chi(\Theta)$ signals the degeneracy of the metric sector, not superluminal signalling: spatial separation ceases to be operational, and the description must change to a purely informational one (cf. Sec. 6.1).
## Quantization of $\Theta$

Finite information principles [Bekenstein, 1973, Wheeler, 1990, Lloyd, 2000, Knuth, 2005] motivate a quantized spectrum

$$ \Theta_n = n \Delta\Theta_{\min}, \quad n = 0, \dots, N_{\max}, \quad 0 \le \Theta \le \frac{\pi}{2}, $$

to avoid unbounded internal state capacity per RTQ. This implies:

* a finite set of admissible phase states;

* $\Theta = 0$ (frozen, purely metric) and $\Theta = \pi/2$ (pure coherence) are attainable orientations, not asymptotic limits;

* a countable family of macroscopic “phase-defined” universes labelled by coarse-grained $\Theta$.

A key consequence is that “extreme” phase regimes are reached through a finite cascade of admissible phase states, rather than as continuous asymptotes. In particular, the strong-curvature limit relevant for horizons corresponds to a finite-step approach toward the saturated phase $\Theta = 0$, which provides a built-in cutoff mechanism against classical singular behavior (see Sec. 4.3.1).

Conservation of $\Lambda_0$ then enforces an exchange rule

$$ \Lambda_0^2 = (\Lambda^P)^2 + (\Lambda^I)^2 = \text{const}, $$

where transfers between $\Lambda^P$ and $\Lambda^I$ represent rearrangements of tension rather than conversion of Joules into “information energy”. Only $\Lambda^P$ couples to the emergent metric and stress-energy tensor, preserving standard thermodynamic constraints (no “Landauer reversal”).

## 2.3 Internal Interaction Dimensions $d_1-d_5$

Each RTQ carries a finite set of internal axes $d_1, \dots, d_5$ along which $\Lambda_0$ can be oriented and projected. These axes are state-space directions, not extra spacetime dimensions, and they govern how $\Lambda^P$ and $\Lambda^I$ are stored, propagated, or locked in different interaction channels [Bohm, 1980, Overduin and Wesson, 1997a, Healey, 2007]. In particular, they are distinct from the spectral dimension $d_s$ of the RTQ network, which will be defined later from diffusion on the underlying hypergraph. The internal axes $d_i$ live in an abstract interaction/state space; the spatial dimensionality emerges from the large-scale connectivity of the RTQ ensemble.

In this work we reserve $d_1, \dots, d_5$ for the internal interaction/state axes carried by each RTQ node. To avoid any ambiguity with emergent spacetime coordinates, we denote the coarse-grained macroscopic spacetime as $d_6-d_9$, with $(d_6, d_7, d_8) \equiv (x, y, z)$ and $d_9 \equiv t$. These emergent coordinates do not exist at the microscopic RTQ level; they arise only after coarse-graining the structural adjacency graph in the $\Lambda^P$-dominated regime.

In that sense, the causal/CPT internal axis $d_4$ contributes indirectly to spacetime emergence by enabling a globally consistent causal ordering of updates. When the metric sector degenerates ($\Lambda^P \to 0$), spatial separation becomes non-operational and adjacency must be redefined in informational terms (phase/coherence-based neighbourhood rather than metric neighbourhood).

* $d_1$: U(1) axis — electromagnetic phase and electric charge.

* $d_2$: SU(2) axis — weak sector and inertial mass projection.

* $d_3$: SU(3) axis — colour connectivity and confinement.

* $d_4$: causal/CPT axis — local arrow of time and matter-antimatter mapping.

* $d_5 \equiv \Theta$: global $\Lambda$-phase — regulator of the $\Lambda^P/\Lambda^I$ balance and, hence, of effective constants.

Rotations along these axes change how tension is distributed (masses, charges, coherences, arrows of time), not where RTQs are in emergent spacetime.
### 2.3.1 d₁ (U(1)): Charge and Electromagnetic Phase

Axis $d_1$ carries a U(1) internal phase $\phi_{d_1}$ associated with electromagnetism [Lyre, 2004, Navas et al., 2024]. The corresponding connection $A_\mu^{(d_1)}$ defines a curvature

$$F_{\mu\nu}^{(d_1)} = \partial_{\mu}A_{\nu}^{(d_1)} - \partial_{\nu}A_{\mu}^{(d_1)},$$

which reproduces the electromagnetic field strength in the continuum limit.

* Electric field and magnetic field can be written schematically as

$$E_{\mu} \sim -\partial_{\mu}\phi_{d_1}, \quad B_{\mu} \sim (\nabla \times A_{d_1})_{\mu}.$$

#### 2.3.1.1 Charge as winding

Electric charge arises as a topological winding number of

$$q = \frac{e}{2\pi} \oint d\phi_{d_1}. \qquad (1)$$

so that charge conservation is a consequence of topological invariance [Thouless, 1998, Zhang et al., 2020]. Stable charged excitations correspond to U(1) loops in the network, reminiscent of Skyrme-type solitons [Skyrme, 1961, Ma and Yang, 2016].

#### 2.3.1.2 Normalization and the appearance of fractional charges (confinement-first)

Eq. (1) determines electric charge as a topological invariant, but it fixes it only up to an overall unit. We therefore separate the integer winding number from the charge unit by writing

$$n_{d_1} \equiv \frac{1}{2\pi} \oint d\phi_{d_1} \in \mathbb{Z}, \quad Q = n_{d_1} q_0,$$

and ask what minimal choice of $q_0$ allows leptons and quarks to be described within a single U(1) winding mechanism.

In MCP, the closure principle comes from confinement (Sec. 2.3.3): a single coloured excitation carries a non-trivial $Z_3$ obstruction (the centre of SU(3) on $d_3$) and is therefore not an asymptotic closed object, whereas colour-neutral composites close naturally in triads. In a confinement-first reading, this selects the minimal bookkeeping unit

$$q_0 = \frac{e}{3}, \quad \Rightarrow \quad Q = n_{d_1} \frac{e}{3}, \quad n_{d_1} \in \mathbb{Z}.$$

Leptonic unit charge then corresponds to three elementary winding units, while quark charges appear as partial U(1) windings that are only globally admissible within colour-neutral composites. In this view, quarks are not “fractionally charged objects” in isolation: they are non-closable motifs unless the SU(3) obstruction is neutralized (Sec. 2.3.3).

With this convention one has, for example,

$$Q_u = +2e/3, \quad Q_d = -e/3, \quad Q_{e-} = -e,$$

and additivity of windings on closed motifs reproduces composite charges (e.g. $Q_{uud} = +e$).
<table><caption>Table 3: Charge bookkeeping from U(1) winding with MCP normalization.</caption><thead><tr><th>Particle</th><th>n<sub>d1</sub></th><th>Q = n<sub>d1</sub> e/3</th><th>Observed Q/e</th></tr></thead><tbody><tr><td>ν</td><td>0</td><td>0</td><td>0</td></tr><tr><td>d</td><td>-1</td><td>-e/3</td><td>-1/3</td></tr><tr><td>u</td><td>+2</td><td>+2e/3</td><td>+2/3</td></tr><tr><td>e<sup>-</sup></td><td>-3</td><td>-e</td><td>-1</td></tr><tr><td>e<sup>+</sup></td><td>+3</td><td>+e</td><td>+1</td></tr></tbody></table>

### 2.3.2 d₂ (SU(2)): Inertia, Mass Generation and Weak Sector

Axis $d_2$ encodes SU(2) orientation and plays the role of an *inertial axis* for RTQ motifs [Erler and Freitas, 2020]. A spinor state $\psi_i = (\psi_L, \psi_R)$ lives on SU(2); only its *orientation* relative to $d_2$ matters.

Let $\hat{n}$ denote the (unit) internal orientation of a motif in SU(2) state space, and let $\hat{e}_{d_2}$ denote the unit inertial axis associated with $d_2$. We define the projection angle $\theta_{d_2}$ by

$$ \cos \theta_{d_2} \equiv \hat{n} \cdot \hat{e}_{d_2} \iff \theta_{d_2} = \arccos(\hat{n} \cdot \hat{e}_{d_2}). $$

A schematic mass projection law is then

$$ m \propto \Lambda^P (\hat{n} \cdot \hat{e}_{d_2}) = \Lambda^P \cos \theta_{d_2}. $$

To connect this projection law with the Standard Model mapping, we write the fermion mass as an effective relation

$$ m_f = y_f \Lambda^P \cos \theta_{d_2}, $$

where $y_f$ plays the phenomenological role of a Yukawa coupling, but is reinterpreted in MCP as an *effective topological stiffness* (or geometric sensitivity) of the fermionic motif relative to its anchoring on $d_2$, rather than as an arbitrary numerical residue.

In this view, mass hierarchies reflect differences in how “tight” or “rigid” the corresponding closed RTQ motifs are in state space (their internal loop complexity and anchoring efficiency on $d_2$), while the projection factor $\cos \theta_{d_2}$ captures the orientation dependence emphasized above. At the present stage, $y_f$ is an effective parameter to be derived from microscopic motif geometry in the fully discrete description.

* $\theta_{d_2} = \pi/2$: no projection $\Rightarrow$ massless excitations (photon, gluon).

* $\theta_{d_2} < \pi/2$: stored torsion $\Rightarrow$ massive modes (fermions, W/Z).

This geometrises the Higgs mechanism: electroweak fields are rotated into massless and massive modes by an internal angle, now interpreted as the orientation of tension in ($d_1, d_2$)-space [Erler and Freitas, 2023].

Identifying $d_2$ with a Higgs axis $d_H$, the Higgs boson corresponds to collective *angular vibrations* $\delta\hat{e}_{d_2}$ around the vacuum orientation, analogous to amplitude (Higgs) modes in condensed matter [Boroojerdian, 2013, Gneiting et al., 2017, Pekker and Varma, 2020].

Spatial gradients of the same projection $\Lambda^P \cos \theta_{d_2}$ act as sources of effective curvature, tying together inertia and gravitation as two scales of a single angular configuration of $\Lambda^P$.

The Weinberg–Salam model, which contextualizes the weak sector within an emergent spacetime framework, further supports the idea that mass generation can be understood in terms of angular projections in the SU(2) space, as detailed [Niemi and Slizovskiy, 2010]. Their work connects the geometrization of the weak sector with the concept of emergent spacetime, reinforcing the analogy between gauge symmetries and curvature dynamics.
### 2.3.3 $d_3$ (SU(3)): Strong Confinement and Colour Connectivity

Axis $d_3$ encodes SU(3) colour orientation [Ryan, 2017, Birse, 2018, Dehghan and Faber, 2024]. Colour-neutral configurations satisfy

$$ \sum_{i=1}^{3} \Lambda_{(i)}^{P} = 0, $$

where the subscript $i$ labels colour components. Strong interactions correspond to how $\Lambda^P$ is stored and transported in non-abelian flux structures.

In the network picture:

* colour flux is the component of $\Lambda^P$ aligned with $d_3$;

* confinement arises from the requirement that colour flux forms closed node loops with

$$ \oint \Lambda_{(d_3)}^{P} = 0, $$

#### 2.3.3.1 A one-line closure test: quarks are non-asymptotic, baryons close.

The colour axis $d_3$ carries a $Z_3$ centre constraint: a single coloured motif belongs to a non-trivial $Z_3$ class and therefore cannot define an asymptotic closed excitation on $d_3$ (confinement-first). Colour-neutral states are precisely those for which the combined $d_3$ holonomy lies in the trivial $Z_3$ class, with the minimal closure being triadic (baryons) or quark-antiquark (mesons). Once this closure is satisfied, the U(1) winding bookkeeping of Sec. 2.3.1 becomes globally admissible and reproduces the observed composite charges (see Sec. 2.3.3.3 for the worked uud example).

#### 2.3.3.2 A minimal “mechanism-level” closure rule (electron, quarks, proton)

To make the fermion sector explicit (beyond a purely geometric description), one may write a discrete RTQ loop $\Gamma$ with three parallel transports, one per internal gauge axis:

$$ \hat{u}_k \equiv (\chi_k, e^{i\theta_k}, g_k), \quad \chi_k \in \text{SU}(2) (d_2-d_4), \quad e^{i\theta_k} \in \text{U}(1) (d_1), \quad g_k \in \text{SU}(3) (d_3). $$

The loop invariants are then the holonomies

$$ U_{\Gamma}^{(24)} = \prod_{k} U_{k}^{(24)}, \quad \Delta\theta_{1,\Gamma} = \sum_{k} \Delta\theta_{1,k}, \quad U_{\Gamma}^{(3)} = \prod_{k} U_{k}^{(3)}. $$

A minimal MCP-quantization package can be stated as:

$$ U_{\Gamma}^{(24)} = -I_2 \quad (\text{spinorial closure} / 4\pi\text{-double cover}), $$

$$ \Delta\theta_{1,\Gamma} = 2\pi w_1, \quad Q = \frac{e}{3} w_1 \quad (\text{U(1) winding} \rightarrow \text{charge}), $$

$$ U_{\Gamma}^{(3)} = \omega^t I_3, \quad \omega = e^{i2\pi/3}, \quad t \in \{0, 1, 2\} \quad (\text{colour } Z_3 \text{ obstruction}). $$

Colour-singlet states correspond to $t=0$. A single quark corresponds to $t \neq 0$ and therefore cannot be an asymptotic closed excitation on $d_3$ (confinement-first). The baryonic closure rule is then simply:

$$ t_1 + t_2 + t_3 \equiv 0 \pmod{3} \iff U_{\Gamma_1}^{(3)} U_{\Gamma_2}^{(3)} U_{\Gamma_3}^{(3)} = I_3. $$
### 2.3.3.3 Worked bookkeeping

With the normalization $Q = (e/3)w_1$, an electron is a colour-singlet motif with $t = 0$ and $w_1 = -3$. A proton is a colour-singlet triplet $(u, u, d)$ whose charge is the sum of the three U(1) windings,

$$w_1(u) = +2, \quad w_1(d) = -1 \implies Q_p = \frac{e}{3}(2+2-1) = +e,$$

while colour-neutrality is enforced by the $Z_3$ closure of the three $d_3$ holonomies.

A fuller construction (explicit step budgets, minimal $N$ for each motif, and “braided” realizations of $uud$) is given in the dedicated Appendix E.

leading to string-like potentials $V(r) \simeq kr$ consistent with lattice QCD and dual-superconductor models.

Gluons correspond to propagating colour-realignment modes:

$$\text{gluon} \sim \delta\phi_{d_3} = \partial_t(\Delta\phi_{d_3}),$$

with non-abelian curvature

$$F_{\mu\nu}^{(d_3)} = \partial_{\mu}A_{\nu}^{(d_3)} - \partial_{\nu}A_{\mu}^{(d_3)} + [A_{\mu}^{(d_3)}, A_{\nu}^{(d_3)}],$$

and stored physical tension scaling like $\langle F_{\mu\nu}^{(d_3)}F^{(d_3)\mu\nu} \rangle$. Again, $\Lambda^P$ tracks the crystallized tension in these loops; $\Lambda^I$ controls the coherence of colour alignment. Hadrons are stable “tension knots” in the $d_3$ sector.

### 2.3.4 $d_4$: Topological / Causal Axis (Arrow of Time, CPT)

Axis $d_4$ encodes causal orientation and CPT structure [Lüders, 1954, Pauli, 1955, Beisert, 2019]. Each RTQ carries a local causal phase $\theta_i$ and an effective timelike vector $T_{\text{local}}$ in $d_4$-space.

A Kuramoto-type synchronization equation,

$$\frac{d\theta_i}{dt} = \omega_i + K \sum_j \sin(\theta_j - \theta_i),$$

with coupling $K \propto \Lambda^P$, captures the emergence of a global arrow of time [Acebrón et al., 2005, Rodrigues et al., 2016, Barbour et al., 2014]:

* below a critical $K$: causal phases remain disordered $\to$ reversible, pre-temporal regime;

* above threshold: phases lock globally $\to$ emergent time arrow, with a well-defined macroscopic $T_\mu$ satisfying $T_\mu T^\mu = -1$.

In this view, quantum superposition corresponds to local desynchronization in $d_4$; decoherence corresponds to resynchronization enforced by the environment. $\Lambda^P$ acts as a synchronizing tension: larger $\Lambda^P$ enhances temporal ordering and irreversibility.

CPT symmetry is implemented as a global inversion

$$(d_1, d_2, d_3, d_4) \to (-d_1, -d_2, -d_3, -d_4),$$

naturally leading to a pair of conjugated sheets:

* $M^+$: our sheet, with arrow $+T$;

* $M^-$: mirror sheet, with arrow $-T$ and conjugate charges,

linked by a global constraint $\delta E_{(+)} + \delta E_{(-)} = 0$. Matter and antimatter appear as opposite causal orientations in this twin-sheet structure.
### 2.3.5 $d_5$: Global Phase Dimension $\Theta$ (Angular Regulator of Universes)

Axis $d_5$ is identified with the global tension phase $\Theta$:

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

$d_5$ is not spatial; it re-orient the same invariant $\Lambda_0$ between:

* a crystallized, curvature-bearing mode ($\Lambda^P$),

* and a coherence-bearing mode ($\Lambda^I$).

Regimes:

* $\Theta \simeq 0$: $\Lambda^P$-dominated $\rightarrow$ stiff, curvature-heavy network; slow coherence propagation.

* $\Theta \simeq \pi/4$: $\Lambda^P \simeq \Lambda^I \rightarrow$ balanced sheet, candidate regime for our Universe.

* $\Theta \rightarrow \pi/2$: $\Lambda^I$-dominated $\rightarrow$ coherence regime with vanishing metric stiffness; a coherence boundary or purely informational limit (in a strictly operational, non-metaphysical sense).

Within the domain $\Theta < \pi/2$, all derived effective constants will remain finite; the singular behavior at $\Theta \rightarrow \pi/2$ represents the breakdown of metric-based description rather than a physical divergence.

In later sections (3 and 5), $d_5$ will be used to express familiar constants {$c, \hbar, G, \alpha, \dots$} as angular projections of $\Lambda_0$ and to organize phase-defined cosmological scenarios.

## 2.4 Synthesis: $\Lambda_0$ and $\Theta$ as the Minimal Generative Pair

The axiomatic content of the $\Lambda$-framework can now be summarized in terms of a minimal generative pair ($\Lambda_0, \Theta$) acting on a finite set of internal axes $d_1 - d_5$. $\Lambda_0$ is a bounded invariant tension; $\Theta$ sets the balance between its physical and informational projections. The axes $d_1 - d_4$ organize how this tension appears as charges, masses, confinement, and causal ordering; $d_5$ regulates global regimes and, ultimately, the hierarchy of effective constants.

<table><caption>Table 4: A compact summary roles of $\Lambda^P$, $\Lambda^I$, $\Theta$ and $d_1-d_5$ is given below</caption><thead><tr><th>Symbol / axis</th><th>Domain / symmetry</th><th>Primary role in Λ-framework</th><th>Typical emergent manifestations</th></tr></thead><tbody><tr><td>Λ<sup>P</sup></td><td>–</td><td>crystallized, curvature-bearing tension; couples to metric</td><td>Inertia, gravitating energy, cosmological constant</td></tr><tr><td>Λ<sup>I</sup></td><td>–</td><td>Coherence-bearing tension; stores negentropic order</td><td>Phase coherence, correlation, high-order structure</td></tr><tr><td>Θ</td><td>d<sub>5</sub> (global phase)</td><td>Sets ratio Λ<sup>I</sup>/Λ<sup>P</sup> = tan Θ; regime selector</td><td>Angular control of {c, h, G, α, ...}; phase-defined universes</td></tr><tr><td>d<sub>1</sub></td><td>U(1)</td><td>Electromagnetic phase / charge winding</td><td>Photons, electric charge, EM fields</td></tr><tr><td>d<sub>2</sub></td><td>SU(2)</td><td>Inertial axis; mass projection and weak sector</td><td>Fermion/W/Z masses, Higgs-like modes</td></tr><tr><td>d<sub>3</sub></td><td>SU(3)</td><td>Colour orientation; confinement loops</td><td>Gluons, hadrons, nuclear matter</td></tr><tr><td>d<sub>4</sub></td><td>Causal / CPT</td><td>Local arrow of time; matter-antimatter mapping, twin sheets</td><td>Time's arrow, decoherence, M<sup>+</sup>/M<sup>-</sup> pairing</td></tr><tr><td>d<sub>5</sub></td><td>Global Λ-phase Θ</td><td>Global balance between Λ<sup>P</sup> and Λ<sup>I</sup></td><td>Regimes from metric-dominated to coherence-dominated</td></tr></tbody></table>
Read as a grammar:

• $\Lambda_0$ sets the *bounded resource* of tension in the network.

• $\Theta$ partitions this resource between physical and informational modes.

• $d_1 - d_3$ distribute $\Lambda^P$ and $\Lambda^I$ across gauge channels (U(1), SU(2), SU(3)).

• $d_4$ organizes causal coherence and implements CPT symmetry and sheet pairing.

• $d_5$ determines the global regime in which all of the above operate and constrains how constants can vary (coherently) across cosmological histories.

