Section 7 — Emergent Gravity and Lepton Phase Volume (TFP, v11.1)
Geometry as Response to Mass Density
By John Gavel
7.0 Overview
Gravity in TFP is not primitive. It emerges as a geometric response of the finite handshake substrate to local mass processing. Curvature arises because mass consumes handshake capacity, reducing local handshake density \(H(x)\) and producing triangulation strain. Mass density \(M(x)\) is proportional to local mesoscopic tension \(T(x)\). In the continuum limit, Einstein-like relations appear as statements about correlation-induced metrics and loop holonomy. No independent gravitational constant is postulated; \(G\) emerges from substrate grains, cosmic coherence scales, and handshake bookkeeping.
7.1 Mass as the Source of Curvature
Mesoscopic tension (coarse-grained from Section 2): \[ T(x) \approx k_\text{avg} \cdot [1 - A(x)^2] + \frac{k_\text{avg} \cdot a^2}{2} |\nabla A(x)|^2 \] In aligned domains (\(|A| \approx 1\)), the bulk term is negligible; gradient term dominates: \[ T(x) \approx \frac{k_\text{avg} \cdot a^2}{2} |\nabla A(x)|^2 \] Local mass density (operational): \[ M(x) = \gamma_M \cdot T(x) \] Interpretation: Mass measures flip/reflection rate, sourcing gradients and geometric strain (A2, A6, A9).
7.2 Curvature from Handshake Gradients
Local handshake density \(H(x)\): number of usable handshake addresses at \(x\). Ideal flat substrate: \(H_\text{ideal} = H = 132\). Fractional deficit: \[ \Delta H(x) = H_\text{ideal} - H(x), \quad \text{fractional\_deficit} = \frac{\Delta H(x)}{H_\text{ideal}} \] Ricci-like scalar (operational): \[ R(x) \propto \frac{H_\text{ideal} - H(x)}{H_\text{ideal}} \] Interpretation: Mass consumes handshake budget; curvature is proportional to \(M(x)\). Equivalence principle (TFP): acceleration and local handshake scarcity are indistinguishable at the motif level.
7.2.1 Refined Geometric Origin
\(\Delta H(x)\) arises because mass-processing motifs lock handshake slots for persistent cycles. Increased local handshake usage → adjacent triangulation coherence drops → loop holonomy accumulates → operational curvature.
7.3 Emergent Metric from Correlations and Holonomy
Operational distance (Section 3.2): \[ d_{ij} = - \ln |C_{ij}(0)| \] where \(C_{ij}(0)\) is the equal-time correlation between sites \(i\) and \(j\). Continuum identification for small separations: \[ ds^2 = g_{\mu\nu} dx^\mu dx^\nu = d_{ij}^2 \] Ricci tensor from loop holonomy (sum over small loops \(p\) containing \(x\)): \[ H_p = \sum_{\text{edges } u \to v \in p} \tau_{u \to v}, \quad R_{\mu\nu}(x) = \frac{1}{\langle A_p \rangle} \sum_{p \ni x} H_p \hat{e}_\mu \hat{e}_\nu \] \(\langle A_p \rangle\) = average plaquette alignment, \(\hat{e}_\mu\) = loop-edge unit vectors. Metric and curvature are coarse-grained encodings of loop-level handshake phase accumulation and mismatch.
7.4 Einstein-like Relation (Continuum Limit)
Stress-energy proxy from motif 4-velocity \(u_\mu\): \[ T_{\mu\nu}(x) = M(x) \cdot u_\mu u_\nu \] Geometric left-hand side: \[ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \] Emergent coupling: \[ G_{\mu\nu} = \kappa_G \cdot T_{\mu\nu} \] \(\kappa_G\) depends on substrate grains, cosmic coherence, and handshake dilution (no independent constant).
