Section 5 — Dimensional Correspondence and Physical Anchoring (TFP, v11.1)
From Substrate to Unified Mass Laws
By John Gavel
5.0 Overview
Section 4 described particles, mass, charge, and spin entirely in dimensionless lattice units of the K = 12 adjacency substrate. Section 5 bridges these to physical units, showing how dimensionless quantities (\(H\), \(K\), \(\delta\), \(\tau_0\), \(L_c\), etc.) map onto measurable energy, time, and length scales.
To assign absolute units to the substrate, one fixed physical scale is required: the proton anchor \(\Psi\). This is necessary and sufficient:
- Proton is stable, volumetric, and universal
- Saturates the handshake budget \(H = 132\)
- Defines the energy scale of maximal single-particle coherence
Once \(\Psi\) is fixed, all other particle masses (leptons, mesons, baryons) follow from lattice combinatorics alone — no free parameters.
Mass perspective: Mass is a measure of resistance to recursive handshake flow. Per-step impedance \(Q\) represents the delay or latency of causal updates propagating through the lattice:
\[ m \propto \tau_\text{eff} \propto \frac{1}{\text{update rate}} \]
5.1 Justification for the Proton Anchor \(\Psi\)
Dimensionless lattice quantities determine only ratios, not absolute scales. Examples:
- Proton–electron mass ratio: \(\frac{M_p}{M_e} \approx \frac{H_\text{effective} \cdot 1.25}{\delta}\)
- Baryon/meson hierarchy arises from volumetric vs. surface recursions
Mapping to MeV requires a reference point: the proton.
Definition: \(\Psi = 938.272 \text{ MeV}\) represents the lattice proton motif.
All masses derive from \(\Psi\) via combinatorics and recursive impedance \(Q\).
Interpretation: \(\Psi\) encodes the physical manifestation of the substrate’s maximum single-particle handshake occupancy.
5.2 The Unified Substrate Constant
- Adjacency per node: \(K = 12\)
- Combinatorial address space: \(K^2 = 144\)
- Handshake budget: \(H = 132\) (maximum causal updates per node per tick)
Note: H is not derived from K. Handshakes are causal tokens constrained by concurrency. 3D volumetric extension: additional 8 states for corner addressing → \(H_\text{total} = H + 8 = 140\)
Substrate efficiency: \[ \eta_\text{sub} = \frac{H}{K^2} = \frac{11}{12} \approx 0.9167 \]
Interpretation: \(K^2\) = combinatorial adjacency possibilities per node, \(H\) = usable handshake flux per tick, \(\eta_\text{sub}\) = global coherence efficiency affecting mass, binding, and gravitational bookkeeping.
5.3 The Unified Mass Law (Q-Unified Form)
Mass is resistance to recursive substrate flow, quantified by per-step impedance \(Q\).
- Hadrons (additive Q): \[ M_\text{hadron} = \Psi \times Q_\text{hadron}, \quad Q_\text{hadron} = \sum Q_\text{quark} + Q_\text{phase volume} \]
- Leptons (resonant Q): \[ M_\text{lepton}(d) = \frac{\Psi}{1836} \exp(Q(d)) \] \[ Q(d) = \text{Linear\_Term}(d) - \text{Quadratic\_Term}(d) \] \[ \text{Linear\_Term}(d) = C_L \cdot (d-1), \quad \text{Quadratic\_Term}(d) = C_Q \cdot (d-1)^2 \]
5.3.1 Additive Gap Expression (Hadrons)
- Baryons: \(\text{Topology}_\text{baryon} = \left(\frac{N_\text{total}}{N_\text{core}}\right)^{1.25}, N_\text{core} = 6\)
- Exponent encodes volumetric closure penalty; torque = synchronization cost between nested shells
- Mesons: \(\text{Topology}_\text{meson} = \frac{G_\text{sum}}{H} \cdot \frac{\chi}{\pi_s}\)
5.3.2 Reciprocal Resonance (Leptons)
Axis-centered resonances of handshake flow. Mass arises from exponential accumulation of Q per recursive step.