In the next section, this axiomatic structure will be connected explicitly to the constants dataset introduced in Section 1. By treating $\{c, \hbar, G, \alpha, \dots\}$ as angular projections of $\Lambda_0$ along the axes $d_1 - d_5$, we obtain a compact map where interaction strengths and scales emerge from a small set of geometric parameters rather than from independent primitives.
# 3 Fundamental Constants as Angular Projections of $\Lambda$ (Theory II)

In the $\Lambda$-framework, “fundamental constants” are not independent inputs but responses of a discrete tension network oriented by the global phase $\Theta$ and internal axes $d_i$. Dimensionless couplings $(\alpha, \alpha_w, \alpha_s, \dots)$ arise as angular coefficients; dimensionful constants $(c, \hbar, G, \dots)$ are rescaling parameters that relate internal dynamics to SI units via the Planck grain $(\ell_P, t_P)$ and tension scale $\Lambda_0$.

## 3.1 General Principle: Constants as Projections on Internal Axes

Physical reality is modeled as a network of RTQs with bipolar tension

$$ (\Lambda^P, \Lambda^I) = \Lambda_0(\cos \Theta, \sin \Theta), $$

defined on internal axes $d_1-d_4$ (U(1), SU(2), SU(3), causal/CPT) plus a global phase axis $d_5 \equiv \Theta$ (Sections 1–2). Here, $\Lambda^P$ represents curvature-bearing stiffness, while $\Lambda^I$ represents coherence and negentropy [Schrödinger, 1944, Shannon, 1948, Jaynes, 1957, Landauer, 1961].

In this setting, each constant associated with an interaction channel $i$ can be decomposed as an effective resistivity of the network:

$$ C_i = \mathcal{G}_i \times \mathcal{A}_i(\Theta, \{\theta_j\}) \times \mathcal{S}_i(\Lambda_0, \ell_P, t_P, \dots), $$

where:

* $\mathcal{G}_i$: a geometric factor (e.g. $1/4\pi$, $1/8\pi$) fixed by embedding conventions (solid angles, spherical surfaces, etc.);

* $\mathcal{A}_i$: an angular factor (dimensionless), built from the global phase $\Theta$ and local internal angles $\theta_j$ on the axes $d_1, d_2, d_3, d_4$;

* $\mathcal{S}_i$: a dimensional carrier, collecting all SI dimensions (kg, m, s) through combinations of $\Lambda_0$ and the Planck grain, or through reference scales (masses, momenta, densities).

The resulting constants dataset – including, schematically, $c, \hbar, G$ on the carrier side and $\alpha, \alpha_w, \alpha_s, \alpha_G$ on the alignment side – is thus a projection grammar of $\Lambda_0$ onto specific channels, not a list of unrelated parameters.

### 3.1.1 Dimensional vs Dimensionless Factors

In standard practice, one distinguishes dimensionless constants (pure numbers like $\alpha$) from dimensionful constants (scales like $c, \hbar, G$). The $\Lambda$-framework sharpens this split by assigning them distinct structural roles:

#### 1. Dimensionless constants as pure angular functions.

Couplings such as $\alpha, \alpha_w, \alpha_s$ and the gravitational alignment $\alpha_G$ belong to the alignment sector:

$$ \alpha, \alpha_w, \alpha_s, \alpha_G, \dots \in \mathcal{A}_i(\Theta, \{\theta_j\}). $$

They encode relative orientations and misalignments along the internal axes $d_1, d_2, d_3$ and their embedding within the global phase $\Theta$.

#### 2. Dimensionful constants as carriers of the tension scale.

Constants like $c, \hbar, G$ appear primarily through the carrier sector $\mathcal{S}_i$. They provide the SI-calibrated projection channels from the microscopic RTQ scales $(\ell_0, t_0)$ and the invariant tension $\Lambda_0$.

Based on the dimensional analysis of the network (see Appendix A), the schematic scaling laws are:

$$ \hbar \sim \Lambda_0 \ell_0 t_0, \quad c \sim \frac{\ell_0}{t_0} \chi(\Theta), \quad G \sim \frac{\ell_0^2}{\langle \rho \rangle \Lambda^P}, $$

with $\chi(\Theta) \equiv \Lambda^I/\Lambda^P = \tan \Theta$ the internal propagation tilt, and $\langle \rho \rangle$ an effective node density. (In the gravitational sector we will refine $\Lambda^P$ to its structural part $\Lambda_{struct}^P$; see Secs. 3.2.1 and 3.4)
When we quote Planck-domain magnitudes, this is only an *a posteriori* CODATA calibration choice for reader orientation; it does not enter the audit or any numerical extraction of parameters.

Crucially, because $\Lambda^P$ and $\Lambda^I$ depend on $\Theta$, these “constants” are dynamically coupled. The specific functional form of this dependence — and how it unifies propagation, quantisation, and gravity — is derived in the following sections (3.2–3.4) and synthesized in the Unified Angular Map (Sec. 3.5).

## 3.2 Propagation Constant and Planck Structure

The speed of light and Planck units emerge from how the invariant tension $\Lambda_0$ is partitioned between structural stiffness and propagating/ confined modes.

### 3.2.1 Internal Propagation Parameter $\chi(\Theta)$ as Structural Tilt

Sec. 3.1 decomposed $\Lambda^P$ into three channels:

$$ \Lambda^P = \Lambda_{\text{struct}}^P + \Lambda_{\text{manif}}^P, \quad \Lambda_{\text{confined}}^P \subset \Lambda_{\text{manif}}^P. $$

* $\Lambda_{\text{struct}}^P$: sheet-wide structural stiffness with no associated energy, no curvature, no information. It sets the causal cone and maximal update rate.

* $\Lambda_{\text{manif}}^P$: manifested tension in propagating modes (gauge fields, elastic curvature modes).

* $\Lambda_{\text{confined}}^P$: torsion locked in closed motifs (mass channel).

The structural component is taken as

$$ \Lambda_{\text{struct}}^P = \Lambda_0 \cos \Theta, $$

and one can associate to it a formal propagation bound

$$ v_{\text{bound}}^2 = \frac{1}{\Lambda_{\text{struct}}^P t_P^2} = \frac{1}{\Lambda_0 \cos \Theta} \frac{1}{t_P^2}. $$

Using $\Lambda^P = \Lambda_0 \cos \Theta$ and $\Lambda^I = \Lambda_0 \sin \Theta$, it is more convenient to work with the dimensionless structural tilt

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

which lives entirely in internal tension space and is not a metric quantity.

A schematic phase-controlled speed scale is

$$ v_*(\Theta) \propto (\ell_0/t_0) \chi(\Theta), $$

which maps to the observed light speed $c$ once a specific sheet with $\Theta = \Theta_*$ is selected.

### 3.2.2 Qualitative regimes

$$ \Theta \approx 0: \Lambda^P \text{ dominant} \Rightarrow \chi(\Theta) \to 0 \quad (\text{propagation strongly suppressed}). $$

$$ \Theta \to \frac{\pi}{2}: \Lambda^P \to 0 \Rightarrow \chi(\Theta) \to \infty \quad (\text{metric channel collapses; no operational superluminality}). $$

Within a given sheet, $\Theta = \Theta_*$ is fixed, so the corresponding light speed $c$ is constant; varying $\Theta$ compares different phase-defined universes, not time-varying light speeds inside one.

The “$\Lambda^P-c-E$” paradox disappears once the tripartition is kept explicit:
* c depends only on $\Lambda_{struct}^P$.

* Mass $m$ depends only on $\Lambda_{confined}^P$ and its projection on $d_2$.

* Radiative/field energy depends on $\Lambda_{manif}^P$.

In $E = mc^2$, $m$ and $c^2$ are parameters from *independent layers* of the same tension field.

### 3.2.3 Planck Units as Network Rescaling Parameters

We denote by $(\ell_0, t_0)$ the microscopic RTQ grain, i.e. the characteristic adjacency-update and tension-reorientation scales of the discrete substrate. These are *internal network scales* and are not introduced through $G, \hbar, c$, nor used as inputs in the constants audit. Rather, they serve as a conceptual micro-scale for discussing how a discrete update process can underwrite continuum carriers [Smolin, 2006, Barbour, 2009].

When we occasionally quote Planck-domain magnitudes, this is *only* a reader-facing CODATA calibration [NIST CODATA, 2022a,b] used for orientation:

$$ \ell_P \approx 1.616 \times 10^{-35} m, \quad t_P \approx 5.391 \times 10^{-44} s. $$

This corresponds to choosing a familiar numerical anchor for $(\ell_0, t_0)$ after the fact; it does not define the theory and it does not enter any inference.

With that understood, one may write schematic dimensional guidance relations (not used as definitions) such as

$$ \hbar \sim \Lambda_0 \ell_0 t_0, $$

interpreted as a minimal action scale associated with a discrete orientation update [Wheeler, 1989, Lloyd, 2000, Knuth, 2005]. Likewise, speed scales can be discussed through $\ell_0/t_0$ together with the phase structure $\chi(\Theta)$ (Section 3.2.1). All audited calculations and carrier checks in this manuscript remain strictly in explicit SI (no $\hbar = c = 1$ convention).

## 3.3 Gauge Couplings and Fine-Structure Relations

Gauge couplings quantify how easily the network reconfigures its internal phases along the axes $d_1$ (U(1)), $d_2$ (SU(2)), and $d_3$ (SU(3)). They are interpreted as *channel-specific resistivities* of $\Lambda_0$ to reorientation.

### 3.3.1 Electromagnetic Coupling $\alpha$ and the $d_1$ projection

On the U(1) axis $d_1$, each RTQ carries a phase $\phi_{d_1}$. Phase gradients and windings define EM fields and charges [Lyre, 2004, Gonzalez-Garcia and M. Yokoyama, 2019]:

* Electric field $E_\mu \sim -\partial_\mu \phi_{d_1}$,

* Magnetic field $B_\mu \sim (\nabla \times A_{d_1})_\mu$,

* Electric charge $q = \frac{e}{2\pi} \oint d\phi_{d_1}$ [Thouless, 1998, Zhang, 2024].

The fine-structure constant $\alpha$ is understood as the *typical misalignment* of U(1) phases between neighboring RTQs. For a representative angle $\theta_e$ on $d_1$,

$$ \alpha \approx \tan^2\theta_e, $$

up to channel-dependent factors. Small $\alpha \sim 1/137$ corresponds to nearly-aligned U(1) phases; EM interactions then represent gentle corrections to phase coherence.

In the resistivity decomposition, the effective Coulomb constant can be written as

$$ k_e = \frac{1}{4\pi} \alpha_y \hbar c = \left(\frac{1}{4\pi}\right) \times \underbrace{(4\pi\alpha)}_{\alpha_y} \times \underbrace{(\hbar c)}_{S_{EM}}, $$
so that the EM determinant

$$C_{\text{EM}} = \frac{1}{4\pi} \alpha_y \hbar c = \alpha \hbar c \equiv e_*^2$$

has dimensions $[kg m^3 s^{-2}]$. Here: $G_{\text{EM}} = 1/4\pi$, $A_{\text{EM}} = \alpha_y = 4\pi\alpha$, $S_{\text{EM}} = \hbar c$.

α itself remains a pure angular quantity; $\hbar c$ carries the dimensions.

### 3.3.2 Weak and Strong Couplings ($\alpha_w, \alpha_s$) and $d_2-d_3$ Geometry

For the weak sector ($d_2$, SU(2)), the Fermi constant can be decomposed as

$$G_F = 1 \times \alpha_w \times M_W^{-2},$$

where $\alpha_w$ is a dimensionless angular factor depending on the electroweak mixing angle $\theta_W$ and the orientation of the Higgs axis $d_2$, while $M_W^{-2}$ is the carrier with units $[kg^{-2} m^{-4} s^4]$. One may think schematically of

$$\alpha_w \sim f(\theta_W, \theta_{d_2}, \Theta),$$

with $\theta_{d_2}$ the typical projection angle of mass-bearing motifs on $d_2$.

For the strong sector ($d_3$, SU(3)), the running coupling $\alpha_s(Q^2)$ depends on both the internal angular configuration on $d_3$ and the probing scale $Q$ [Birse, 2018, Dehghan and Faber, 2024]. A schematic resistivity form is

$$G_s = \frac{1}{2\sqrt{\pi}} \alpha_s(Q^2) S_{\text{strong}}(Q^2),$$

with $S_{\text{strong}}(Q^2)$ a dimensional carrier built from $Q$ and $\hbar c$ (e.g. $\propto \hbar c/Q^2$ depending on conventions). Here:

$$\mathcal{G}_{\text{strong}} = \frac{1}{2\sqrt{\pi}}, \quad \mathcal{A}_{\text{strong}} = \alpha_s(Q^2), \quad \mathcal{S}_{\text{strong}} = \hbar c$$

Angular interpretation:

* $\alpha_s$ measures the ease of colour reorientation on $d_3$, i.e. how sensitive the network is to SU(3) phase misalignments at scale $Q$.

* Confinement corresponds to a regime where reorientations that try to separate colour charges greatly increase $\Lambda^P$ in flux tubes, making isolated colour states energetically inaccessible.

In both cases, $\alpha_w$ and $\alpha_s$ sit entirely in the angular factor $A_i$. Their running with scale $Q$ reflects how internal angles effectively change when probing different motifs and loop configurations in the network, without altering the underlying discrete grain.
## 3.4 Gravitational Constant as Network Rigidity

Gravity in $\Lambda$-theory is emergent hydrodynamics of the tension field, not a fundamental force with its own primitive constant $G$ [Sakharov, 1967, Jacobson, 1995, Padmanabhan, 2007, Visser, 2002, Volovik, 2003].

### 3.4.1 $G(\Theta)$ and Mean Connectivity $\langle \rho \rangle$

At the microscopic level, the balance at node $i$ reads

$$ \sum_j I_{ij} = \Lambda^P \rho_i + E_i, $$

where $\sum_j I_{ij}$ encodes combinatorial curvature of adjacency, $\rho_i$ is the local density of crystallized tension, and $E_i$ is local stored energy. Coarse-graining yields an effective equation of state analogous to Einstein's equation:

$$ \langle \sum_j I_{ij} \rangle = \langle \Lambda^P \rho_i \rangle + \langle E_i \rangle \leftrightarrow G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}. $$

In this mapping:

* $G$ emerges as the conversion factor between discrete tension/curvature densities and the continuum stress-energy tensor.

* The geometric factor $8\pi$ arises from spherical integrations in a smooth 3D manifold; it has no fundamental status in a discrete relational setting [Rovelli and Smolin, 1995].

A natural scaling is

$$ G \propto \frac{1}{\langle \rho \rangle \Lambda^P} \ell_P^2, $$

with $\langle \rho \rangle$ a coarse-grained node density and $\Lambda^P = \Lambda_0 \cos \Theta$. A sparser network ($\langle \rho \rangle$ small) or smaller $\Lambda^P$ yields a larger effective $G$: the same tension gradients produce stronger curvature.

In resistivity form:

$$ G = \frac{1}{8\pi} A_{\text{grav}}(\Theta) S_{\text{grav}}(\langle \rho \rangle), $$

with:

$$ G_{\text{grav}} = \frac{1}{8\pi}, \quad A_{\text{grav}}(\Theta) = \alpha_G(\Theta), \quad S_{\text{grav}} = \frac{\ell_0^2}{\Lambda_0 \langle \rho \rangle}. $$

$G$ is thus *network rigidity*: an effective elastic coefficient for the $\Lambda$-field at macroscopic scales.

### 3.4.2 $\Lambda^P$ as Fundamental Resistivity of the Network

Collecting the channels:

* Gauge sectors ($U(1)$, $SU(2)$, $SU(3)$) describe *internal reorientations* of $\Lambda^P$ along $d_1, d_2, d_3$.

* The metric sector describes *spatial gradients* of $\Lambda^P$ in emergent dimensions $d_6-d_9$.

The same quantity $\Lambda^P$ appears as:

* local curvature generator (through $\nabla \Lambda^P$),

* substrate stiffness for propagation (through $\Lambda_{\text{struct}}^P$),

* reservoir for mass (through $\Lambda_{\text{confined}}^P$ and projection on $d_2$).

From this viewpoint, *gravity is the most constrained channel*:
* U(1) allows relatively free reorientation (weakest resistivity).

* SU(2) and SU(3) are more constrained (non-abelian structure, confinement).

* The metric channel is the most constrained: large-scale curvature changes involve coordinated reorientations across many RTQs, yielding a very small effective coupling G.

### 3.4.3 Angular Dependence and Scaling with $\Theta$

Having identified the sector-specific carriers and the tripartition $\Lambda^P = \Lambda_{struct}^P + \Lambda_{manif}^P$ with $\Lambda_0^2 = (\Lambda^P)^2 + (\Lambda^I)^2$, we can now make explicit how the fundamental couplings depend on the global phase $\Theta$. Since the modulus $\Lambda_0$ is invariant, any change in the structural angle $\Theta$ corresponds to a redistribution of a fixed tension budget between stiffness ($\Lambda^P$) and coherence ($\Lambda^I$). The “constants” are therefore not independent inputs, but coupled projections of this partition.

#### 3.4.3.1 Causal Tilt and Propagation ($\chi$)

The effective propagation bound is controlled by the ratio of available coherence to structural stiffness. This defines the internal “tilt” of the causal cone in tension space:

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

This parameter organizes the kinematic regime of the network:

* **Frozen limit ($\Theta \to 0$):** stiffness dominates ($\Lambda^P \gg \Lambda^I$), so $\chi \to 0$. The network is maximally rigid and causal propagation is strongly suppressed.

* **Non-local limit ($\Theta \to \pi/2$):** stiffness vanishes while coherence is maximal, so $\chi(\Theta)$ diverges. This divergence signals the breakdown of the metric light cone and the onset of a regime where connectivity is purely informational rather than metric.

#### 3.4.3.2 Quantum of Action ($\hbar$)

Quantum behavior is driven by the informational component $\Lambda^I$. The capacity of the network to store and exchange quantum phase scales with this tension channel. At leading order, we therefore take the effective action quantum to scale linearly with the available informational tension:

$$ \hbar(\Theta) \propto \Lambda^I(\Theta) = \Lambda_0 \sin \Theta. $$

This scaling enforces a natural correspondence with the classical limit. If the informational component vanishes ($\Theta \to 0$), the action quantum tends to zero, $\hbar \to 0$, recovering a purely classical vacuum described by a metric theory of gravity. Conversely, $\hbar(\Theta)$ is maximised in the coherence-dominated regime where $\Lambda^I$ saturates.

#### 3.4.3.3 Gravitational Coupling ($G$)

In the gravitational sector, gravity emerges as the elastic response of the physical tension $\Lambda^P$. A larger $\Lambda^P$ means a stiffer network that resists geometric deformation. The effective gravitational constant $G$, which measures the coupling between matter and curvature, is therefore a **susceptibility**: it is inversely proportional to the stiffness density,

$$ G(\Theta) \propto \frac{1}{\langle \rho \rangle \Lambda^P(\Theta)} = \frac{1}{\langle \rho \rangle \Lambda_0 \cos \Theta}. $$

This dependence resolves the apparent “$\Lambda^P-c-G$” paradox:

* At $\Theta \approx 0$ (high stiffness), the network is rigid and the gravitational susceptibility is small: gravity is weak.
* As $\Theta$ increases towards $\pi/2$, the network softens ($\Lambda^P \to 0$). The effective coupling $G(\Theta)$ formally diverges, indicating that the energy cost to curve the background becomes negligible—a regime of metric transparency where the metric description itself ceases to be valid rather than a physically accessible strong-gravity limit.

#### 3.4.3.4 Gauge Coupling Strength ($\alpha$)

The fine-structure constant $\alpha$ quantifies the strength of the interaction between charged motifs and the electromagnetic field. In the $\Lambda$-framework, a gauge interaction requires the simultaneous presence of:

* metric stiffness $\Lambda^P$ to support field geometry and local propagation, and

* coherent flexibility $\Lambda^I$ to allow phase fluctuations and charge winding.

To leading order in the global phase, the gauge coupling strength is thus driven by the normalized geometric product of the two tension components:

$$ \alpha(\Theta) \propto \frac{\Lambda^P(\Theta) \Lambda^I(\Theta)}{\Lambda_0^2} = \cos\Theta \sin\Theta = \frac{1}{2} \sin(2\Theta), \quad \alpha(\Theta) \sim f_{U(1)}(\theta_e) \frac{1}{2} \sin(2\Theta) $$

This behavior is distinct from that of $G$ or $\hbar$:

* The coupling vanishes in both asymptotic limits $\Theta \to 0$ and $\Theta \to \pi/2$, because either the carrier medium (stiffness) or the fluctuation capacity (coherence) disappears.

* The interaction strength is maximised in the balanced regime $\Theta \approx \pi/4$, suggesting that rich gauge structure and physical complexity are favoured where stiffness and coherence are comparable.

These relations define the leading-order $\Lambda$-grammar: a single angular variable $\Theta$ controls the hierarchy of scales, linking the causal tilt $\chi$, the quantum of action $\hbar$, the strength of gravity $G$, and the intensity of gauge interactions $\alpha$. The resulting unified angular map is summarized in the table Sec. 3.5.1.