7.4.1 Gravitational Constant as Handshake Leakage
TFP expression: \[ G = \frac{a_s^3}{\tau_s^2 M_p} \cdot \frac{a_s}{R_U} \cdot \frac{4 \pi}{H K} \cdot \Gamma \] where:
- \(a_s\) = substrate lattice pitch
- \(\tau_s\) = substrate update tick
- \(M_p\) = proton mass anchor
- \(R_U\) = cosmic coherence radius
- \(H = 132, K = 12\)
- \(\Gamma =\) curvature limit factor (\(\Gamma_c = 4 / \pi \approx 1.2732\))
- \(a_s^3 / (\tau_s^2 M_p)\) → local gravitational scale
- \(a_s / R_U\) → cosmic dilution
- 4π / (H·K) → combinatorial dilution
- \(\Gamma\) → curvature packing correction
7.5 Newtonian Limit (Weak-Field Approximation)
For \(|v| \ll c\), weak curvature: \[ g_{00} \approx -(1 + 2 \Phi_g / c^2), \quad g_{ij} \approx \delta_{ij} \] Operational Laplacian: \[ R \approx \frac{\nabla^2 T}{T} \rightarrow \nabla^2 \Phi_g \propto - \int \frac{M(x')}{|x - x'|} d^3x' \] Conclusion: Newtonian gravity emerges naturally from handshake-based tension distributions.
7.6 Lepton Masses and Universal Phase Volume
Critical correction (v11.1): PHI (raw volumetric phase) is now fully derived from Section 5.3.2, avoiding circularity.
Raw volumetric phase: \[ \text{PHI} = \pi \sqrt{K-1}, \quad \Xi = \frac{\text{PHI}^2}{\sqrt{3}}, \quad \eta = \frac{H}{K^2} = \frac{132}{144} \] Universal phase volume: \[ \Omega = \frac{\text{PHI}}{\Xi \cdot \eta} \approx 7.517 \] Spinor-locking correction: \[ \text{spinor\_lock} = 1 - \frac{1.5}{K}, \quad C_L = \Omega \cdot \text{spinor\_lock} \approx 6.585 \] Lepton mass formula (d = radial recursion depth): \[ M_\text{lepton}(d) = M_e \exp \Big( C_L (d-1) - C_Q (d-1)^2 \Big) \] Quadratic term \(C_Q\) accounts for impedance from prior recursive layers (Section 5.3.2). Interpretation: Lepton masses are phase-volume shadows of lattice geometry; no arbitrary tuning.
7.7 Breakdown of Geometry — Singularities
When \(|C_{ij}| \to 0 \Rightarrow d_{ij} \to \infty \Rightarrow g_{\mu\nu} \to \text{degenerate}\), handshake starvation occurs (local demand > H = 132). TFP black holes: substrate updates freeze; continuum divergence replaced by stalled dynamics.
7.8 Axiomatic Closure Table
| Physical Quantity | Substrate Basis | Source |
|---|---|---|
| Mass \(M\) | flip frequency / tension | A2, A6, A9 |
| Curvature \(R\) | Laplacian(T)/T | A2, A3, A6 |
| Metric \(g_{\mu\nu}\) | correlation distance \(d_{ij}\) | A2, A3 |
| Einstein-like relation | continuum limit of discrete dynamics | Sections 2–3 |
| Newtonian limit | weak-field expansion, Section 5 anchors | Sections 5, 7.5 |
| G | residual handshake deficit / cosmic dilution | Sections 4, 7.4 |
| Lepton hierarchy | phase volume \(\Omega\), spinor lock | Section 5.3.2, Section 7.6 |
7.9 Bridge to Section 8 — Fields and Unified Action
Section 6: forces emerge as handshake-gradient phenomena. Section 7: gravity emerges as curvature from handshake exhaustion. Section 8 unifies alignment field \(A(x,t)\) and metric \(g_{\mu\nu}\) as classical variables, constructing an action functional that encodes:
- Tension minimization
- Electromagnetic bookkeeping
- Geometric response
All derived from the same finite H = 132-bit substrate.
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