| Generation (d) | delta = d−1 | Linear_Term(d) | Quadratic_Term(d) | Q(d) = Linear−Quadratic | M_lepton(d) [MeV] |
|---|---|---|---|---|---|
| 1 (Electron) | 0 | 0 | 0 | 0 | 0.511 |
| 2 (Muon) | 1 | 6.585 | 1.255 | 5.330 | 105.49 |
| 3 (Tau) | 2 | 13.170 | 5.020 | 8.150 | 1769.64 |
Formula applied: \[ M_\text{lepton}(d) = \frac{\Psi}{1836} \exp(Q(d)) \]
5.3.3 Mass Scaling via Jacobian and Phase Coherence
Local Jacobian \(J\) sets stability, eigenvalues \(\lambda_j\) set coherence exponent \(\Omega\):
\[ \Omega = -\frac{1}{N_\text{path}} \sum_j \ln|\lambda_j|, \quad \text{Omega_factor} = \exp(\pi (\Omega_0 - \Omega)) \]
Combined symbolic mass: \[ M_\text{particle} = \Psi \times Q_\text{particle} \times \text{Omega_factor} \]
5.3.4 Fine-Structure Constant \(\alpha\) as Residual Q
\(\alpha\) measures the fraction of handshake flux lost to geometric projection and holonomy. It is a residual impedance factor in the substrate:
\[ \alpha^{-1} = \frac{K \cdot \Psi_\text{sph}}{H \cdot (K-1)} + (2\pi + \Phi) \]
Stepwise numeric result: \(\alpha^{-1} \approx 137.036\)
5.4 Structural Synchrony — Axial Projection Factor
\[ \Xi = \frac{\Phi^2}{\sqrt{3}}, \quad \Xi_\text{effective} = \Xi \cdot \eta_\text{sub} \]
Torque = \((\pi/2) \sqrt{\text{Gear_Ratio}} \cdot \Xi_\text{effective} \cdot (1 + 1/K)\)
5.5 Saturation and Curvature Limit
\(\Gamma_c = 4/\pi \approx 1.2732\) — Reflects pentagonal deficit in 3D packing, setting baryonic core ceiling.
5.6 Derived Substrate Scales (Explicit Anchoring)
| Quantity | Value / Formula |
|---|---|
| \(\Psi_J\) | 1.503 × 10^-10 J |
| f_s | \(\Psi / h \approx 2.268 × 10^{23} Hz\) |
| \(\tau_s\) | 1 / f_s ≈ 4.408 × 10^-24 s |
| a_s | c_char × τ_s ≈ 1.321 × 10^-15 m |
| c_char | L_c / τ_s (emergent from substrate updates) |
5.7 Nuclear Magic Numbers and Binding (Revised v12.4)
Shell capacity (TFP combinatorial): \[ N_n = 10 n^2 + 2 \] n = 1 → 12, n = 2 → 42, n = 3 → 92
Cumulative filled count: \[ A(n) = \frac{1}{3} (2n+1)(n^2+n+3) \]
Handshake-minimization principle: \[ S = \sum_i \left[ H - \frac{1}{K} \sum_{j\sim i} \sigma_{ij} \right] \]
Alpha-cluster recursion: \[ N_\alpha(k) = \frac{k(k+1)(k+2)}{6}, \quad k=1→1, k=2→4, k=3→10, k=4→20 \]
Deuteron binding: \[ B_d = \frac{1}{3} \frac{\Psi}{\alpha_\text{inverse}} \approx 2.282 \text{ MeV} \] Neutron–proton mass splitting: \[ \Delta M_{np} = \Psi \frac{1}{H} \kappa \approx 1.29 \text{ MeV}, \quad \kappa \approx 0.18 \]
Gravity (symbolic, emergent from Q / substrate latency): \[ G_\text{dim}(l) \propto \frac{a_s^3}{\tau_s^2 M_p} \frac{a_s}{R_U} \frac{4 \pi}{H K} \Gamma_\text{correction} \]
5.8 Causal Chain Table (Numeric, Q-unified)
| Step | Input / Anchor | Q contribution | Output / Mass [MeV] |
|---|---|---|---|
| Proton | \(\Psi = 938.272\) | N/A | 938.272 |
| Electron | \(\Psi / 1836\) | Q(1)=0 | 0.511 |
| Muon | \(\Psi / 1836\) | Q(2)=5.330 | 105.49 |
| Tau | \(\Psi / 1836\) | Q(3)=8.150 | 1769.64 |
| Neutron | \(\Psi\), additive Q_hadron | sum Q_quarks + phase | 940.64 |
| Lambda | \(\Psi\), additive Q_hadron | sum Q_quarks + phase | 1162.97 |
| Omega- | \(\Psi\), additive Q_hadron | sum Q_quarks + phase | 1638.94 |
| Deuteron | \(\Psi / \alpha_\text{inverse}\) | residual Q | 2.282 |
| Gravity | a_s³ / (τ_s² M_p) × a_s / R_U × 4π / (H K) | cumulative residual Q | 6.674×10⁻¹¹ |
5.9 Axiomatic Closure Map
| Physical Quantity | Substrate Basis | Source |
|---|---|---|
| Mass | Handshake latency (H_total, Q) | A10 / Section 3 |
| Length / Time | Lattice pitch a_s, tick τ_s | A9 / Section 2 |
| Alpha | Residual per-step Q (η_sub, R_geom−H) | Section 5.3.4 |
| Gravity | Handshake exhaustion + Q residual | Section 5.7 |
| Nuclear binding | Partial handshake sharing, volumetric Q | Section 5.7 |
5.10 Bridge to Section 6
With \(\eta_\text{sub}\), \(\Phi\), Projective_Offset, \(\tau_s\), \(a_s\), \(\Gamma_c\), and \(\Psi\) defined, force dynamics emerge as gradients of recursive handshake flow \(Q\).
Electromagnetism, strong, and weak interactions arise naturally as bookkeeping consequences of finite \(H = 132\) and \(K = 12\) adjacency.
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