### 3.4.4 Numerical spinor closure: the pass-count $N_{pass} \simeq 2$

*Motivation.* Section 3.4.4 discusses fermionic motifs and the spinor nature of 1/2 representations (double covering, $4\pi$ closure). Here we add a compact numerical closure check: the electron loop estimate requires approximately two passes to return to the same internal state, as expected for a spinor.

#### 3.4.4.1 Definition

Let $N_{loop}^{(Compton)}$ be the metric (line-like) estimate of the electron loop count, and let $N_{loop}^{(Causal-corr)}$ be the causal-depth estimate corrected by the geometric reconciliation factor (Appendix B). Define the pass-count

$$ N_{pass} \equiv \frac{N_{loop}^{(Compton)}}{N_{loop}^{(Causal-corr)}} $$

#### 3.4.4.2 Result (closure check)

Evaluating both estimators on our sheet (taking $\alpha_G \simeq 1$ as in the audit naturalness assumption) yields

$$ N_{pass} \approx 2.0 \quad (\text{numerically} \sim 2.03 \text{ in Appendix E}). $$

*Interpretation.* A scalar loop would satisfy $N_{pass} \simeq 1$ (single $2\pi$ closure). The MCP loop instead requires (within percent accuracy) a double winding to return to the same internal state, consistent with spin-1/2 topology (a $4\pi$ closure condition). Thus, the spinor character is not postulated: it is encoded in the closure structure of the $\Lambda$-tension loop itself.
## 3.5 Unified Angular Map of Constants

We now summarize how the observable constants emerge from the interplay between the invariant tension modulus $\Lambda_0$, the global phase $\Theta$, and the internal axes.

### 3.5.1 Observable constants

<table><caption>Table 5: Summary Table: {$c, \hbar, \alpha, \alpha_w, \alpha_s, G, \Lambda_{GR}$} as Functions of $\Theta$</caption><thead><tr><th>Symbol</th><th>Angular sector</th><th>Dimensional carrier (Θ-independent)</th><th>Angular factor f(Θ)</th><th>Λ-framework expression (schematic)</th></tr></thead><tbody><tr><td>c</td><td>Global phase axis d<sub>5</sub> = Θ; structural vs informational tilt</td><td>l<sub>0</sub>/t<sub>0</sub> (base propagation scale of the RTQ network)</td><td>χ(Θ) = tan Θ</td><td>c(Θ) ~ (l<sub>0</sub>/t<sub>0</sub>) χ(Θ)</td></tr><tr><td>ħ</td><td>Tension modulus Λ<sub>0</sub>; informational component along global phase</td><td>Λ<sub>0</sub> l<sub>0</sub> t<sub>0</sub> (minimal RTQ action cell)</td><td>sin Θ</td><td>ħ(Θ) ~ (Λ<sub>0</sub> l<sub>0</sub> t<sub>0</sub>) sin Θ</td></tr><tr><td>G</td><td>Metric sector; global phase Θ and coarse-grained connectivity ⟨ρ⟩</td><td>l<sub>0</sub><sup>2</sup>/Λ<sub>0</sub>⟨ρ⟩ (*)</td><td>(cos Θ)<sup>-1</sup></td><td>G(Θ) ~ <sup>1</sup>/<sub>8π</sub> α<sub>G</sub> <sup>l<sub>0</sub><sup>2</sup></sup>/<sub>Λ<sub>0</sub>⟨ρ⟩</sub> (cos Θ)<sup>-1</sup> ∝ <sup>1</sup>/<sub>⟨ρ⟩</sub> Λ<sub>0</sub> cos Θ</td></tr><tr><td>α<sub>w</sub></td><td>SU(2) / weak axis d<sub>2</sub>; electroweak mixing angle θ<sub>W</sub>; global phase Θ</td><td>M<sub>W</sub><sup>-2</sup> (Fermi-scale carrier in G<sub>F</sub>)</td><td>f(θ<sub>W</sub>, Θ)</td><td>G<sub>F</sub> = π/√2 α<sub>w</sub>(Θ) M<sub>W</sub><sup>-2</sup></td></tr><tr><td>α<sub>s</sub></td><td>SU(3) axis d<sub>3</sub>; QCD sector; renormalization scale Q</td><td>1 (dimensionless; hadronic scales sit in the carrier)</td><td>g(θ<sub>d<sub>3</sub></sub>, Θ; Q<sup>2</sup>)</td><td>α<sub>s</sub>(Q<sup>2</sup>, Θ) ~ g(θ<sub>d<sub>3</sub></sub>, Θ; Q<sup>2</sup>)</td></tr><tr><td>α<sub>G</sub></td><td>Metric / gravitational response (rigidity channel)</td><td>1 (purely dimensionless alignment)</td><td>1</td><td>α<sub>G</sub> ~ O(1) (naturalness; no fine-tuning), taken Θ-invariant in the audit; any Θ-reading is model-level.</td></tr><tr><td>Λ<sub>GR</sub></td><td>Large-scale average of Λ<sup>P</sup> along emergent metric directions</td><td>Sheet-specific curvature-density scale (e.g. 1/l<sub>P</sub><sup>2</sup> in our sheet)</td><td>α cos Θ</td><td>Λ<sub>GR</sub>(Θ) ~ ⟨Λ<sup>P</sup>⟩<sub>macro</sub> ∝ Λ<sub>0</sub> cos Θ</td></tr></tbody><tfoot><tr><td colspan="5">(*) The gravitational carrier can be equivalently written, after SI calibration on a given sheet, as c<sup>3</sup>/⟨ħρ⟩; the model-level angular dependence of G is captured here by the explicit (cos Θ)<sup>-1</sup> factor (Sec. 3.4).</td></tr></tfoot></table>

## Reading the map

### Stiffness-dominated limit $Θ \to 0$

The network is frozen: $\Lambda^P \to \Lambda_0$, $\Lambda^I \to 0$.

One finds $c(Θ) \to 0$, $ħ(Θ) \to 0$, $α(Θ) \to 0$.

Gravity becomes weak in the sense that the metric is maximally rigid: the susceptibility $G(Θ) \propto 1/(\Lambda^P)$ is minimal.

### Information-dominated limit $Θ \to \pi/2$

The metric sector collapses: $\Lambda^P \to 0$, $\Lambda^I \to \Lambda_0$.

The quantum of action saturates, $ħ(Θ) \to \Lambda_0 l_0 t_0$, while the effective $c(Θ)$ and $G(Θ)$ diverge formally, signalling the breakdown of any local metric description in favour of non-local connectivity.
Balanced regime $\Theta \approx \pi/4$

Stiffness and coherence are comparable. The gauge coupling $\alpha(\Theta)$ is maximised (up to the $f_{U(1)}(\theta_e)$ factor), suggesting that the richness of field interactions is favoured in phases where curvature and coherence are in near-balance.

The notation $\alpha_G(\Theta)$ replaces the provisional $\lambda^P$: it denotes the dimensionless gravitational alignment coefficient in the resistivity factorization, and controls the angular part of the gravitational response.

### 3.5.2 Two graphs: audit vs model

To prevent ambiguity between standard physics and the $\Lambda$-hypothesis, it is useful to distinguish two graph objects with different roles.

#### (G1) Constants identity graph – the audit object

It is the *audit graph*, detailed in Appendix A. Its nodes are standard constants and sector quantities ($c, \hbar, G, k_B, e, \alpha, \dots$), and its edges are textbook identities and definitions (e.g. $\alpha = e^2/4\pi\varepsilon_0\hbar c$, Planck-unit relations, cosmological density identities, etc.).

* It contains no $\Theta$-dependence.

* Its only purpose is *representational hygiene and SI traceability*: to ensure that all equivalent parametrizations of a given quantity reduce to the same carrier $S_i$ under declared mappings.

* It is not a physical substrate and contains no RTQ degrees of freedom.

#### (G2) $\Lambda$-projection wiring diagram – the theoretical map

It is the *interpretive wiring diagram* (Fig. 1). It summarizes how the $\Lambda$-framework maps onto the constants appearing in (G1):

* *Input*: the invariant tension $\Lambda_0$ and the global phase $\Theta$, plus the internal axes $d_1, d_2, d_3$.

* *Operation*: angular projection on these axes, splitting the tension between curvature-bearing and coherence-bearing components.

* *Output*: the effective values $\{c(\Theta), \hbar(\Theta), \alpha(\Theta), G(\Theta), \Lambda_{GR}(\Theta), \dots\}$ for a given sheet.

Edges in (G2) are *not* additional textbook identities. They represent:

* geometric factors (e.g. $1/4\pi$, $1/8\pi$),

* angular functions (e.g. $\tan\Theta$, $\cos\Theta\sin\Theta$, mixing angles),

* and carriers (Planck scales, masses, densities, renormalization scales).
### 3.5.3 Reference Table: The Lambda Nomenclature

<table><caption>Table 6: To avoid ambiguity between internal tension components, effective couplings, and standard notation, the following table summarizes the symbols used throughout the Λ-framework.</caption><thead><tr><th>Notation</th><th>Type / Physical Role</th><th>Key Relations & Scaling</th></tr></thead><tbody><tr><td>Λ<sub>0</sub></td><td>Total invariant tension.<br>Conserved modulus, source of all<br>tension budgets. Fundamental scale<br>of the network.</td><td>Λ<sub>0</sub><sup>2</sup> = (Λ<sup>P</sup>)<sup>2</sup> + (Λ<sup>I</sup>)<sup>2</sup>.</td></tr><tr><td>Λ</td><td>Complex tension field.<br>Local<br>state of the network, Λ = Λ<sup>P</sup> + iΛ<sup>I</sup>.</td><td>Effective action term (schematic):<br>S<sub>eff</sub> ⊃ −<sup>Z(Λ)</sup>2 (∂Λ)<sup>2</sup>.<br>Coherence current: J<sub>μ</sub> = Λ<sup>P</sup> ∂<sub>μ</sub>Λ<sup>I</sup>.</td></tr><tr><td>Λ<sup>P</sup></td><td>Physical tension (real part).<br>Rigidity, inertia, curvature source.</td><td>Projection on the phase axis: Λ<sup>P</sup> = Λ<sub>0</sub> cos Θ.<br>Emergent gravity (schematic): a ≈ −c<sup>2</sup>2 ∇lnΛ<sup>P</sup>.</td></tr><tr><td>Λ<sup>I</sup></td><td>Informational tension<br>(imaginary part).<br>Coherence,<br>order, quantum capacity.</td><td>Projection: Λ<sup>I</sup> = Λ<sub>0</sub> sin Θ.<br>Internal propagation tilt: χ(Θ) ≡ Λ<sup>I</sup>/Λ<sup>P</sup> = tan Θ.</td></tr><tr><td>Λ<sup>P</sup><sub>struct</sub></td><td>Structural tension.<br>Vacuum<br>rigidity; sets the causal cone (hence<br>c) and enters the gravitational<br>stiffness (hence G).</td><td>Formal bound on propagation:<br>v<sub>bound</sub> ∝ (Λ<sup>P</sup><sub>struct</sub>)<sup>-1/2</sup>.<br>In vacuum, Λ<sup>P</sup><sub>struct</sub> = Λ<sub>0</sub> cos Θ is the dominant<br>contribution to Λ<sup>P</sup>.</td></tr><tr><td>Λ<sup>P</sup><sub>manif</sub></td><td>Manifested tension.<br>Propagating<br>modes on the manifold (waves,<br>gauge fields, elastic curvature).</td><td>Decomposition of the physical channel:<br>Λ<sup>P</sup> = Λ<sup>P</sup><sub>struct</sub> + Λ<sup>P</sup><sub>manif</sub>.<br>Contributes to the dynamical Λ-Lagrangian (Sec. 4).</td></tr><tr><td>Λ<sup>P</sup><sub>confined</sub></td><td>Confined tension.<br>Inertial mass<br>stored in closed torsion loops.</td><td>Mass channel (schematic): m ∝ Λ<sup>P</sup><sub>confined</sub> cos(θ<sub>d2</sub>).<br>Inclusion: Λ<sup>P</sup><sub>confined</sub> ⊂ Λ<sup>P</sup><sub>manif</sub>.</td></tr><tr><td>Λ<sub>GR</sub></td><td>Cosmological constant (GR).<br>Effective large-scale curvature<br>density.</td><td>Einstein equations: G<sub>μν</sub> + Λ<sub>GR</sub>g<sub>μν</sub> = 8πGT<sub>μν</sub>.<br>Macroscopic average: Λ<sub>GR</sub> ~ 〈Λ<sup>P</sup>〉<sub>macro</sub>.</td></tr><tr><td>λ</td><td>Lagrange multiplier in the<br>effective action. Enforces the norm<br>of the emergent time vector T<sup>μ</sup>.</td><td>Time/coherence sector: ℓ<sub>T</sub> ⊃ λ(T<sup>μ</sup>T<sub>μ</sub> + 1).<br>Ensures that T<sup>μ</sup> remains time-like and<br>unit-normalized.</td></tr><tr><td>λ<sub>laser</sub></td><td>Wavelength (contextual notation).<br>Standard optical parameter in<br>interferometry tests.</td><td>Induced phase shift (schematic):<br>Δφ ≃ <sup>2πL</sup>2 λ<sub>laser</sub> η<sub>EM</sub> ΔΛ<sup>P</sup>.<br>Used in interferometric constraints of Sec. 6.</td></tr></tbody></table>
### 3.5.4 Back-of-the-envelope closure: Bridging network scales and Planck units

#### 3.5.4.1 Network notation connecting to Planck units

A conceptual distinction must be made between the fundamental network scales and the observed Planck units:

* $(\ell_0, t_0, \Lambda_0)$ are intrinsic scales of the discrete RTQ substrate (the “grain” of the graph). At the fundamental level they are dimensionless adjacency/update scales, which only acquire SI units once a sheet and a calibration are chosen.

* $(\ell_P, t_P, m_P)$ are effective relativistic units defined inside a metric manifold, from the emergent constants $(G, \hbar, c)$ of a given sheet.

They are linked by the angular projection. For example, the measured quantum of action in a given sheet satisfies schematically

$$ \hbar(\Theta) \sim \Lambda_0 \ell_0 t_0 \sin \Theta, $$

so that the observed $\hbar_{obs}$ is the value frozen at our cosmological phase $\Theta_*$.

Similar relations hold for $c(\Theta)$ and $G(\Theta)$: once $\Theta$ is fixed, the combination of $(\Lambda_0, \ell_0, t_0, \langle\rho\rangle)$ that reproduces the observed $(c, \hbar, G)$ defines the corresponding Planck units for that sheet.

We provide a reader-facing closure check. Throughout this manuscript, Planck units are not used as inputs: they are derived inside the emergent metric manifold from the measured $(G, \hbar, c)$ of a given sheet. This is crucial for the audit contract. Nevertheless, once a sheet is fixed (i.e. once $\Theta = \Theta_*$ is fixed), one can invert the gravitational carrier to obtain a transparent order-of-magnitude closure for the microscopic RTQ grain and the invariant tension. This does not add fitted relations; it is a reader-facing consistency check.

#### 3.5.4.2 From the gravitational carrier to the RTQ grain $\ell_0$

In the resistivity factorization, the Newton constant is written as

$$ G(\Theta) = \frac{1}{8\pi} \alpha_G (\cos \Theta)^{-1} c^3 \hbar^{-1} \langle \rho \rangle^{-1}, \qquad (2) $$

with $\alpha_G$ the (dimensionless) gravitational alignment coefficient (taken $\Theta$-invariant in the audit, $\alpha_G \sim \mathcal{O}(1)$ by naturalness) and $\langle\rho\rangle$ the coarse-grained carrier inferred by SI audit to have dimension $[\langle\rho\rangle] = m^{-2}$ (inverse-area carrier). Using the definition of the Planck length,

$$ \ell_P^2 = \frac{\hbar G}{c^3}, $$

Eq. (2) implies

$$ \langle\rho\rangle^{-1} = \frac{8\pi}{\alpha_G} (\cos \Theta_*) \frac{\hbar G}{c^3} = \frac{8\pi}{\alpha_G} (\cos \Theta_*) \ell_P^2. $$

Identifying the carrier scale with an RTQ grain through

$$ \langle\rho\rangle \equiv \ell_0^{-2}, $$

one obtains the compact estimate

$$ \ell_0 = \sqrt{\frac{8\pi \cos \Theta_*}{\alpha_G}} \ell_P = 5.013 \sqrt{\frac{\cos \Theta_*}{\alpha_G}} \ell_P. \qquad (3) $$

Thus, unless the gravitational alignment is pathologically small or large, the RTQ grain is mechanically driven to be within a few Planck lengths on our sheet.
### 3.5.4.3 From $\ell_0$ and $\Theta_*$ to the invariant tension $\Lambda_0$.

Let $\kappa \equiv \ell_0/\ell_P$ denote the sheet-dependent Planck proximity. The coupling-coefficient map yields the simple closure (in SI calibration)

$$ \Lambda_0 \approx \frac{1}{\kappa^2} \frac{\cos \Theta_*}{\sin^2 \Theta_*} F_P, \quad F_P \equiv \frac{c^4}{G}, $$

so that for $\alpha_G \approx 1$ and $\Theta_* \approx 0.36$ rad (the structural value derived from dark-sector fractions in Sec. 5.3.2),

$$ \Lambda_0 = \mathcal{O}(10^{-1}) F_P, \qquad (4) $$

i.e. the invariant tension sits at a fixed fraction of the Planck force.

$$ \Lambda_0 \approx 0.30 F_P \approx 3.7 \times 10^{43} \text{ N.} $$

### 3.5.4.4 Interpretation – why the RTQ grain and tension sit near Planck scales

Equations (3)–(4) are not used anywhere as defining relations; they merely explain why, once the SI carriers and the angular grammar are fixed, the inferred RTQ micro-scales fall numerically close to the Planck domain. This is exactly the type of “closure check” a referee expects when a discrete substrate is claimed to underlie $(G, \hbar, c)$.

## 3.6 Conclusion

With this map, the disparate list of numerical constants is replaced by a unified angular grammar:

* constants are organized as either angular alignment factors or dimensional carriers of a single tension field,

* no constant is “fundamental” in isolation,

* the observed pattern $\{c, \hbar, \alpha, \alpha_w, \alpha_s, G, \Lambda_{GR}, \dots\}$ is interpreted as the IR projection—through a declared coarse-graining map—of a discrete RTQ/Λ substrate with fixed $\Lambda_0$, phase $\Theta_*$, and a finite set of internal axes.

In later sections (4–6), this structure will be confronted with data: any drift in $\Theta$ must induce correlated shifts in $G$, $\alpha$, and $c$, providing a concrete empirical handle on the Λ-grammar.
# 4 Λ-Dynamics: Time, Entropy and Gravity (Theory III)

In the Λ-framework, spacetime emerges as a coarse-grained regime of an RTQ hypergraph with a complex tension field $\Psi = \Lambda_0 e^{i\Theta} = \Lambda^P + i\Lambda^I$, where $(\Lambda^P)^2 + (\Lambda^I)^2 = \Lambda_0^2$. No manifold is assumed microscopically, and metric, curvature, and horizons emerge from the statistics of $\Lambda^P$ and $\Lambda^I$ under local update rules. This framework aligns with the emergent gravity and information-based physics perspectives advocated by [Jacobson, 1995], [Padmanabhan, 2007, 2010], and [Verlinde, 2011], who argue that gravitational phenomena are emergent properties of deeper informational processes rather than fundamental interactions. Their works on thermodynamic interpretations of gravity and entropy resonate with the Λ-framework's vision, where curvature is seen as a thermodynamic / elastic response to the coherence dynamics in the system.

## 4.1 Causal Axis $d_4$ and Temporal Order

Each RTQ carries a local causal orientation along the internal axis $d_4$, $T_{local} \in d_4$, which, under coarse-graining, becomes an “Einstein–Æther-like” unit time-like direction $T^\mu$, as described by [Eling et al., 2004, Jacobson, 2008]. The macroscopic arrow of time is identified with the ordered phase of this orientation field. Define the order parameter:

$$T^\mu := \langle T_{local} \rangle, \quad T^\mu T_\mu = -1.$$

Two phases are distinguished:

* **Disordered**: $\langle T_{local} \rangle = 0 \rightarrow$ no macroscopic arrow, effectively reversible dynamics.

* **Ordered**: $\langle T_{local} \rangle \neq 0 \rightarrow$ directed time, causal asymmetry [Prigogine, 1986, Bergson, 1907, Whitehead, 1929].

Temporal order is modeled as a synchronization transition of local phases $\theta_i$ on $d_4$, analogous to Kuramoto-type coupling [Acebrón et al., 2005, Rodrigues et al., 2016]:

$$\dot{\theta}_i = \omega_i + K \sum_j \sin(\theta_j - \theta_i), \quad K \propto \Lambda^P.$$

Above a critical stiffness, $\Lambda^P > \Lambda_c^P \Leftrightarrow K > K_c \Rightarrow$ global phase locking, and a synchronization front propagates as a logical phase wave with:

$$v_{sync} \lesssim c, v_{sync} \sim \sqrt{K/D},$$

where $D$ plays the role of an informational viscosity. Early cosmology can thus be pictured as growing domains of aligned $T_{local}$ with residual misaligned “temporal defects” (Barbour et al. 2014).

The role of $\Lambda^P$ is then described as the “stiffness of time”:

$$\Lambda^P \downarrow \Rightarrow \text{soft network} \Rightarrow \text{temporal disorder}, \Lambda^P \uparrow \Rightarrow \text{rigid network} \Rightarrow \text{stable arrow}.$$

Quantum superpositions correspond to local desynchronizations of $T_{local}$; decoherence is partial realignment. The quantum–classical transition becomes a transition in temporal coherence rather than a change of fundamental law.

## 4.2 Entropy, Misalignment and Λ-Exchange

Axis $d_4$ is the logical carrier of causal parity: each RTQ has a binary orientation, and coarse-graining yields $T^\mu$ as emergent preferred direction. In this picture, entropy measures misalignment of $T_{local}$ along $d_4$ [Internet Encyclopedia of Philosophy, 2014, Barbour et al., 2014, Ebtekar, 2022]:

$$\frac{dS}{dt} \propto \nabla \cdot (\text{misalignment of } T_{local}).$$
Low $S$: aligned $T_{local}$, strong causal coherence. High $S$: disordered orientations, weak coherence. The arrow is not a consequence of entropy, but the direction along which misalignment statistically spreads.

At the RTQ scale, update rules are assumed CPT-symmetric; reversing the macroscopic arrow would require flipping a macroscopically large fraction of orientations. Irreversibility is thus entropic: synchronized configurations occupy a negligible fraction of state space.

The alignment rate obeys schematically

$$ \partial_t \langle T_{local} \rangle \propto \Lambda^P \nabla_{d_4} (\text{phase alignment}), $$

so large $\Lambda^P$ yields strong causal rigidity, whereas small $\Lambda^P$ yields local reversibility and quantum-like variability.

The two components of $\Lambda$ play complementary roles:

$$ \Lambda^I \text{ propagates coherence, } \Lambda^P \text{ stores its footprint.} $$

Each $\Lambda^I$-driven alignment leaves an irreversible structural residue in $\Lambda^P$; erasing it would require reconfiguring extended regions. The macroscopic field $T^\mu$ is the average of these accumulated imprints. The arrow of time is thus the gradient of “coherence memory”.

## 4.3 Gravity as Curvature from Coherence Loss

Gravity is reinterpreted as the macroscopic response of the network to spatial gradients of structural tension $\Lambda^P$, induced by coherence loss. Curvature is thermodynamic / elastic in origin [Sakharov, 1967, Jacobson, 1995, Visser, 2002, Padmanabhan, 2007, Verlinde, 2011].

Consider coherence along $d_4$: high $\Lambda^I$, minimal $\Lambda^P$. Local decoherence yields

$$ \Delta\Lambda^I < 0 \Rightarrow \Delta\Lambda^P > 0, (\Lambda^P)^2 + (\Lambda^I)^2 = \Lambda_0^2. $$

This induces spatial gradients

$$ \nabla_\mu \Lambda^P(x) \neq 0, $$

which distort adjacency relations and thus the effective metric:

$$ g_{\mu\nu}^{\text{grav}} \sim f(\nabla_\mu \Lambda^P). $$

Regions of strong temporal disorder (large coherence loss) correspond to increased $\Lambda^P$ and curvature wells, echoing Jacobson's derivation of Einstein's equations from $\delta Q = T dS$ on local horizons [Jacobson, 1995]. Uniform $\Lambda^P$ yields flat spacetime; heterogeneity yields curvature, schematically

$$ G_{\mu\nu} \propto \nabla_\mu \Lambda^P \nabla_\nu \Lambda^P, $$

with $G_{\mu\nu}$ the coarse-grained Einstein tensor.

Inertial and gravitational masses probe the same structural quantity $\Lambda^P_{confined}$ (cf. Part 3). In $\Lambda$-language,

$$ m_i = m_g = f(\Lambda^P, \nabla \Lambda^P), $$

where $m_i$ is resistance to changing the internal torsion of confined $\Lambda^P$, and $m_g$ is response of the same confined torsion to external $\nabla \Lambda^P$. The Equivalence Principle emerges because all motifs live in and respond to the same $\Lambda^P$ field. Microscopically, decoherence increases $\Lambda^P$ and produces an effective “coherence contraction” (shorter geodesic distances); coarse-graining yields a smooth curvature field.
In extreme regimes, $\Lambda$-dynamics reproduces horizon thermodynamics and holography. Bulk saturation corresponds to

$$ \Lambda_{\text{bulk}}^{P} \rightarrow \Lambda_{0}, \quad \Lambda_{\text{bulk}}^{I} \rightarrow 0, \quad \Theta \rightarrow 0, $$

a maximally rigid, informationally frozen interior with no coherent $\Lambda^I$ propagation. Boundary RTQs retain $\Lambda^I$,

$$ \Lambda_{\text{surface}}^{I} \neq 0, \quad \Lambda_{\text{bulk}}^{I} = 0, $$

so all effective degrees of freedom become surface-encoded, recovering an area law [Bekenstein, 1973, Hawking, 1975] and the holographic principle ['t Hooft, 1993, Susskind, 1995]. Hawking-like radiation arises as surface-driven relaxation of $\Lambda^P$ saturation: stochastic reconfigurations of boundary $\Lambda^I$ under steep $\nabla\Lambda^P$ emit low-amplitude excitations outward. Thermality reflects statistical equilibrium of surface modes; temperature encodes the resistance of $\Lambda_{\text{struct}}^P$ to additional alignment [Hawking, 1975]. The interior stays at $\Theta = 0$ with $\Lambda^I = 0$; evaporation is a boundary phenomenon.

### 4.3.1 Finite-step collapse and the absence of singularities

The horizon picture above becomes mechanically complete once combined with the finite-state hypothesis for the internal phase. As stated in Sec. 2.2.3, $\Theta$ is taken to admit a quantized spectrum

$$ \Theta_n = n \Delta\Theta_{\text{min}}, \quad n = 0, \dots, N_{\text{max}}, \quad 0 \le \Theta \le \frac{\pi}{2}, $$

motivated by finite informational capacity per RTQ. This has a direct implication for gravitational collapse in the $\Lambda$-framework:

* In classical continuum descriptions, “extreme” regimes (horizon approach, curvature blow-up) often appear as asymptotic limits, raising the familiar question of whether singular behavior is a genuine physical endpoint or an artifact of extrapolating a smooth manifold description beyond its domain.

* Here, the strong-curvature interior is identified with bulk saturation ($\Lambda^P \rightarrow \Lambda_0$, $\Lambda^I \rightarrow 0$, $\Theta \rightarrow 0$). With quantized $\Theta$, the approach to this regime is not a continuous drift but a finite cascade of phase steps:

$$ \Theta_n \rightarrow \Theta_{n-1} \rightarrow \dots \rightarrow \Theta_0. $$

Thus $\Theta = 0$ is an attainable state reached after a finite number of admissible updates, not an unreachable limit.

* Once the $\Theta_0$ phase is reached locally, the interior is informationally frozen ($\Lambda^I = 0$) and no further “coherence budget” remains available for additional internal reconfiguration. In this sense, the collapse endpoint is a saturated phase domain rather than a divergent geometric point: the metric description has already reached its maximal stiffness realization compatible with the conserved modulus $\Lambda_0$.

* The remaining physics is therefore pushed to the boundary degrees of freedom, consistent with the area-law behavior discussed above: evaporation and relaxation are surface phenomena mediated by boundary RTQs with $\Lambda^I \neq 0$, while the bulk remains locked in $\Theta = 0$.

In summary, the $\Theta$-quantization postulate upgrades the horizon mechanism from a qualitative picture to a finite-step process: the framework replaces classical singular endpoints by a saturated, finite-state phase domain of the same underlying RTQ network, with dynamics continuing only through boundary coherence modes.

The role of $\Lambda^P$ in generating curvature in this framework is consistent with previous treatments of the entropy of black holes and emergent gravity models [Kleinert, 2005, Bousso, 1999], which propose that spacetime geometry, including gravitational dynamics, can be derived from the flow of relational tension and coherence loss in a background-independent manner.
## 4.4 Curvature–Coherence Complementarity and Time-Crystal Regimes

The complex tension

$$ \Psi = \Lambda^P + i\Lambda^I = \Lambda_0 e^{i\Theta}, \quad \Lambda^P = \Lambda_0 \cos\Theta, \quad \Lambda^I = \Lambda_0 \sin\Theta $$

encompasses Wheeler’s *it-from-bit* perspective [Wheeler, 1990, Padmanabhan, 2007, Visser, 2002]: $\Lambda^I$ encodes coherence, causal alignment and informational capacity, while $\Lambda^P$ encodes rigidity, curvature and the mechanical cost of sustaining alignment. The angle $\Theta$ sets the local balance.

Conservation

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

implies that coherence loss

$$ \Delta\Lambda^I < 0 \Rightarrow \Delta\Lambda^P > 0 $$

strengthens curvature (gravity), whereas curvature relaxation

$$ \Delta\Lambda^P < 0 \Rightarrow \Delta\Lambda^I > 0 $$

increases temporal coherence and weakens gravity. Entropy along $d_4$ measures dispersion of causal coherence:

$$ \Delta S_{d_4} > 0 \Rightarrow \Delta\Lambda^I < 0 \Rightarrow \Delta\Lambda^P > 0. $$

*Gravity becomes the spatial signature of coherence degradation; entropy increase is the temporal signature of the same $\Lambda$-exchange.*

Differentiating with respect to $\Theta$,

$$ \frac{d\Lambda^P}{d\Theta} = -\Lambda_0 \sin\Theta = -\Lambda^I, \quad \frac{d\Lambda^I}{d\Theta} = \Lambda_0 \cos\Theta = \Lambda^P, $$

so each component is the angular derivative of the other. Gradients of $\Lambda^I$ induce changes in $\Lambda^P$ (curvature), and conversely (temporal flow). The field behaves as a phase-balanced oscillator.

The coupled dynamics can be summarized by a conserved “informational–geometric” current

$$ J_{\mu} = \Lambda^{P} \nabla_{\mu} \Lambda^{I}, \quad \nabla^{\mu} J_{\mu} = 0, $$

which plays the role of a Noether-like continuity law for internal phase rotations $\Theta$ (internal U(1): $\Psi \to e^{i\delta\Theta}\Psi$). Solutions range from:

* *Stationary*: $\Lambda^P, \Lambda^I$ constant $\to$ frozen / timeless regimes.

* *Propagating*: gradual exchange $\to$ ordinary time evolution.

* *Resonant*: coherent oscillations $\Lambda^P \leftrightarrow \Lambda^I \to$ time-crystal-like regimes.

Time crystals correspond to phases where the $\Lambda$-field oscillates at fixed modulus [Shapere and Wilczek, 2012, Sacha and Zakrzewski, 2018]:

$$ \Psi(t) = \Lambda_0 e^{i(\Theta_0 + \omega t)}, \quad \Lambda^P(t) = \Lambda_0 \cos(\Theta_0 + \omega t), \quad \Lambda^I(t) = \Lambda_0 \sin(\Theta_0 + \omega t), $$

with equations of motion

$$ \frac{d\Lambda^P}{dt} = -\omega\Lambda^I, \quad \frac{d\Lambda^I}{dt} = +\omega\Lambda^P, $$

describing harmonic exchange at constant $|\Psi|$.
• sufficient rigidity ($\Lambda^P$ large enough to support a stable oscillatory frame, but not fully saturated),

• sufficient coherence ($\Lambda^I$ large enough to sustain phase-locking),

• phase confinement around some $\Theta_0$, with a particularly symmetric regime near

$$ \Theta_0 \approx \frac{\pi}{4}(\Lambda^P \approx \Lambda^I). $$

This defines three qualitative regimes of temporal structure:

<table><caption>Table 7: Qualitative regimes of temporal structure</caption><thead><tr><th>Regime</th><th>Λ<sup>P</sup>/Λ<sup>I</sup></th><th>Temporal behavior</th><th>Analogy</th></tr></thead><tbody><tr><td>Frozen</td><td>Λ<sup>P</sup> ≫ Λ<sup>I</sup></td><td>static / frozen time</td><td>horizon interior, strong gravity</td></tr><tr><td>Crystalline</td><td>Λ<sup>P</sup> ≈ Λ<sup>I</sup></td><td>periodic time (time crystal)</td><td>Floquet crystals, Josephson arrays</td></tr><tr><td>Instantaneous</td><td>Λ<sup>I</sup> ≫ Λ<sup>P</sup></td><td>instantaneous / timeless</td><td>coherence-boundary / pure phase limit</td></tr></tbody></table>

Time-crystal regimes are standing waves of the $\Lambda$-cycle: curvature-coherence exchange in a closed loop. They exemplify that “time” is not an external parameter but a phase-ordered property of the tension field. A transient $\Lambda$-oscillatory phase may have preceded the stabilization of a global arrow in the early Universe, with residual micro-oscillations surviving as subtle coherence modulations of the vacuum.

To avoid any ambiguity, the framework remains atemporal at the microscopic level: no fundamental time parameter is postulated. In the continuum limit, we nevertheless use a covariant 4D description to interface with GR/QFT. The corresponding “clock/arrow” structure is not assumed but encoded by the emergent coherence field $T^\mu$, which becomes time-like only in the ordered phase.

## 4.5 Continuum Interface: Minimal IR Effective Action (IR Template)

### 4.5.1 Effective IR Action (Continuum Bridge)

A potential confusion is best removed upfront. Although the effective action is written as an integral over $d^4x$, this does *not* introduce a fundamental time variable: it is a covariant continuum bookkeeping for the coarse-grained regime. In the $\Lambda$-framework, the operational clock/arrow structure is carried by the emergent causal alignment field $T^\mu$. When $T^\mu$ is stably time-like, it defines an operational proper-time parameter along aligned trajectories; when coherence is lost (disordered $T^\mu$) or becomes purely phase-like, that interpretation breaks down—exactly as expected from the underlying atemporal, discrete update dynamics.

$$ u^{\mu} \equiv \frac{T^{\mu}}{\sqrt{-T_{\nu}T^{\nu}}}. $$

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

Here $\tau$ is a derived, operational parameter (constructed from $T^\mu$), not a primitive variable of the microscopic theory.

### 4.5.2 Minimal Covariant IR Effective Action (Continuum Interface)

Although the $\Lambda$-framework is fundamentally discrete and background-independent, its macroscopic predictions must be confronted with the continuum low-energy language used in gravitation and field theory. We therefore introduce a minimal covariant IR effective action as a *continuum interface*. Crucially, the role of this interface is not to add independent structure, but to encode the *IR universality requirement*: in any ordered, statistically isotropic phase of the RTQ ensemble, the leading two-derivative operator governing small excitations must reduce to a *unique Lorentzian wave operator* (up to an overall normalization), with any Lorentz-violating (LV) corrections parametrically suppressed. This requirement is directly tied to the must-pass propagation checks stated later (Sec. 7.2).
**Box — IR universality requirement: from discrete RTQ operators to a single relativistic cone**
At the template level, the continuum interface is required to exhibit a unique second-order hyperbolic operator in the ordered phase. Denote by $\Delta_{RTQ}$ the (weighted) graph Laplacian associated with local RTQ connectivity. Under statistical isotropy of links at coarse-graining scale $L \gg \ell_0$, one has the standard isotropic limit

$$ \Delta_{RTQ} f = \sum_{j \in \mathcal{N}(i)} W_{ij} (f_j - f_i) \rightarrow -Z_s \nabla^2 f + \mathcal{O}\left(\frac{\ell_0^2}{L^2}\right), $$

while the discrete update-cost term yields a quadratic time-like stiffness $Z_t \partial_t^2$ in the ordered regime (where the operational clock $t$ is reconstructed from $T^\mu$). The resulting leading operator for linearized excitations is therefore

$$ \mathcal{D}_{IR} = Z_t \partial_t^2 - Z_s \nabla^2 = Z_t (\partial_t^2 - c^2 \nabla^2), \quad c^2 \equiv \frac{Z_s}{Z_t}. $$

Any residual Lorentz-violating (LV) terms appear as anisotropic corrections $\delta D_{ij} \partial_i \partial_j$ and higher-derivative operators; in the $\Lambda$-framework they are required to be coarse-graining suppressed by statistical averaging (see Check 2 and Appendix D.2).

Three macroscopic fields capture the coarse-grained state:

* $g_{\mu\nu}$: the emergent metric (statistical orientation structure of the network),

* $T^\mu$: a unit time-like coherence/causal-alignment field (ordered phase only),

* $\Lambda(x)$: the coarse-grained physical tension (with $\Lambda^P, \Lambda^I$ as its complementary components through $\Psi = \Lambda_0 e^{i\Theta}$). In this sense, $\Lambda$ does not “add a new force on top of gravity”: it is the microscopic carrier whose coarse-grained response is encoded geometrically by $g_{\mu\nu}$ in the IR.

**IR universality requirement (unique relativistic operator).** In the ordered phase (stably time-like $T^\mu$) and under statistical isotropy of the coarse-grained neighbourhood, the long-wavelength generator of small excitations has the universal form

$$ \mathcal{D}_{IR} = Z_t \partial_t^2 - Z_s \nabla^2 + \mathcal{O}(\ell_0^2 \nabla^4), \quad c^2 \equiv Z_s/Z_t, $$

i.e. a single d'Alembertian cone in the IR (after rescaling). Any additional direction-dependent quadratic operator would show up as LV dispersion/anisotropy and is therefore constrained by the negative-check suite (Sec. 7.2).

**Quantitative suppression of LV terms (statistical isotropy).** Let $D_{ab}$ denote the coarse-grained spatial diffusion tensor induced by the RTQ neighbourhood, schematically $D_{ab} \propto \sum_{j \in \mathcal{N}(i)} W_{ij} \delta x_{ij}^a \delta x_{ij}^b$. In a statistically isotropic disordered phase, $\langle D_{ab} \rangle = D \delta_{ab}$, while the residual anisotropic part $A_{ab} \equiv D_{ab} - \frac{1}{d} \text{tr}(D) \delta_{ab}$ obeys a central-limit scaling

$$ \frac{\|A\|}{\text{tr}(D)} \sim \frac{1}{\sqrt{N_{\text{eff}}}}, \quad N_{\text{eff}} \sim \left(\frac{L}{\ell_0}\right)^d, $$

for a coarse-graining volume of linear size $L$ containing $N_{\text{eff}}$ independent microscopic contributions. Therefore the LV coefficients entering the effective two-derivative operator are parametrically suppressed as $\mathcal{O}(N_{\text{eff}}^{-1/2})$, and become negligible at laboratory/astrophysical scales. This scaling is made operational in the Lorentz/dispersion checks (Sec. 7.2) and the coarse-graining appendix.
A minimal IR action consistent with these ingredients is

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

Here $\bar{M}_{Pl}$ is the Einstein-frame (background) Planck scale used in the coarse-grained description. In the $\Lambda$-framework it should be viewed as a calibrated parameter fixed by the homogeneous structural stiffness of the present epoch, i.e. by the background values ($\Lambda_0, \Theta_{\text{struct}}$) (and, if needed, by the coarse-grained connectivity scale entering the IR calibration). Local departures from homogeneity are encoded in the scalar sector and in the conformal matter coupling $A(\Lambda)$, and become explicit upon rewriting the theory in a Jordan-like frame (Sec. 4.5.3).

$Z(\Lambda)$ and $V(\Lambda)$ encode the effective stiffness and self-interaction of the tension field, while $S_m$ is the matter action. The universal coupling through the conformally rescaled metric $A^2(\Lambda)g_{\mu\nu}$ enforces a common geometric response of all excitations, providing an IR route to the Equivalence Principle.

The field $T^\mu$ is not “time”; it is a coherence/causal-order parameter. A minimal choice for its dynamics is:

$$\ell_T(g, T) = -\frac{M_T^2}{2} \nabla_\mu T_\nu \nabla^\mu T^\nu + \lambda(T_\mu T^\mu + 1),$$

where $\lambda$ enforces the unit time-like condition when the ordered phase exists. Operational “proper time” is then a constructed parameter along the coherence flow:

$$\text{In the ordered phase, } T_{\mu}T^{\mu} = -1 \Rightarrow u^{\mu} = T^{\mu}, \quad d\tau \equiv -u_{\mu}dx^{\mu},$$

which is meaningful only when $T^\mu$ is stably time-like (ordered regime). In disordered or phase-dominated regimes, the construction degrades exactly as described in Secs. 4.1–4.4.

A compact construction of the coarse-grained tension field $\Lambda(x)$, the coherence current $J^\mu$, and emergent proper time from the RTQ ensemble is given in Appendix B.4.

This EFT sketch is sufficient for: (i) defining the IR degrees of freedom unambiguously, (ii) stating what is meant by “emergent time” in a covariant formalism, and (iii) setting up explicit consistency checks (screening/GR limit, weak-field limit, horizon thermodynamics).

### 4.5.2.1 Angular specialization ($\Lambda \to \Theta$ dictionary)
In the angular realization used throughout this paper, the modulus is fixed ($|\Lambda| = \Lambda_0$), so the scalar sector is most naturally parameterized by the phase $\Theta(x)$ (or equivalently by its real structural projection $\Lambda^P$). We therefore use the change of variables

$$\Lambda^P(x) = \Lambda_0 \cos \Theta(x), \quad \nabla_\mu \Lambda^P = - \Lambda_0 \sin \Theta \nabla_\mu \Theta.$$

With this identification, the $\Lambda$-sector of Eq. (4.5.2) can be rewritten in terms of $\Theta$ with

$$K_{\Theta}(\Theta) \equiv Z(\Lambda^P(\Theta)) \Lambda_0^2 \sin^2 \Theta, \quad V(\Theta) \equiv V(\Lambda^P(\Theta)), \quad A(\Theta) \equiv A(\Lambda^P(\Theta)).$$

This makes explicit that the $\Theta$-field used below is not an additional ingredient, but a reparametrization of the same coarse-grained scalar channel.

A fuller treatment (explicit black-hole solutions; detailed forms for $Z, V, A$; and a derivation route from update rules) is deferred to a dedicated follow-up.

### 4.5.3 Background–perturbation split and screened coupling (from $\Theta_{\text{struct}}$ to $\Theta(x)$)

Using the angular dictionary introduced above, the scalar sector of Sec. 4.5.2 can be rewritten equivalently in terms of the phase field $\Theta(x)$, which we now treat as the fundamental IR scalar variable for the screening discussion.
In the Λ-framework, Θ is a genuine scalar field whose homogeneous component sets the cosmological attractor angle, while local inhomogeneities encode “texture”:

$$ \Theta(x) = \Theta_{\text{struct}}(t) + \delta\Theta(x), \quad |\delta\Theta| \ll 1 \text{ on large scales.} $$

In the Einstein-frame EFT, matter couples universally to the conformally related metric

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

so that the strength of the scalar-mediated fifth force is controlled by the canonical coupling (not by the geometric factor alone). Introducing the canonically normalized field

$$ \frac{d\varphi}{d\Theta} = \sqrt{K_{\Theta}(\Theta)}, $$

the dimensionless coupling is

$$ \alpha(\varphi) \equiv \frac{d \ln A}{d \varphi} = \frac{1}{\sqrt{K_{\Theta}(\Theta)}} \frac{d \ln A}{d \Theta} \equiv \alpha_{\text{phys}}(\Theta). $$

For the MCP angular-stiffness dictionary (Sec. 5.1), one may use

$$ A^2(\Theta) = \frac{\cos \Theta_{\text{struct}}}{\cos \Theta} \implies \frac{d \ln A}{d \Theta} = \frac{1}{2} \tan \Theta, \quad \alpha_{\text{phys}}(\Theta) = \frac{1}{2} \frac{\tan \Theta}{\sqrt{K_{\Theta}(\Theta)}}. $$

Equivalently, writing the equation of motion directly in the original variable Θ gives

$$ \nabla_{\mu} (K_{\Theta}(\Theta) \nabla^{\mu} \Theta) = \frac{dV}{d\Theta} - \frac{d \ln A}{d\Theta} T, $$

where $T \equiv T^\mu_\mu$ is the trace of the matter stress-energy tensor (for non-relativistic matter, $T \simeq -\rho$).

Varying the effective action with respect to $\varphi$ yields the field equation (EOM)

$$ \Box \varphi = \frac{dV}{d\varphi} - \alpha(\varphi) T, $$

where $T \equiv T^\mu_\mu$ is the trace of the matter stress-energy tensor (for non-relativistic matter, $T \simeq -\rho$). Equivalently, the dynamics can be expressed through a density-dependent effective potential

$$ V_{\text{eff}}(\Theta; \rho) = V(\Theta) + \rho \ln A(\Theta), \quad \frac{dV_{\text{eff}}}{d\varphi} = 0 \iff \Box\varphi = 0 \text{ (static limit).} $$

In high-density environments, the curvature of $V_{\text{eff}}$ at its minimum increases, endowing the scalar with a large effective mass

$$ m_{\text{eff}}^2(\rho) \equiv \frac{d^2 V_{\text{eff}}}{d\varphi^2} \bigg|_{\Theta_{\text{min}}(\rho)}, $$

so that gradients are suppressed (“high-density locking”) and the fifth force becomes short-ranged (Yukawa screening). Conversely, in low-density regions (galactic outskirts, voids), $m_{\text{eff}}$ can remain small, allowing $\Theta(x)$ to vary over large scales and enabling environmentally dependent phenomenology while remaining compatible with Solar-System bounds.
### 4.5.4 Relation to dilaton and scalar–tensor fields

In standard scalar–tensor and Kaluza–Klein models, a dilaton field is introduced as a scalar degree of freedom which controls the local scale or effective gravitational coupling. In these frameworks, the dilaton is typically added on top of an already defined spacetime manifold: one specifies a metric $g_{\mu\nu}$ and a scalar $\phi$, and the dynamics follows from an action of the schematic form

$$S_{\text{dil}} = \int d^4 x \sqrt{-g} \left[ \frac{1}{2} (\partial \phi)^2 + V(\phi) + f(\phi) R + \dots \right],$$

where $V(\phi)$ encodes self-interaction and $f(\phi)$ controls the scalar–tensor coupling to curvature [Fujii and Maeda, 2003, Overduin and Wesson, 1997b]. In an appropriate frame (Jordan or Einstein frame), gravity can often be cast as GR plus a scalar, and one may speak of a “dilaton” as a bosonic mode that modulates the effective gravitational coupling and/or the local length scale.

A closer relative of the present framework is provided by dilatonic gravity in lower and higher dimensions, where the scalar field is non-minimally coupled to curvature from the outset. In the $D \to 2$ limit of GR, the Einstein–Hilbert action reduces to a dilaton gravity action in which a scalar $\Phi$ multiplies $R$ [Mann and Ross, 1993]. This structure can be lifted back to 3 + 1 dimensions: in the canonical reduction for dilatonic gravity studied by Scott et al. [2016], the starting point is an action of the form

$$S[g, \Phi] = \int d^4 x \sqrt{-g} (\Phi R + \dots) + S_m[g, \Phi, \psi],$$

where the dilaton $\Phi$ couples directly to the Ricci scalar rather than being a minimally coupled spectator. After an ADM decomposition and canonical reduction, one finds that the quantisation can be carried by the dilaton alone while the background metric remains classical; the dilaton obeys an effective logarithmic Schrödinger equation, with a log–nonlinearity related to the underlying geometric term and to an Everett–Hirschman entropy functional. This viewpoint is further developed in Zloshchastiev [2019] and in the superfluid vacuum interpretation of MOND–like regimes discussed by Scott [2023], where a scalar order parameter and its logarithmic dynamics encode both gravitational response and emergent thermodynamic structure.

By contrast, in the $\Lambda$-framework the invariant tension $\Lambda_0$ and its projections $\Lambda^P, \Lambda^I$ are not added on top of a pre–existing metric: they are the microscopic substrate out of which the effective metric emerges. Coarse–graining the RTQ network yields an emergent $g_{\mu\nu}$ and an effective scalar field $\Lambda(x)$ that encodes long–wavelength fluctuations of the real tension component. In this sense, $\Lambda(x)$ is not an extra field living on top of GR, but the continuum representative of the underlying tension degrees of freedom.

At the microscopic level, the UV RTQ action enforces a fixed modulus $|\Lambda| = \Lambda_0$ together with an internal phase potential $U(\Theta)$ and link terms that penalize phase misalignment. Upon coarse graining, this structure translates into an effective elastic response whose stiffness is proportional to the structural projection $\Lambda_{\text{struct}}^P$, so that the local gravitational rigidity is directly inherited from the underlying tension network. Recall from Sec. 3.2.1 that the real projection of the tension splits as

$$\Lambda^P = \Lambda_{\text{struct}}^P + \Lambda_{\text{manif}}^P, \quad \Lambda_{\text{confined}}^P \subset \Lambda_{\text{manif}}^P,$$

where $\Lambda_{\text{struct}}^P$ sets the sheet–wide structural stiffness and the causal cone, $\Lambda_{\text{manif}}^P$ carries manifested propagating curvature/gauge modes, and $\Lambda_{\text{confined}}^P$ encodes torsion locked in mass motifs. In the gravitational sector, the background Planck scale and effective Newton constant depend only on $\Lambda_{\text{struct}}^P$, while $\Lambda_{\text{manif}}^P$ and $\Lambda_{\text{confined}}^P$ feed the matter and mass channels.

At the level of the IR effective theory used in this paper, we choose to work in an Einstein–frame representation in which the Planck mass is constant and the scalar $\Lambda$ couples conformally to matter (see Sec. 4.5.2):

$$S_{\text{eff}} = \int d^4 x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R - \frac{Z(\Lambda)}{2} \nabla_{\mu} \Lambda \nabla^{\mu} \Lambda - V(\Lambda) + \ell_T(g, T) \right] + S_m(A^2(\Lambda) g_{\mu\nu}, \psi).$$
In this frame the scalar sector can be written as

$$S_{\Lambda} \supset \int d^4 x \sqrt{-g} \left[ -\frac{Z(\Lambda)}{2} \nabla_{\mu} \Lambda \nabla^{\mu} \Lambda - V(\Lambda) \right],$$

with a universal coupling of matter through the conformally rescaled metric $A^2(\Lambda)g_{\mu\nu}$ (Sec. 4.5.2). Small scalar fluctuations around a homogeneous background — which, in the gravitational sector, are dominated by variations of $\Lambda_{\text{struct}}^P$ — then behave as an effective bosonic mode that modulates the local stiffness of spacetime and the effective gravitational coupling.

The underlying MCP interpretation is that this stiffness is controlled by the real structural projection $\Lambda_{\text{struct}}^P$ of the tension field, or equivalently by the internal angle $\Theta$ such that

$$\Lambda_{\text{struct}}^P = \Lambda_0 \cos \Theta,$$

whereas $\Lambda_{\text{manif}}^P$ and $\Lambda_{\text{confined}}^P$ encode respectively propagating and confined excitations. In particular, the effective gravitational coupling is inversely proportional to the structural stiffness, schematically $G_{\text{eff}}^{-1} \propto \Lambda_{\text{struct}}^P$. This can be made manifest by going to a Jordan-like frame in which the Planck mass becomes a function of $\Lambda_{\text{struct}}^P$. Introducing the effective stiffness (or Planck mass squared) as

$$\Phi(\Lambda) \equiv M_{Pl}^2(\Lambda) = \bar{M}_{Pl}^2 F(\Theta) \approx \bar{M}_{Pl}^2 \frac{\Lambda_{\text{struct}}^P}{\Lambda_{\text{struct}}^P},$$

where the last expression holds to first order in fluctuations around a homogeneous FLRW background with $\Lambda_{\text{struct}}^P = \bar{\Lambda}_{\text{struct}}^P$, one may rewrite the gravitational part of the action in a Jordan frame as

$$S_J = \int d^4 x \sqrt{-g} \left[ \frac{1}{2} \Phi(\Lambda) R - \frac{Z_J(\Lambda)}{2} \nabla_{\mu} \Lambda \nabla^{\mu} \Lambda - V_J(\Lambda) + \ell_T(g, T) \right] + S_m[g_{\mu\nu}, \psi],$$

where $Z_J$ and $V_J$ are the Jordan-frame kinetic function and potential. In this representation the $\Lambda$-sector is exactly of dilaton / scalar-tensor type: the effective stiffness field $\Phi(\Lambda) \propto \Lambda_{\text{struct}}^P$ multiplies the Ricci scalar and plays the rôle of a dilaton controlling $G_{\text{eff}}$. The Einstein-frame form $S_{\text{eff}}$ used elsewhere in the paper is obtained by a standard Weyl rescaling $g_{\mu\nu}^{(E)} = \Omega^2(\Lambda) g_{\mu\nu}^{(J)}$ with $\Omega^2(\Lambda) = \Phi(\Lambda)/\bar{M}_{Pl}^2$, which trades the non-minimal coupling $\Phi(\Lambda)R$ for the conformal matter coupling $A^2(\Lambda)g_{\mu\nu}$.

From a structural standpoint, this places the MCP stiffness mode $\Phi(\Lambda)$ in the same broad family as the dilaton fields used in Einstein-Yang-Mills-dilaton theories [Kleihaus and Kunz, 1997, Scott et al., 2016, Scott, 2023]: in all cases a scalar field multiplies $R$ and controls an effective gravitational response. The key ontological difference is that, in conventional scalar-tensor models, $(g_{\mu\nu}, \phi)$ are independent continuum fields, and one can in principle move to a frame where gravity takes the GR form plus a scalar dilaton. In the $\Lambda$-framework, by contrast, $\Lambda^P = \Lambda_{\text{struct}}^P + \Lambda_{\text{manif}}^P$ and $\Lambda^I$ are primitive tension components on the RTQ network, and the metric $g_{\mu\nu}$ is a derived, statistical object. The field $\Phi(\Lambda)$ is not an ad hoc additional degree of freedom, but a coarse-grained representation of the network stiffness encoded in $\Lambda_{\text{struct}}^P$. There is no conformal frame in which $\Lambda$ can be “gauged away”, because the metric itself is a functional of $\Lambda$ and of the network geometry.

This distinction is also reflected in the rôle played by internal axes. As discussed in Sec. 3, the internal axis $d_2$ carries the mass / Higgs channel: mass arises from confined torsion $\Lambda_{\text{confined}}^P$ and its projection on $d_2$, so that a “Higgs-like” degree of freedom can be viewed as an angular mode (orientation of the internal axis) controlling how tension is locked into inertial mass. By contrast, a dilaton-like degree of freedom corresponds to an amplitude / scale mode in the $\Lambda_{\text{struct}}^P$ channel, controlling how much stiffness is available to support curvature and propagation in a given region.

Schematically, in the continuum limit one may distinguish:

* a Higgs-like sector, associated with angular reorientations of confined $\Lambda_{\text{confined}}^P$ along $d_2$ (mass generation by internal alignment),

* a dilaton-like sector, associated with scalar fluctuations $\delta\Lambda_{\text{struct}}^P$ around a background $\Lambda_{\text{struct},0}^P$, which modulate the effective gravitational stiffness and appear as scalar modes in the IR action.
Thus, while the Λ–framework does not posit a fundamental dilaton in the usual sense, its continuum limit naturally gives rise to an effective scalar degree of freedom with dilaton–like properties. In particular, the dependence of the effective Planck stiffness on Λ<sub>struct</sub><sup>P</sup> is “hidden” when one works in the Einstein frame, but becomes explicit in the Jordan frame via Φ(Λ) R. From the standpoint of existing dilaton and scalar–tensor theories, the MCP “dilaton” is simply the continuum stiffness mode of the RTQ lattice: a macroscopic reparametrization of how the underlying tension budget is allocated between structural, propagating, and confined channels.

The Einstein–frame form $S_{eff}$ used elsewhere in the paper is obtained by the Weyl rescaling $g_{\mu\nu}^{(E)} = \Omega^2(\Lambda) g_{\mu\nu}^{(J)}$ with $\Omega^2(\Lambda) = \Phi(\Lambda)/\bar{M}_{Pl}^2$. As a result, matter becomes universally coupled to the conformally related metric $\tilde{g}_{\mu\nu} = A^2(\Lambda) g_{\mu\nu}^{(E)}$ with $A^2(\Lambda) = \Omega^{-2}(\Lambda) = \bar{M}_{Pl}^2/\Phi(\Lambda)$: in the Einstein frame $M_{Pl}$ is fixed in the gravitational action, and departures from GR are encoded in the scalar-mediated channel through $A(\Lambda)$ rather than through a varying Planck mass.

## 4.6 Λ-framework construction pipeline (summary and cross-links)

For the benefit of the reader and the referee, we collect here the logical construction pipeline of the Λ-framework, from the RTQ substrate to cosmology and empirical tests.

Each step is specified by: (i) its input assumptions, (ii) its output structures, and (iii) where it is implemented in the article and appendices. This makes explicit that no additional hidden hypotheses are introduced between the axioms and the phenomenological predictions.

The pipeline can be summarized in eight steps:

1. RTQ hypergraph and spectral dimension

2. Constants graph and resistivity grammar

3. Dimensional audit and selection of $d = 3$

4. Minimal IR effective action

5. Angular cosmology

6. Environment-dependent Hubble rate and Λ-texture

7. Empirical tiers and falsifiability checks

We now describe each step and its formal location.

### 4.6.1 Step 1: RTQ substrate and spectral dimension

#### Input

Axioms of Sec. 2:

* background-free network of RTQs (hypergraph substrate, no a priori manifold, metric or coordinates),

* complex tension field $\Lambda = \Lambda^P + i\Lambda^I$ with fixed modulus $\Lambda_0$,

* internal axes $d_1, \dots, d_5$ organizing interaction channels and global phase.

#### Construction

The RTQ substrate is represented as an oriented hypergraph $\mathcal{H} = (V, E)$ with adjacency matrix or Laplacian $\Delta$. We treat the effective spatial dimensionality as a spectral dimension $d_s$, defined from the scaling of the average return probability $P(\tau)$ of a diffusion process on the RTQ hypergraph.

A diffusion process (random walk or heat kernel) is defined on $\mathcal{H}$; its return probability $P(\tau)$ is used to extract the spectral dimension

$$d_s(\tau) = -2 \frac{d \ln P(\tau)}{d \ln \tau}.$$
On intermediate scales (well above the Planck grain, well below cosmological curvature), $d_s(\tau)$ exhibits a plateau

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

interpreted as an effective spatial dimension for the emergent continuum description.

### Output

A measured (not assumed) effective spatial dimension $d_s \approx 3$ on the RTQ substrate.

A coarse-graining map from hypergraph data to local coordinate charts $x^\mu$ and metric $g_{\mu\nu}$ in a $d_s+1$-dimensional spacetime.

### Location

Axioms: Secs. 2.1–2.3.

RTQ hypergraph and diffusion operator: a compact construction of $d_s(\tau)$ and the associated plateau $d_s \approx 3$ is given in Appendices B.2–B.3.

This step is deliberately “data-driven”: the dimension is inferred from diffusion on the network rather than postulated.

## 4.6.2 Step 2: constants graph and resistivity grammar

### Input

* Constants dataset and graph introduced in Secs. 1.2–1.3.

* Axiomatic $\Lambda$-structure of Sec. 2 and projection grammar of Sec. 3

### Construction

Build the constants graph: nodes are standard constants $\{c, \hbar, G, k_B, e, \alpha, \alpha_s, m_W, \Lambda_{GR}, \dots\}$; edges are textbook relations (Planck-unit definitions, expressions of $\alpha$ in terms of $e, \hbar, c, \varepsilon_0$, cosmological identities, etc.).

Impose the audit rules (Appendix A):

* only standard identities and definitions are allowed (no fitted “numerology”),

* explicit SI dimensional tracking for all carriers,

* invariance under equivalent parametrizations (e.g., $\alpha$ vs. $e$, different normalizations of couplings),

* exclusion of ambiguous or representation-dependent carriers (negative controls).

Under these constraints, reorganize the “final determinants” of the four interaction sectors into the three-layer form

$$C_i = \mathcal{G}_i \times \mathcal{A}_i \times \mathcal{S}_i,$$

where $\mathcal{G}_i$ is purely geometric, $\mathcal{A}_i$ is a dimensionless alignment/resistivity factor, and $\mathcal{S}_i$ is a dimensional carrier.

### Output

The formal resistivity table (Table 1), with

* EM, weak, strong and gravitational sectors all written in the same $\mathcal{G} \times \mathcal{A} \times \mathcal{S}$ structure,

* identification of the gravitational carrier

$$\mathcal{S}_G = c^3 \hbar^{-1} \langle \rho \rangle^{-1},$$

where $\langle \rho \rangle$ is an effective node-density carrier,

* unit-declared carriers for all sectors (Appendix A.3).
A compact “constants grammar” that interprets the usual constants as angular projections and carriers of a single tension field, rather than as independent primitives.

### Location

Definitions and Table 1: Sec. 1.3 and Secs. 3.1–3.5.

Audit details and negative controls: Appendices A.1–A.4.

### 4.6.3 Step 3: dimensional audit and selection of d = 3

This step connects the RTQ substrate (Steps 1) and the constants graph (Step 2) by requiring that audited carriers remain well-defined when one allows the number of spatial dimensions d to vary formally.

#### Input

* Effective spectral dimension $d_s \simeq 3$ from diffusion (Step 1).

* Constants graph and resistivity decomposition, with explicitly audited carriers (Step 2, Appendix A.3).

* Standard dimensional scaling of the gravitational and weak couplings in d spatial dimensions.

#### Construction

##### Gravitational carrier

Start from the gravitational row of Table 1, with carrier

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

This yields

$$[S_G] = kg^{-1} m^3 s^{-2}.$$

Requiring $[S_G]$ to match the d-dimensional gravitational constant $[G_d]$ selects a unique spatial dimensionality. With the usual d-dependent scaling of $G_d$, equality is only possible for

$$d = 3.$$

##### Weak-sector carrier

In d spatial dimensions, the Fermi coupling has dimension

$$[G_F^{(d)}] = kg^{1-d} m^{2-2d} s^{2d-2},$$

so that the associated carrier

$$S_W^{(d)} = [G_F^{(d)}]^{1/2}$$

carries

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

This coincides with the audited weak carrier in Appendix A.3 only for $d = 3$. For $d \neq 3$, one cannot simultaneously keep $G_F$ in its standard form and preserve the three-layer decomposition with the same carrier interpretation.

##### SM-like renormalizability

In d spatial dimensions, the canonical mass dimensions of gauge fields and scalars imply that:

* gauge couplings $g$,
* Yukawa couplings $y$,

* quartic scalar couplings $\lambda$

are all dimensionless (in mass units) only for $d = 3$. For $d \neq 3$, at least one of these couplings acquires a non-zero mass dimension, violating the standard renormalizability pattern of the SM and spoiling the clean separation between angular factors $A_i$ and carriers $S_i$.

### Output

A dimensional consistency constraint: the simultaneous validity of

* the audited gravitational carrier $S_G$,

* the audited weak carrier $S_W$,

* and SM-like dimensionless gauge/Yukawa/quartic couplings

is only possible for $d = 3$.

Combined with the spectral plateau $d_s \simeq 3$ from Steps 1, this singles out three spatial dimensions in two independent ways: via diffusion on the substrate and via dimensional consistency of the constants grammar.

A compact summary is:

Combining the spectral plateau $d_s \simeq 3$ with the dimensionality constraints on the audited carriers and SM-like couplings selects $d = 3$ as the only spatial dimensionality compatible with both the constants graph and the RTQ substrate.

### Location

Gravitational and weak-sector carriers: Table 1, Secs. 3.3–3.4, Appendix A.3.

Dimensional analysis and $d$-selection: Appendix B.5 (explicit $[G_d]$, $[S_W^{(d)}]$, and mass dimensions of gauge/Yukawa/scalar couplings).

### 4.6.4 Step 4: minimal IR effective action

#### Input

Axioms and internal structure (Sec. 2).

Dimensional selection $d = 3$ (Step 3).

Continuum interface requirements (compatibility with GR/QFT tests).

#### Construction

Introduce continuum fields:

* emergent metric $g_{\mu\nu}$ as coarse-grained orientation structure,

* coherence/causal-alignment field $T^\mu$,

* coarse-grained tension field $\Lambda(x)$ (real scalar encoding $\Lambda^P$, with $\Lambda^I$ captured via phase $\Theta$ and current $J_\mu$).

Write the minimal covariant IR effective action

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

with $\ell_T$ enforcing $T_\mu T^\mu = -1$ in the ordered phase and $A(\Lambda)$ implementing universal coupling.

Define operational proper time only in the ordered phase,

$$u^\mu \equiv T^\mu, d\tau \equiv -u_\mu dx^\mu,$$
and explicitly state that this is a *derived* object, not a microscopic parameter.

**Output**

A continuum bridge compatible with standard GR/QFT tests while keeping the microscopic theory atemporal and discrete.

A well-defined IR template for deriving cosmological and gravitational predictions, used in Sec. 5 and in the test architecture of Sec. 7.

**Location**

Conceptual bridge: Sec. 4.5.1.

Minimal IR action and $T^\mu$ sector: Sec. 4.5.2.

This step is explicitly presented as an *effective* description, not as a fully derived action from RTQ update rules.

### 4.6.5 Step 5: angular cosmology (dual sheets and dark sector)

**Input**

Complex tension field $\Lambda = \Lambda^P + i\Lambda^I = \Lambda_0 e^{i\Theta}$ (Sec. 2–4).

IR effective description (Step 4).

Dimension $d = 3$ fixed by Step 3.

**Construction**

Interpret cosmology as an angular organization of $\Lambda_0$ in the $(\Lambda^P, \Lambda^I)$-plane, with:

* metric sheet $M^+$ (our Universe),

* mirror sheet $M^-$ at phase $\Theta + \pi/2$,

* coherence boundary $N$ in the limit $\Theta \to \pi/2$.

Define the structural angle $\Theta_{struct}$ governing the split between:

* dark energy (sheet imbalance / curvature fraction),

* dark matter (gravitational shadow of $M^-$),

* visible matter (small coupling $\epsilon$ in the Poisson equation).

Show (in Sec. 5.3.2) that the observed fractions $\rho_{DE} \sim 69\%$, $\rho_{DM} \sim 26\%$, $\rho_{vis} \sim 5\%$ can be reproduced with a single structural angle $\Theta_{struct} \approx 0.36 \text{ rad}$ and one angular coupling parameter $\epsilon$.

**Output**

Interpretation of dark energy as inter-sheet curvature-coherence imbalance.

Interpretation of dark matter as the gravitational effect of mirror-sheet curvature.

A parameter-light angular description of the dark sector (one angle + one coupling) consistent with current fractions.

**Location**

Angular cosmology and sheets: Sec. 5.3.2.

Dark energy and $\Theta(t)$: Sec. 5.2.

Dark matter and structural angle fit: Sec. 5.3.
### 4.6.6 Step 6: Hubble tension and Λ-texture

#### Input

Angular cosmology (Step 5).

Spectral/structural heterogeneity of the RTQ network: spatial variability of node density $n(x)$ and phase $\Theta(x)$.

#### Construction

Relate the local Hubble rate to the local curvature fraction,

$$H_0(x, t) \propto \sqrt{\Lambda_P^{(+)}(x, t)} = \sqrt{\Lambda_0 \cos \Theta(x, t)}.$$

Express small deviations around $\Theta_{struct}$ as

$$\frac{\delta H_0}{H_0} \approx -\frac{1}{2} \tan \Theta_{struct} \delta\Theta,$$

so that modest spatial variations of $\Theta(x)$ induce few–10% variations in the inferred $H_0$, compatible with the local vs. CMB discrepancy.

Parametrize directional dependence $H_0(\Omega)$ via a coherence index $C(\Omega) = \cos \Theta(\Omega)$ linked to large-scale node density $n(\Omega)$, providing a natural source for reported anisotropies.

#### Output

Reinterpretation of the $H_0$ tension as a probe of Λ-texture (heterogeneous $\Theta(x), n(x)$), not necessarily as a breakdown of ΛCDM.

Concrete handle for correlating Hubble anisotropies with large-scale structure surveys.

#### Location

Local vs. CMB $H_0$ split: Sec. 5.4.1.

Directional variability: Sec. 5.4.2.

### 4.6.7 Step 7: empirical tiers and falsifiability architecture

#### Input

Full Λ-framework up to the cosmological sector (Steps 0–6).

Need for a *transparent, independently auditable* separation between:

* internal consistency checks,

* parameter constraints,

* genuine falsification channels.

#### Construction

Organize tests into empirical tiers (detailed in Sec. 7 and Appendices C–E):

* Tier A: negative consistency checks (no-dispersion bounds on $v_{GW}$, constancy of dimensionless ratios, standard GR and QFT limits).

* Tier B: structural compatibility tests (e.g., dark-sector fractions, qualitative Hubble tension behavior).

* Tier C: positive falsification channels and “Λ-signature” searches (correlated drifts of constants, anisotropic Hubble maps consistent with Λ-texture, high-coherence regimes).

For each check, specify:

* which step(s) of the pipeline it constrains or could falsify,
* which additional parameters (if any) are involved,

* whether it is already constrained by current data or requires future observations.

**Output**

An explicit mapping "assumption → prediction → test", avoiding the common pitfall where speculative frameworks only reproduce existing phenomena without clear falsifiability criteria.

A roadmap for future work: which parts of the Λ-framework are robust under existing data, and which are open to being ruled out.

**Location**

Narrative description and list of checks: Sec. 7.

Detailed formulae and data interfaces: Appendices C–E.

**4.6.8 Logical status of the pipeline**

The eight steps above define the *logical backbone* of the paper:

* Steps 1 (RTQ substrate and spectral dimension) and Step 2 (constants graph) are essentially structural and data-driven, constrained by diffusion properties on the network and by the audited constants dataset.

* Step 3 (dimensional audit) is a sharp consistency condition: if any alternative $d \neq 3$ were compatible with both the carriers and SM-like renormalizability, the framework would have to be generalized accordingly.

* Step 4 (IR effective action) is explicitly labelled as a minimal continuum interface, not as a claim of fully derived dynamics.

* Steps 5-6 (angular cosmology and Hubble tension) produce phenomenological predictions whose status is clearly separated: dark-sector fractions (structural angle), Hubble tension (Λ-texture), anisotropies (directional $\Theta(\Omega)$).

* Step 8 (empirical tiers) codifies the falsifiability architecture.

In particular, the combination of:

* the measured spectral plateau $d_s \simeq 3$,

* the dimensional audit of carriers and couplings selecting $d=3$, and

* the ability to express $\{c, \hbar, G, \alpha, \alpha_w, \alpha_s, \Lambda_{GR}\}$ as angular projections and carriers of a single invariant $\Lambda_0$,

constitutes the *minimal validation core* of the Λ-grammar. If any of these three pillars is invalidated by data or by a more careful dimensional audit, the framework in its present form is ruled out
# 5 Angular Cosmology: Dual Sheets and the Dark Sector (Methods)

With the complex tension field $\Lambda(\Theta) = \Lambda^P(\Theta) + i\Lambda^I(\Theta)$ introduced in Sec. 4, cosmology is treated as the large-scale angular organization of a single modulus $\Lambda_0$. The real part $\Lambda^P$ encodes curvature/inertia, the imaginary part $\Lambda^I$ encodes coherence/information. What is usually grouped under the “dark sector”, together with the $H_0$ tension, is then read as the way this single tension is partitioned between angular sectors, rather than as evidence for extra fluids or fields.

In this section we introduce the angular architecture of the Universe — metric sheet $M^+$, mirror sheet $M^-$ and coherence boundary $N$— and show how dark energy, dark matter and $H_0$ variability arise as different projections of the same $\Lambda$-structure.

## 5.1 The Universe as an Angular Projection of $\Lambda$

### 5.1.1 Metric and mirror sheets as angular sectors

In the $\Lambda$-framework, a “sheet” is not an extra spacetime but an angular sector of the RTQ network in the $(\Lambda^P, \Lambda^I)$ plane. We write

$$ \Lambda(\Theta) = \Lambda^P(\Theta) + i\Lambda^I(\Theta), \quad \Lambda^P(\Theta) = \Lambda_0 \cos \Theta, \Lambda^I(\Theta) = \Lambda_0 \sin \Theta, $$

and define the internal tilt

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

which controls the formal propagation bound; the physical light speed in our sheet corresponds to $\chi(\Theta_*)$ at a fixed structural angle $\Theta_*$ (Sec. 2, Sec. 4).

We distinguish two phase-conjugate sectors:

* $M^+$ (metric / physical sheet): curvature-dominated regime with $0 < \Theta = \Theta_{\text{struct}} < \frac{\pi}{2}$ and $\Lambda^P > \Lambda^I$.

This is the sector where we identify the observed metric, ordinary matter and light. Effective masses and temporal order are measured along the internal mass axis $d_2$ and causal axis $d_4$ (Sec. 4).

* $M^-$ (mirror / coherence-shifted sheet): conjugate sector at angle $\Theta + \frac{\pi}{2}$, where the same modulus $\Lambda_0$ is projected onto a rotated internal frame and curvature/coherence swap roles in the $\Lambda$-plane.

Observers in $M^+$ and $M^-$ each use their own ($d_2, d_4$) axes and see positive local inertia and forward time. However, when fields are re-expressed in the coordinates of the opposite sheet, effective mass polarities and stress-energy signs can flip, in line with bimetric and mirror-matter constructions [Boyle et al., 2018, Foot, 2004, 2014, Ciarcelluti, 2009, Yin et al., 2020].

At cosmological scales, a nearly homogeneous structural angle $\Theta_{\text{struct}}$ describes the large-scale “composition angle” of the Universe, while a local field $\Theta(x, t)$ encodes spatial and temporal modulation:

$$ (\Lambda^P, \Lambda^I)(x, t) = \Lambda_0(\cos \Theta(x, t), \sin \Theta(x, t)). $$

Folds or kinks of $\Theta(x, t)$ behave as phase solitons carrying bounded curvature/coherence distortions, with Sine-Gordon-like dynamics, but no extra fields beyond $\Lambda$ are introduced.

### 5.1.2 $\pi/2$ pairing and resonant double-sheet dynamics

The duality between $M^+$ and $M^-$ is implemented as a rigid quarter-turn in the $\Lambda$-plane:

$$ (\Lambda^P, \Lambda^I)_{M^-} = (\Lambda^I, -\Lambda^P)_{M^+}, \quad \Lambda(\Theta + \frac{\pi}{2}) = i\Lambda(\Theta). $$
At the coarse-grained level the pair behaves as a phase-conjugate oscillator,

$$ \frac{d\Lambda^P}{dt} = -\omega \Lambda^I, \quad \frac{d\Lambda^I}{dt} = +\omega \Lambda^P, $$

with $\omega$ an effective exchange frequency set by the underlying RTQ update rules (Sec. 3). This keeps the modulus $\Lambda_0$ fixed while allowing curvature and coherence to exchange continuously.

Other relative phase shifts either damp the oscillation or generate beats and secular drifts. A simple $\pi$ rotation,

$$ (\Lambda^P, \Lambda^I) \rightarrow (-\Lambda^P, -\Lambda^I), $$

just inverts the field and does not support a robust exchange. Only the $\pi/2$ pairing maintains permanent quadrature, constant amplitude and minimal entropy production in the coupled evolution of the two sheets [Hassan and Rosen, 2012, Mishra et al., 2023, Brizuela et al., 2024].

Cosmologically, each stable pair $(\Theta, \Theta + \frac{\pi}{2})$ defines a *bimetric resonance*: $M^+$ and $M^-$ exchange curvature and coherence through $\Lambda^I$-mediated phase dynamics without requiring any local negative inertial mass ($m < 0$) or associated runaway instabilities; the apparent sign flips arise from cross-sheet re-expression in an $M^+$ bookkeeping (Table 8), while the total tension $\Lambda_0$ remains conserved.

The constant quadrature relation prevents runaway behavior: curvature increases in one sheet correspond to coherence-weighted adjustments in the conjugate sheet, maintaining global equilibrium. This stability mechanism is structurally comparable to the steady angular coupling observed in earlier bimetric cosmological constructions like the Janus model [Petit and D'Agostini, 2014, Petit et al., 2019], but arises here directly from the single-field phase constraint without invoking multiple independent geometric degrees of freedom.

*Clarification.* Although the phenomenology of a conjugate mass sector is reminiscent of earlier bimetric proposals, the present construction does not adopt a Janus-like modification of the Einstein field equations with two fundamental metrics coupled by a fixed sign-flip operator. Here $M^+$ and $M^-$ are $\pi/2$-conjugate angular sectors of a single tension field $\Lambda$, and any apparent “negative” sign arises only when re-expressing one sector in the bookkeeping coordinates of the other (cross-projection), not from locally negative inertial matter ($m < 0$) in our sheet.

*Ontological distinction from bimetric gravity.* While the phenomenology involves two coupled sectors, the construction is constrained: $M^+$ and $M^-$ are $\pi/2$-conjugate angular projections of a single tension field $\Lambda$ at fixed modulus $\Lambda_0$, rather than two independent dynamical metrics. As a result, the effective two-sector description carries fewer free propagating degrees of freedom than generic bimetric gravity. A dedicated EFT analysis is still required to establish the absence of ghost/instability modes in any specific covariant completion.

In the angular cosmology that follows, dark energy, dark matter and $H_0$ variability are read as different projections of how $\Lambda_0$ is split between $\Lambda^P$ and $\Lambda^I$ across this conjugate pair, rather than as evidence for independent dark fluids.

### 5.1.3 Informational limit $N(\Theta \rightarrow \pi/2)$ as coherence boundary

We denote by $N$ the coherence-dominated boundary phase of the $\Lambda$-field, obtained in the angular limit

$$ \Theta \rightarrow \frac{\pi}{2}, \quad \Lambda^P \rightarrow 0, \quad \Lambda^I \rightarrow \Lambda_0. $$

In this regime, metric curvature vanishes and the usual spacetime structure becomes degenerate: the notion of spatial distance controlled by $\Lambda^P$ ceases to be operational, and only the topological organization of coherence survives. Formally, an effective propagation speed

$$ \chi(\Theta) \propto \tan \Theta $$
would diverge at $\Theta = \pi/2$, not because signals outrun light, but because the metric collapses: the ratio “distance / time” loses meaning once “distance” is no longer a geometric invariant. i.e. the divergence is a coordinate/operational one (metric degeneracy), not a physical superluminal signal speed.

Adjacency in $N$ is therefore defined by *phase similarity* rather than spatial proximity. Two nodes $i, j$ are “close” in the coherence sense if

$$ \mathrm{Adj}(i, j) \iff |\varphi_i - \varphi_j| < \varepsilon_{\mathrm{coh}}, $$

with $\varphi$ a local coherence phase and $\varepsilon_{\mathrm{coh}}$ a resolution threshold. A convenient weight on links is

$$ w_{ij} = \exp[-(\Delta\varphi_{ij}/\sigma)^2], \quad d_{ij}^{(\Lambda^I)} = 1 - w_{ij}, $$

which defines a purely informational “distance” $d_{ij}^{(\Lambda^I)}$ on the same underlying RTQ network.

In this sense, $N$ is not an extra metaphysical layer but the informational limit of the same $\Lambda$-field that generates curvature and particles: a non-metric completion where correlations are organized by coherence rather than by spacetime separation, in the spirit of entropic and information-based approaches to physics [Shannon, 1948, Jaynes, 1957, Landauer, 1961, Verlinde, 2011]. Across all domains, curvature and coherence remain complementary expressions of a single conserved tension $\Lambda_0$.

In metric sectors ($M^+, M^-$), $\Lambda^P$ governs inertia, energy and local causality; at the boundary $N$, $\Lambda^I$ governs memory and non-metric correlations. In Secs. 5.2–5.5, dark energy, dark matter and $H_0$ anomalies are interpreted as consequences of how $\Lambda^P$ and $\Lambda^I$ are distributed between $M^+, M^-$ and $N$, without adding new fields.

### 5.1.4 Curvature-coherence exchange and retroactive $\Lambda$-loop

At fixed modulus $\Lambda_0$, curvature and coherence are treated as dynamically complementary components of the same tension. The basic exchange reads

$$ \frac{d\Lambda^P}{dt} \approx -\frac{d\Lambda^I}{dt}, $$

expressing a $\Lambda$-exchange channel: curvature-based tension $\Lambda^P$ can be continuously converted into coherence-based tension $\Lambda^I$, and vice versa, without introducing additional sources beyond $\Lambda_0$.

At the coarse-grained cosmological level, this exchange is organized as a minimal three-domain loop:

* $M^+$ (metric sheet): observed domain with $(\Lambda_+^P, \Lambda_+^I)$,

* $N$ (coherence boundary): informational limit where $\Lambda^I \to \Lambda_0$, $\Lambda^P \to 0$,

* $M^-$ (mirror sheet): conjugate domain with $(\Lambda_-^P, \Lambda_-^I)$.

Qualitatively:

* $M^+ \to N$: part of the entropy-producing dynamics in $M^+$ can be re-encoded as long-lived coherence patterns at the boundary $N$ (curvature “dissipation” mapped to structured $\Lambda^I$);

* $N \to M^-$: only patterns compatible with the coherence manifold of $N$ are projected back as diffuse mirror curvature $\Lambda_-^P$ in $M^-$;

* $M^- \to M^+$: mirror curvature feeds back into $M^+$, tending to reduce discrepancies between $\Lambda_+^P$ and $\Lambda_-^P$ and stabilize the structural angle $\Theta_{\mathrm{struct}}$.

From a thermodynamic viewpoint, the $\Lambda^P \leftrightarrow \Lambda^I$ channel acts as a *structured bookkeeping of dissipation*: entropy production in $M^+$ is not “undone”, but part of its effect is stored as coherence in $N$ and as smooth curvature in $M^-$, in the spirit of information-negentropy links [Brillouin, 1953, Jaynes, 1957, Prigogine and Stengers, 1984]. Energy conservation remains explicit: only the representation of tension (curvature vs coherence, metric vs non-metric) is changed.
Local subsystems can then operate near a quasi-steady exchange,

$$ \frac{d\Lambda_{\text{loc}}^{P}}{dt} \approx -\frac{d\Lambda_{\text{loc}}^{I}}{dt}, $$

maintaining an approximately constant $\Lambda_0^2 = (\Lambda_{\text{loc}}^P)^2 + (\Lambda_{\text{loc}}^I)^2$ while continuously cycling between curvature and coherence. Cosmologically, this same $\Lambda$-loop provides the common background for:

* the angular interpretation of dark energy as sheet imbalance (Sec. 5.2),

* dark matter as gravitational shadow of $M^-$ (Sec. 5.3),

* $H_0$ variability from spatial heterogeneity of $\Theta(x)$ and node density (Sec. 5.4),

* the locking transition that selects $\Theta_{\text{struct}}$ (Sec. 5.5).

Sections 5.2–5.5 thus appear as specific coarse-grained realizations of a single, $\Lambda$-conserving curvature–coherence cycle.

### 5.1.5 Resonant dimension $d_{10}$ and hierarchical coherence (quantum to cosmological)

The angular structure of the $\Lambda$-field involves two distinct internal roles:

* an instantaneous angular coordinate $d_5$, specifying the position on the $\Lambda$-circle through

$$ \Lambda = \Lambda_0(\cos \Theta + i \sin \Theta), $$

* a resonant dynamical degree of freedom $d_{10}$, encoding how the pair $(\Lambda^P, \Lambda^I)$ rotates at fixed modulus:

$$ d_{10} \equiv \frac{\partial(\Lambda^P, \Lambda^I)}{\partial t} \Big|_{\Lambda_0=\text{const}} = \Lambda_0 \dot{\Theta} (-\sin \Theta, \cos \Theta). $$

In short, $d_5$ tells where the system sits on the $\Lambda$-circle (the value of $\Theta$), while $d_{10}$ encodes how it progresses along this circle at fixed $\Lambda_0$. The parameter $t$ remains the emergent physical time associated with the causal axis $d_4$; $d_{10}$ is not a second time, nor a spatial direction, but a non-metric, dynamical dimension that tracks the continuity of the curvature–coherence loop.

At microscopic scales, quantum systems already exploit a controlled tradeoff between phase information and energy: interference, decoherence and entanglement can be read as different ways of distributing constraints between “phase-like” and “energy-like” degrees of freedom, in line with information-based readings of statistical mechanics [Shannon, 1948, Jaynes, 1957]. In the $\Lambda$-language, such processes correspond to finely tuned $\Lambda^P \leftrightarrow \Lambda^I$ exchanges with small $\Lambda_0$, monitored by $d_{10}$.

At cosmological scales, the same $\Lambda$-cycle, now coarse-grained over many RTQs, governs the dark-sector split, the effective $\Lambda_{\text{eff}}(\Theta)$ and the variability of $H_0$. The role of $d_{10}$ is simply to ensure phase continuity across scales: it enforces that curvature–coherence exchanges in local subsystems and in the cosmological background remain compatible with the global $\Lambda_0$ constraint. Some driven, far-from-equilibrium subsystems (e.g. engineered quantum media, non-linear materials) are able to maintain a comparatively high ratio

$$ \eta \equiv \frac{|\Lambda^I|}{|\Lambda^P|} $$

over long times: they form local “pockets” where coherence remains organized against a curvature-dominated environment. In the present article, such regimes are only used as existence proofs that hierarchical coherence is compatible with the $\Lambda$-cycle.
### 5.1.6 From the angular field $\Theta$ to a screened scalar-tensor channel and MOND-like phenomenology

Motivation. Section 4.5.4 showed that the coarse-grained stiffness mode of the RTQ lattice admits a Jordan-like scalar-tensor representation of the form $\Phi(\cdot) R$. Here we make this correspondence explicit starting from the angular field $\Theta$ defined in Sec. 5.1, and we isolate three consequences: (i) a universal scalar-tensor (dilaton-like) channel with a screened coupling in dense environments, (ii) a natural route to MOND-like long-range behaviour in diffuse regimes, and (iii) a conceptual bridge to the bimetric/MOND literature (BiMOND shows the conceptual bridge exists).

#### 5.1.6.1 Jordan stiffness as an angular projection

In the angular cosmology, the structural stiffness in our sheet is $\Lambda_{\text{struct}}^P(\Theta) = \Lambda_0 \cos\Theta$. Accordingly, we define an effective Jordan-frame Planck stiffness field

$$ \Phi(\Theta) \equiv M_{Pl}^2(\Theta) = \bar{M}_{Pl}^2 \frac{\Lambda_{\text{struct}}^P(\Theta)}{\bar{\Lambda}_{\text{struct}}^P} = \bar{M}_{Pl}^2 \frac{\cos\Theta}{\cos\Theta_{\text{struct}}} $$

where $\bar{\Lambda}_{\text{struct}}^P = \Lambda_0 \cos \Theta_{\text{struct}}$ is the homogeneous background value for the present epoch (see Sec. 5.3.2 for the determination of $\Theta_{\text{struct}}$). Small angular texture then implies

$$ \frac{\delta\Phi}{\Phi} \approx -\tan\Theta \delta\Theta, \quad \frac{\delta G}{G} \approx -\frac{\delta\Phi}{\Phi} = +\tan\Theta \delta\Theta. $$

Thus a single field $\Theta(x, t)$ simultaneously controls the dark-sector partition (Secs. 5.2–5.3) and a scalar-tensor modulation of the effective gravitational stiffness.

#### 5.1.6.2 Einstein-frame coupling and the geometric prefactor

In the Einstein-frame representation used elsewhere in the paper (Sec. 4.5.2), matter couples universally through

$$ S_m[g_{\mu\nu}^{(J)}, \psi] = S_m[\Omega^{-2}(\Theta) g_{\mu\nu}^{(E)}, \psi] \equiv S_m[A^2(\Theta) g_{\mu\nu}^{(E)}, \psi], \quad A^2(\Theta) = \Omega^{-2}(\Theta) = \frac{\bar{M}_{Pl}^2}{\Phi(\Theta)}. $$

With the above $\Phi(\Theta)$, this gives the explicit angular form

$$ A^2(\Theta) = \frac{\cos\Theta_{\text{struct}}}{\cos\Theta}, \quad \frac{d\ln A}{d\Theta} = +\frac{1}{2}\tan\Theta. $$

We define the purely geometric prefactor

$$ \alpha_{\text{geom}}(\Theta) \equiv \frac{1}{2} \tan \Theta. $$

At the structural angle $\Theta_{\text{struct}} \approx 0.359 \text{ rad} \approx 20.6^\circ$, one has

$$ \alpha_{\text{geom}}(\Theta_{\text{struct}}) \approx \frac{1}{2} \tan(20.6^\circ) \approx 0.19, $$

which will be used below as the “corset-force” prefactor.
### 5.1.6.3 Screening by high-density locking (chameleon-like regime)

Local tests constrain long-range scalar forces. In the MCP, a natural suppression mechanism is already present in the stiffness sector: in high-density environments the confined component saturates local network capacity, producing a steep effective potential for $\Theta$ and suppressing gradients. At the EFT level this can be summarized by a density-dependent effective potential

$$V_{\text{eff}}(\Theta; \rho) = V(\Theta) + \rho \ln A(\Theta),$$

whose curvature near its minimum increases with $\rho$ ("high-density locking"). Consequently the field becomes short-ranged and

$$\nabla\Theta \approx 0 \quad \Rightarrow \quad \text{fifth-force effects are screened locally, recovering GR limits.}$$

### 5.1.6.4 MOND-like long-range limit and the “corset force”

In the quasi-static weak-field regime, the scalar-mediated acceleration is, to leading order,

$$\mathbf{a}_{\Theta} \approx -c^2 \nabla \ln A(\Theta) = -c^2 \frac{d \ln A}{d\Theta} \nabla\Theta = -\alpha_{\text{geom}}(\Theta) c^2 \nabla\Theta.$$

We will refer to this as the *corset-force form*: a purely geometric prefactor multiplying the stiffness-gradient. In low-density environments (galactic outskirts, voids), where screening is ineffective and the angular mode can remain light, this channel provides a natural route to MOND-like long-range behaviour (developed in Sec. 5.3.3). Conceptually, this also resonates with bimetric-MOND constructions: *BiMOND shows the conceptual bridge exists* between a two-metric viewpoint and MOND-type galactic phenomenology.

### 5.1.6.5 Application: numerical locking of the angular texture

The corset-force relation can be used in reverse to estimate how tightly the angular field must be locked in screened regions. Let $\Delta a$ denote the maximum allowed anomalous acceleration at Solar-System scales, and evaluate the geometric prefactor at the structural angle, $\alpha_{\text{geom}}(\Theta_{\text{struct}}) \approx 0.20$. A conservative bound is

$$|\nabla\Theta| \lesssim \frac{|\Delta a|}{\alpha_{\text{geom}}(\Theta_{\text{struct}}) c^2}.$$

Over a characteristic length $L$ this translates into a locking of the angular texture,

$$|\Delta\Theta| \sim |\nabla\Theta| L \lesssim \frac{|\Delta a| L}{\alpha_{\text{geom}}(\Theta_{\text{struct}}) c^2} \sim 10^{-6},$$

where the last estimate is indicative (order-of-magnitude) and illustrates the phenomenological point: *screened environments require extremely small angular excursions*, consistent with the high-density locking picture above.

*Scope note.* This section establishes a compact dictionary and a phenomenological pathway. A quantitative prediction of galaxy-scale profiles and the precise screening law requires specifying the IR coarse-graining that fixes $K_{\Theta}(\Theta)$ and $V(\Theta)$.

## 5.2 Dark Energy as Inter-Sheet Cosmic Osmosis

### 5.2.1 Angular Origin of the Cosmological Constant

We assign to each sheet an effective curvature fraction

$$\Lambda_P^{(+)} = \Lambda_0 \cos \Theta, \quad \Lambda_P^{(-)} = -\Lambda_0 \sin \Theta, \quad \Delta\Lambda(\Theta) = \Lambda_P^{(+)} - \Lambda_P^{(-)} = \Lambda_0(\cos \Theta + \sin \Theta).$$
$\Delta\Lambda$ measures the curvature imbalance between the two sheets. In $M^+$, this imbalance appears as an effective cosmological constant $\Lambda_{eff}$ that drives accelerated expansion. Rather than postulating a separate dark-energy fluid, $\Lambda$-Theory interprets dark energy as the visible effect of slow angular re-equilibration between the curvature-dominated and coherence-dominated sectors [Di Valentino et al., 2018, Valdivia et al., 2023].

A minimal phenomenological dynamics is

$$ \frac{d\Lambda_P}{dt} = -k(\Lambda_P - \Lambda_I), \quad 0 < k \ll H_0, $$

so that curvature slowly relaxes toward equal sharing with coherence. At fixed $\Lambda_0$, this is equivalent to a slow drift of $\Theta(t)$, and therefore of $\Lambda_{eff}(\Theta(t))$, similar in spirit to emergent dark-energy scenarios [Li and Shafieloo, 2019]. When $\Lambda_P^{(+)} > \Lambda_P^{(-)}$ our sheet expands; the mirror sheet is then in a complementary “contracting” phase in its own metric description.

### 5.2.2 Evolution of $\Theta(t)$ and Large-Scale Expansion

The angular evolution of the pair can be written as

$$ \Lambda(\Theta(t)) = \Lambda_0 e^{i\Theta(t)}, \quad \text{fracd}\Theta dt = \frac{1}{\Lambda_0^2}(\Lambda_P \dot{\Lambda}_I - \Lambda_I \dot{\Lambda}_P). $$

The Hubble rate in $M^+$ is assumed to scale as

$$ H_0(t) \propto \sqrt{\Lambda_P^{(+)}(t)} = \sqrt{\Lambda_0 \cos\Theta(t)}, $$

so any slow drift of $\Theta(t)$ translates into a mild time-variation of $\Lambda_{eff}$ and $H_0$. The present structural angle $\Theta_{struct} \simeq 0.359 \text{ rad} \approx 20.6^\circ$ will be fixed below from the observed dark-sector fractions; in this picture, $\Theta(t)$ relaxes toward a state where $\cos\Theta$ and $\sin\Theta$ are more evenly balanced, causing dark energy to *fade* gradually with cosmic time rather than remain strictly constant [Peebles and Ratra, 2003, Frieman et al., 2008, Weinberg, 2008].

## 5.3 Dark Matter as the Gravitational Shadow of $M^-$

*Terminology note.* Throughout Secs. 5.3–5.4, we reserve *antimatter* for local antiparticles within a given sheet, and use *mirror sector* ($M^-$) for the $\pi/2$-conjugate branch. Any apparent “negative” sign is a cross-sheet bookkeeping in an $M^+$ frame (projection), not a local $m < 0$ inertial matter.

<table><caption>Table 8: Terminology and sign conventions for antimatter vs. mirror sector (MCP).</caption><thead><tr><th>Term (paper)</th><th>Sheet</th><th>Signature seen from M<sub>+</sub></th><th>Local signature (in its own sheet)</th><th>Appearance in M<sub>+</sub></th></tr></thead><tbody><tr><td>Visible matter</td><td>M<sub>+</sub></td><td>(+d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, +d<sub>4</sub>)</td><td>(+d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, +d<sub>4</sub>)</td><td>baryons / photons / etc.</td></tr><tr><td>Local antimatter (antiparticles)</td><td>M<sub>+</sub></td><td>(-d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, -d<sub>4</sub>)</td><td>(-d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, -d<sub>4</sub>)</td><td>visible if produced locally (annihilation possible)</td></tr><tr><td>Mirror matter (source of effective DM)</td><td>M<sub>-</sub></td><td>(-d<sub>1</sub>, -d<sub>2</sub>, -d<sub>3</sub>, -d<sub>4</sub>)</td><td>(+d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, +d<sub>4</sub>)</td><td>effective dark matter = gravitational shadow of M<sub>-</sub></td></tr><tr><td>Mirror antimatter (antiparticle in M<sub>-</sub>)</td><td>M<sub>-</sub></td><td>(+d<sub>1</sub>, -d<sub>2</sub>, -d<sub>3</sub>, +d<sub>4</sub>)</td><td>(-d<sub>1</sub>, +d<sub>2</sub>, +d<sub>3</sub>, -d<sub>4</sub>)</td><td>still “dark” (no EM/weak coupling), only gravitational via cross-projection</td></tr></tbody></table>

**Legend:** The “signs” in the column *Signature seen from M<sub>+</sub>* describe how the conjugate sector is expressed in the $M_+$ bookkeeping (projection / re-expression), **not** a local $m < 0$ inertial matter in our sheet. Local observers in each sheet still measure positive inertia. In MCP, the observed dark-matter effect in $M_+$ is modeled at coarse grain by $\nabla^2\Phi \propto \rho_+ + \varepsilon\rho_-$, where $\varepsilon\rho_-$ is the *effective* (non-luminous) contribution sourced by the mirror sheet.

With this dictionary fixed, we now derive how $M^-$ contributes to $M^+$ dynamics as an effective, non-luminous gravitational term (“shadow”) rather than as annihilating antimatter.
### 5.3.1 Phase-Shifted Matter and Effective Mass Polarity (as projected in $M_+$)

Matter in $M^-$ is described by the same $\Lambda_0$ but a $\pi/2$-rotated phase:

$$ (\Lambda_P, \Lambda_I)_{M^-} = (\Lambda_I, -\Lambda_P)_{M^+}. $$

Here “polarity” refers to an *effective* cross-sheet bookkeeping in an $M_+$ frame (Table 8), not to any local $m < 0$ matter in our sheet.

Because the internal mass axis $d_2$ and causal axis $d_4$ are also rotated, mirror matter carries positive local inertia and forward time in its own sheet, but appears with effective opposite mass polarity when projected into $M^+$ coordinates:

$$ m_{(-)}(\Theta + \frac{\pi}{2}) \propto -\Lambda_P \sin(\Theta) \text{ (in an } M^+ \text{ frame).} $$

Direct gauge couplings between the two sectors are suppressed: $M^-$ matter does not couple to the U(1) and SU(2) fields that define ordinary photons and weak interactions in $M^+$, and is thus electromagnetically and weakly dark from our viewpoint [Foot, 2004, 2014, Ciarcelluti, 2009, Yin et al., 2020]. However, both sectors curve the same underlying RTQ lattice, so an observer restricted to $M^+$ detects mirror mass only through its *gravitational shadow*.

At coarse graining, the Poisson equation in $M^+$ becomes

$$ \nabla^2 \Phi_{(+)} \propto \rho_{(+)} + \varepsilon \rho_{(-)}, $$

where $\rho_{(+)}$ is visible matter density, $\rho_{(-)}$ the mirror density, and $0 < \varepsilon \ll 1$ an angular coupling parameter set by the degree of phase misalignment between sheets. The $\varepsilon\rho_{(-)}$ term is interpreted as *dark matter*: a non-luminous mass component inferred from rotation curves, lensing and structure formation, but physically realised as curvature sourced by $M^-$ [Foot, 2014, Bousder and Sakhi, 2023].

#### 5.3.1.1 Perturbations and the CMB (why the shadow is CDM-like at linear order)

The present paper focuses on the *background* closure (the angular partition of $\Omega_{DE}$ and $\Omega_{DM}$). A full computation of the CMB power spectrum would require solving the coupled linear-perturbation system across the two sheets within an Einstein-Boltzmann pipeline, which is beyond scope here. However, the *mechanism* by which $M^-$ enters $M^+$ already fixes the qualitative perturbation behaviour: the shadow term sources $\Phi_+$ only through the Poisson channel $\nabla^2\Phi_+ \propto \rho_+ + \varepsilon\rho_-$, with no electromagnetic coupling to the photon-baryon fluid and no radiation pressure in the $M^+$ sector. Therefore, to leading order in the IR (sub-horizon, weak-field regime), the projected contribution $\rho_{sh} \equiv \varepsilon\rho_-$ acts as an effectively pressureless source ($w_{sh} \simeq 0$, $c_{s,sh}^2 \simeq 0$) for structure growth, i.e. it is CDM-like at the level that controls the acoustic-peak phenomenology. Deviations from strict $\Lambda$CDM (e.g. scale-dependent growth or mild potential-slip effects induced by the inter-sheet bookkeeping) are *testable* and belong to the dedicated perturbation follow-up.

### 5.3.2 Quantitative Angular Balance of Visible and Dark Components

Empirically, using Planck PR4/NPIPE 2020–2024 results, present-day cosmological densities are approximately [Planck Collaboration et al., 2020a, Tristram et al., 2024]:

$$ \rho_{DE} \approx 69.2\%, \quad \rho_{DM} \approx 25.97\%, \quad \rho_{vis} \approx 4.83\%. $$

We model the fractions in $M^+$ as angular projections:

$$ \rho_{DE} \propto \cos \Theta, \quad \rho_{DM} \propto \sin \Theta, \quad \rho_{vis} \approx \varepsilon \cos \Theta, $$

with $\Theta = \Theta_{struct}$ and the same $\varepsilon$ as in the Poisson equation. From

$$ \frac{\rho_{DM}}{\rho_{DE}} \approx \frac{0.2597}{0.692} \approx 0.37529 $$
we obtain

$$ \tan \Theta_{struct} \approx 0.37529 \Rightarrow \Theta_{struct} \approx 0.35899 \text{ rad} \approx 20.57^{\circ}, $$

and therefore

$$ \cos \Theta_{struct} \approx 0.9363, \quad \sin \Theta_{struct} \approx 0.3513. $$

Normalizing $\rho_{DM}$ and $\rho_{DE}$ to $\rho_{DM} + \rho_{DE}$ gives the observed ~26/69 split. Choosing $\epsilon \approx 0.0516$ yields

$$ \rho_{vis} \approx \epsilon \cos \Theta_{struct} \approx 0.0516 \times 0.9363 \approx 0.0483, $$

which matches the baryon component exactly. Thus, the triplet $(\rho_{vis}, \rho_{DM}, \rho_{DE})$ emerges from a single structural angle $\Theta_{struct}$ plus an angular coupling $\epsilon \sim 1/20$, rather than three independent cosmological input [Peebles and Ratra, 2003, Frieman et al., 2008, Weinberg, 2008, Planck Collaboration et al., 2020b].

In this view,

* dark matter = quasi-static projection of curvature from M⁻ into M⁺;

* dark energy = time-varying rebalancing of $\Delta\Lambda(\Theta)$.

Both vanish if $\Lambda_P^{(+)} = \Lambda_P^{(-)}$. The “coincidence problem” is rephrased: present-day dark fractions measure where $\Theta(t)$ currently sits between 0 and $\pi/2$.

### 5.3.2.1 PR4 update and “data tightening”

The shift from Planck 2018-era representative values to the final Planck PR4 inference modifies the dark-sector ratio from the previously used $\mathcal{O}(0.40)$ level to

$$ \frac{\Omega_{DM}}{\Omega_{DE}} \approx \frac{0.2597}{0.692} \approx 0.3753, $$

which moves the structural angle from the ~22° ballpark to

$$ \Theta_{struct} \approx 0.3590 \text{ rad} \approx 20.6^{\circ}. $$

This is a modest drift ($\Delta\Theta \sim -1.4^{\circ}$), indicating that the angular closure is stable under improved CMB processing and likelihood choices, while slightly favoring a more $\Lambda^P$-dominated (“stiffer”) present sheet.

Importantly, in the $\Lambda$-grammar the angle $\Theta_{struct}$ is determined by the dark-only ratio $\Omega_{DM}/\Omega_{DE}$, whereas the visible fraction follows from the baryon closure

$$ \Omega_b \approx \epsilon \cos \Theta_{struct} \quad \Rightarrow \quad \epsilon \approx \frac{\Omega_b}{\cos \Theta_{struct}}. $$

Using the PR4 angle above, the value required by the observed $\Omega_b$ is

$$ \epsilon \approx \frac{0.0483}{0.936} \approx 5.16 \times 10^{-2} \approx \frac{1}{20}, $$

which is closer to the baryon budget than the earlier estimate ($\epsilon \sim 0.07$). Thus, sharper cosmological inputs tighten the closure and improve the “one-angle + one-efficiency” description without introducing additional degrees of freedom.
### 5.3.3 Phenomenology: galaxies (MOND-like regime from angular stiffness)

*Low-density, weak-field limit.* In diffuse environments (galactic outskirts, low-$\rho$ regions), the high-density locking mechanism is ineffective and the angular mode can remain light. In this regime, the scalar contribution to the acceleration takes the corset-force form derived in Sec. 5.1.6:

$$ \mathbf{a}_{\Theta} \approx -\alpha_{\text{geom}}(\Theta) c^2 \nabla \Theta, \quad \alpha_{\text{geom}}(\Theta) = \frac{1}{2} \tan \Theta, \quad \alpha_{\text{geom}}(\Theta_{\text{struct}}) \approx 0.20. $$

*Log-profile and flat rotation curves.* If the IR coarse-grained dynamics admits an approximately scale-free exterior solution around a compact baryonic source (e.g. weak potential term $V'(\Theta) \approx 0$ and slowly varying stiffness in the outskirts), then the static field equation naturally supports logarithmic profiles,

$$ \Theta(r) \sim \ln r \quad \Rightarrow \quad |\nabla\Theta| \propto \frac{1}{r}. $$

Inserted into the corset-force relation, this yields

$$ |\mathbf{a}_{\Theta}| \propto \frac{1}{r}, $$

which is precisely the scaling required to generate approximately flat rotation curves in the outer parts of galaxies. The key point is structural: a MOND-like $1/r$ acceleration can arise as a long-range, low-density limit of the same angular stiffness field that is screened in dense environments.

*Acceleration scale and cosmological linkage.* Empirically, MOND phenomenology is often summarized by an acceleration scale

$$ a_0 \sim \eta c H_0, \quad \eta = \mathcal{O}(1), $$

suggesting a link between the onset of galactic modifications and the background cosmological curvature scale. In the $\Lambda$-framework, the late-time expansion rate is itself controlled by the structural stiffness channel, schematically

$$ H_0 \propto \sqrt{\Lambda_0 \cos \Theta_{\text{struct}}}, $$

so the relation $a_0 \sim \eta c H_0$ is not an ad hoc coincidence but a natural place where cosmology and the galactic IR limit can meet: the same background angle $\Theta_{\text{struct}}$ that fixes the dark-sector partition also fixes the stiffness scale entering $H_0$. A first-principles prediction of $\eta$ and of detailed rotation-curve shapes requires specifying the IR coarse-graining that determines $K_{\Theta}(\Theta)$ and $V(\Theta)$, and solving the resulting field equation with realistic baryonic sources.

## 5.4 The $H_0$ Tension as Signature of Network Density Variability

A key observational puzzle is the discrepancy between local and early-Universe determinations of the Hubble parameter:

* Local / JWST ladder: $H_0^{\text{local}} \approx 73 \pm 1 \text{ km s}^{-1} \text{ Mpc}^{-1}$ (Riess and al 2024).

* CMB + $\Lambda$CDM: $H_0^{\text{CMB}} \approx 67 \pm 1 \text{ km s}^{-1} \text{ Mpc}^{-1}$ (Candanosa 2024; Siegel 2025).

$\Lambda$-Theory interprets this as a signature of spatial heterogeneity of $\Theta(x)$ and RTQ density $n(x)$, rather than a breakdown of $\Lambda$CDM per se.

### 5.4.1 Expansion as Projection of $\Lambda^P$ Gradients

In $M^+$, the local expansion rate scales with the local curvature fraction:

$$ H_0(x, t) \propto \sqrt{\Lambda_P^{(+)}(x, t)} = \sqrt{\Lambda_0 \cos \Theta(x, t)}. $$
Gradients in $\Theta(x)$ reflect variations of how $\Lambda_0$ is split between $\Lambda^P$ and $\Lambda^I$, which in turn correlate with node density $n(x)$ (voids vs filaments). A simple heuristic is

$$ \frac{d\Theta}{dx} \propto \frac{1}{n(x)} \frac{dn}{dx}, $$

so that spatial changes in connectivity induce spatial variations in $\Theta$ and thus in $H_0$. Small perturbations around $\Theta_0 = \Theta_{struct}$ give

$$ \frac{\delta H_0}{H_0} \approx -\frac{1}{2} \tan \Theta_0 \delta\Theta, $$

so a modest $\delta\Theta$ across the observable volume translates into the observed $\approx 9\%$ tension between local and CMB-inferred values.

In regions where $\Lambda^P$ is slightly higher (smaller $\Theta$), expansion is slower; in regions with slightly lower $\Lambda^P$ (larger $\Theta$), expansion is faster. Early-Universe probes (CMB) and late-time local probes sample different effective averages of $\Theta(x)$ and $n(x)$, naturally leading to different inferred $H_0$ values.

### 5.4.2 Directional Variability and Node-Density Mapping

Several analyses report directional anisotropies in $H_0$ at the few-percent level from supernovae, clusters and other probes [Krishnan et al., 2023, Migkas, 2024, Boubel et al., 2025, Kashyap et al., 2025]. In the $\Lambda$-framework this is expected: the Universe has a non-trivial $\Lambda$-field texture.

We parametrize

$$ \Lambda_P(\Omega) \propto n(\Omega), \quad \Theta(\Omega) \approx \arctan \frac{\Lambda_I}{\Lambda_P(\Omega)} \approx \arctan \frac{1}{n(\Omega)}, $$

and define a directional coherence index

$$ C(\Omega) := \frac{\langle \Lambda_P(\Omega) \rangle}{\Lambda_0} = \cos \Theta(\Omega), \quad H_0(\Omega) \propto \sqrt{C(\Omega)}. $$

Dense directions (filaments, clusters) have larger $n(\Omega)$, smaller $\Theta(\Omega)$ and thus lower $H_0(\Omega)$; void-dominated directions have the opposite trend. Directional maps of $H_0$ therefore act as tomographs of $\Lambda$-field rigidity, to be correlated with galaxy surveys, void catalogs and possibly large-scale magnetic or polarization structures.

In this context, early Dark Energy (EDE) models, such as those proposed by Poulin et al. [2019], offer a potential alternative to the standard $\Lambda$-texture interpretation by providing a mechanism that could bridge the gap in the $H_0$ tension. These models suggest a transient component that could resolve the discrepancy between early-Universe and local measurements. Furthermore, work on the cosmology of voids Bromley and Geller [2024] complements this by investigating the role of large-scale structures in the variation of $H_0$, with implications for the interpretation of anisotropies and the density distribution of the Universe.

And in the $\Lambda$ framework, the $H_0$ tension becomes a probe of $(\Lambda^P, \Lambda^I, n(x))$ heterogeneity rather than an inconsistency of the cosmological model.

## 5.5 Angular Genesis and Absence of Physical Inflation

$\Lambda$-Theory proposes that cosmological initial conditions arise from an angular phase transition, not from a period of physical super-luminal inflation. Time, metric and the dark-sector fractions are all products of a single locking event of $\Theta(x)$.

### 5.5.1 Terminology lock: what “Big Bang” and “inflation” mean in $\Lambda$-Theory

Because the words *Big Bang* and *inflation* are historically tied to a metric description, we explicitly fix the terminology used in this section.
### 5.5.1.1 Big Bang (in $\Lambda$-Theory)

“Big Bang” refers to the onset of a globally consistent *causal order*: the transition from a pre-temporal RTQ regime with no global time coordinate to an ordered phase of the causal axis $d_4$. In other words, the Big Bang is the *crystallization of macroscopic time* (and of a stable effective manifold description), not a physical singularity in a pre-existing spacetime.

### 5.5.1.2 Inflation (in $\Lambda$-Theory)

“Inflation” is reinterpreted as the rapid relaxation of the angular order parameter during the formation of causal structure. We distinguish:

$$ \Theta(x) \equiv \Theta_{\text{dyn}}(x) \quad \rightarrow \quad \Theta_{\text{struct}} = \text{const}, $$

where $\Theta_{\text{dyn}}$ is the fluctuating pre-temporal phase angle of the complex tension $\Lambda = \Lambda^P + i\Lambda^I$ at fixed modulus $|\Lambda| = \Lambda_0$, and $\Theta_{\text{struct}}$ is the late-time attractor selected by the locking event. Thus, the phenomenon historically modelled as a superluminal metric expansion is here an *angular synchronization / ordering* process in the network state space, not a separate inflaton-driven epoch.

### 5.5.1.3 Why the standard puzzles are addressed without a dedicated inflaton

In the pre-temporal regime there are no light cones and no notion of causal horizon; global correlations can therefore form before the emergence of a macroscopic time axis. Once causal structure crystallizes, the emergent metric regime is already close to homogeneous and flat in the coarse-grained sense (small $\nabla\Lambda^P$), removing the need for a distinct inflaton field.

### 5.5.1.4 Energy budget and the CMB

A key consistency requirement is to explain the origin of the large radiation bath. In $\Lambda$-Theory this is provided by the locking transition itself: excess fluctuations stored in the coherence sector $\Lambda^I$ are released when the system settles into $(\Lambda^P(\Theta_{\text{struct}}), \Lambda^I(\Theta_{\text{struct}}))$. This motivates the interpretation developed in Sec. 5.5.3: the CMB as the latent heat of temporal ordering.

### 5.5.1.5 Scope

This re-interpretation does not modify late-time $\Lambda$CDM geometry as an effective description; it only shifts the *mechanism* behind the early “initial condition” problem from metric expansion to an angular ordering transition.

## 5.5.2 Pre-Temporal Network Sorting and $\Theta$-Crystallization

To avoid confusion between the *relaxing* early-time network state and the *locked* late-time attractor, we distinguish two roles for the $\Lambda$-phase angle:

* $\Theta_{\text{dyn}}(x, t)$ is a coarse-grained, pre-temporal (or weakly causal) order parameter describing how the network partitions its invariant tension $\Lambda_0$ between coherence $\Lambda^I$ and curvature $\Lambda^P$ while the causal order is still forming.

* $\Theta_{\text{struct}}$ is the *structural fixed point* of that relaxation: once the network has locked into a stable causal regime, $\Theta$ no longer drifts as a global degree of freedom and the observed Universe is characterized by a persistent $\Theta_{\text{struct}}$ (the same angle that controls the dark-sector partition in the late-time effective description).

Within this language, “inflation” is not postulated as an extra metric degree of freedom driven by an inflaton field. Instead, it corresponds to a rapid $\Lambda$-phase relaxation $\Theta_{\text{dyn}} \rightarrow \Theta_{\text{struct}}$ during the formation of causal order (time crystallization), i.e. a fast redistribution between $\Lambda^I$ and $\Lambda^P$ while the network synchronizes its internal phases.

The “Big Bang” then refers to the onset of a globally consistent causal arrow (a locked, propagating notion of time) rather than a singular initial condition. In this picture, the CMB is interpreted as the latent heat of the
locking transition: the release of excess coherence during the final approach to $\Theta_{struct}$ when the causal regime becomes stable.

In the earliest, pre-temporal regime (i.e. prior to the crystallization of a global causal order), there is no spacetime manifold and no global time: only RTQs with internal angular coordinates exist Rovelli [1991], Barbour [1999]. Their tension is described by

$$ \Lambda = \Lambda^P + i\Lambda^I, \quad \Theta(x) = \Theta_{dyn}(x), $$

with $|\Lambda| = \Lambda_0$ fixed but $\Theta_{dyn}$ fluctuating. Fermionic torsion loops (mass motifs on $d_2$) have not yet formed; dynamics is reversible and pre-temporal.

As coupling between RTQs increases, an effective Kuramoto-like synchronization takes place in angular space Kuramoto [1975], Acebrón et al. [2005]:

$$ \Theta_{dyn}(x) \rightarrow \Theta_{struct} = \text{const.}, $$

driven by a free-tension functional $\mathcal{F}(\Theta)$ with minima at specific attractors (e.g. $\Theta \simeq 0, \pi/4, \pi/2$) determined by the relative cost of curvature and coherence. When $\Theta_{struct}$ is selected, we obtain:

* fixed decomposition $\Lambda^P(\Theta_{struct}), \Lambda^I(\Theta_{struct})$,

* emergence of a global time axis $d_4$ and of a macroscopic arrow (Part 4),

* bifurcation into two conjugate mass orientations along $d_2$, defining $M^+$ and $M^-$ as internal angular branches, not spatially separated worlds.

The structural angle $\Theta_{struct} \approx 0.36$ rad inferred from dark-sector fractions (Section 5.3.2) is thus the cosmological order parameter of this transition; different values would define universes with different ($\rho_{DE}, \rho_{DM}, \rho_b$).

Because the pre-temporal phase has no light cones, there is no horizon problem: the $\Lambda$-field can correlate globally before time exists. When the causal structure crystallizes, the emerging metric is already nearly homogeneous and flat ($\nabla\Lambda^P$ small), without requiring a separate inflaton field [Guth, 1981, Linde, 1982, 1990, Liddle and Lyth, 2000, Weinberg, 2008].

### 5.5.3 CMB as Latent Heat of Temporal Ordering

Before locking, fluctuations in $\Lambda^I$ (coherence) and $\Lambda^P$ (curvature) are large but constrained by $|\Lambda| = \Lambda_0$. At crystallization, the system settles into $\Lambda^P(\Theta_{struct}), \Lambda^I(\Theta_{struct})$; excess coherence fluctuations must be dissipated. The released energy is of order

$$ \Delta E_{release} \sim \int (\Lambda_{dyn}^I(x) - \Lambda_{struct}^I)^2 dV, $$

analogous to latent heat in a standard phase transition [Goldenfeld, 1992, Landau and Lifshitz, 1980]. At this stage the Universe is boson-only: fermionic torsion loops have not yet stabilized, so only photons and curvature-wave-like modes can carry this energy.

These excitations thermalize into a nearly homogeneous radiation bath, naturally explaining:

* the high photon-to-baryon ratio (baryons appear later, after fermion nucleation),

* the remarkable homogeneity and blackbody nature of the CMB [Planck Collaboration et al., 2020b].

Residual angular inhomogeneities $\delta\Theta_{dyn}(x)$ at locking produce small variations in $\Lambda^P$ and $\Lambda^I$, which map to the observed $\sim 10^{-5}$ CMB anisotropies. After crystallization, the metric phase expands and cools the radiation to the present $T_{CMB} \approx 2.7$ K the CMB becomes the thermal relic of $\Theta$-locking, not necessarily of a baryon-antibaryon annihilation era.

Because correlations are established in the pre-metric phase and preserved during the $\Theta$-transition, $\Lambda$-Theory can reproduce the main successes of inflation (homogeneity, near-flatness, super-horizon correlations) without a physical inflationary field. The distinction from $\Lambda$CDM inflation lies in subtle signatures in primordial
spectra and non-Gaussianities [Kolb and Turner, 1990, Brandenberger, 2017, Planck Collaboration et al., 2020b].

In this angular cosmology, our Universe is one node in a continuum of Θ-universes: its dark-sector composition, $H_0$ anomalies and CMB relics all encode the choice of a single structural angle and its coupling to a $\pi/2$-conjugate sheet, rather than evidence for multiple independent dark substances.

## 5.6 Causal-Depth Closure: The Electron Mass as a Cosmological Scaling Law

The angular genesis described in Sec. 5.5 fixes dimensionless cosmic ratios through the structural angle $\Theta$ (e.g. $\Omega_{DM}/\Omega_{DE}$). A deeper question concerns the absolute scale of matter: why does the electron mass $m_e$ take its observed value relative to the Planck scale?

In the $\Lambda$-framework, microscopic parameters are constrained by the total information content of the Universe. We define the total causal depth $N_{cosmo}$ as the number of fundamental update ticks since the causal origin. Because the carrier audit selects $d = 3$ emergent spatial dimensions, the effective linear capacity available to a 1D topological loop scales as a cubic root. This implies a scaling law linking the macroscopic age of the Universe to the microscopic scale of the lightest charged fermion.

To test this hypothesis, we compare two a priori independent routes for estimating the electron loop length $N_{loop}$: one derived from measured microphysics (Compton route) and one derived purely from the causal depth of the Universe (MCP route).

### 5.6.1 Metric / Compton route

Using the reduced Compton wavelength,

$$ \lambda_{C,e} = \frac{\hbar}{m_e c}, $$

the corresponding “metric” loop count in RTQ units is

$$ N_{loop}^{(C)} \equiv \frac{\lambda_{C,e}}{\ell_0}. $$

With numerical inputs

$$ \lambda_{C,e} \approx 3.86159 \times 10^{-13} \text{ m}, \quad \ell_0 \approx 8.1027 \times 10^{-35} \text{ m}, $$

one obtains

$$ N_{loop}^{(C)} \approx \frac{3.86159 \times 10^{-13}}{8.1027 \times 10^{-35}} \approx 4.766 \times 10^{21}. $$

### 5.6.2 Causal-depth / MCP route

Let the network update tick be

$$ t_0 \equiv \frac{\ell_0}{c} \approx 2.7028 \times 10^{-43} \text{ s}. $$

Over a cosmological duration $T$, the maximal causal depth in network ticks is

$$ N_{cosmo}^{(0)} \equiv \frac{T}{t_0}. $$

Because the carrier audit selects $d = 3$ spatial dimensions, the effective linear capacity available to a 1D loop scales as the cubic root. In the structural sector, adjacency is weighted by $\Lambda^P = \Lambda_0 \cos \Theta$, hence a natural modulation by $\cos \Theta_{struct}$. We therefore write

$$ N_{loop}^{(M)}(t_P) \equiv N_{pass} 4\pi \left(\frac{T}{t_P}\right)^{1/3} \cos \Theta_{struct}, $$
where we use the Planck clock $t_P$ as the reference metric clock for the projection, and $N_{\text{pass}} \approx 2$ (spinor closure).

### 5.6.3 The numerical convergence

We now evaluate the causal-depth prediction using PR4-era inputs and the updated $\Theta_{\text{struct}}$.

Numerical inputs:

$$T \approx 4.3508 \times 10^{17} \text{ s}, \quad t_P \approx 5.3912 \times 10^{-44} \text{ s}, \quad \Theta_{\text{struct}} \approx 0.35899 \text{ rad.}$$

The total causal depth in Planck ticks is

$$N_{\text{cosmo}}^{(P)} \equiv \frac{T}{t_P} \approx 8.07 \times 10^{60},$$

hence the linear capacity is $\eta_P = (N_{\text{cosmo}}^{(P)})^{1/3} \approx 2.006 \times 10^{20}$. With $\cos \Theta_{\text{struct}} \approx 0.9363$ and taking the spinor closure $N_{\text{pass}} = 2$,

$$N_{\text{loop}}^{(M)} = 2 \cdot 4\pi \cdot (2.006 \times 10^{20}) \cdot 0.9363 \approx 4.720 \times 10^{21}.$$

This should be compared to the metric/Compton count derived in Sec. 5.6.1:

$$N_{\text{loop}}^{(C)} \approx 4.766 \times 10^{21}.$$

The two independent routes therefore agree at the percent level (1.009 ratio). Equivalently, this confirms the dimensional projection factor $\kappa = \ell_0/\ell_P \approx 5.01$ (consistent with $\sqrt{8\pi}$), as $\kappa^{1/3} \approx 1.71$ links the network-tick depth to the metric depth. Expressing the causal route in network ticks $t_0$ yields

$$N_{\text{loop}}^{(M)}(t_0) = \frac{N_{\text{loop}}^{(M)}(t_P)}{\kappa^{1/3}} \approx 2.76 \times 10^{21}, \quad \kappa^{1/3} \approx 1.71,$$

so that the previously observed $\sim 1.71$ offset appears exactly as the dimension-projection conversion between $t_0$ and $t_P$, while the physically comparable quantities (Compton vs. Planck-stiffness clock) coincide.

### 5.6.4 Interpretation: an electron-mass scaling law

The Compton estimate is a *metric* quantity and naturally aligns with the Planck-stiffness clock $t_P$, whereas the causal-depth route is defined in *network update* ticks $t_0 = \ell_0/c$. Because the MCP closure explicitly takes a cubic root (3D $\rightarrow$ 1D effective capacity), the conversion between these two clocks necessarily introduces the factor $\kappa^{1/3}$.

Most importantly, setting the two loop counts equal,

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

yields an explicit scaling law for the electron mass:

$$\frac{\hbar}{m_e c \ell_0} = N_{\text{pass}} 4\pi \left(\frac{T}{t_P}\right)^{1/3} \cos \Theta_{\text{struct}},$$

or

$$m_e = \frac{\hbar}{c \ell_0} \frac{1}{N_{\text{pass}} 4\pi} \left(\frac{t_P}{T}\right)^{1/3} \frac{1}{\cos \Theta_{\text{struct}}}.$$
Thus $m_e \propto T^{-1/3}$, with a modulation by the structural angle fixed by dark-sector ratios. The striking point is that the right-hand side depends on cosmological depth and a discrete holonomy multiplicity $N_{\text{pass}}$, yet numerically reproduces the observed electron loop cost $N_{\text{loop}} \sim 10^{21}$ without continuous parameter tuning.