Section 17 — Quantum–Relativistic Simulation Framework For Emergent Gauge Couplings In Temporal Flow Physics (TFP) (v11.0)

Section 17 — Quantum–Relativistic Simulation Framework For Emergent Gauge Couplings In Temporal Flow Physics (TFP) (v11.0)

By John Gavel

17.0 PURPOSE AND SCOPE

This section defines the computational and analytical framework required to simulate the discrete causal evolution of the Temporal Flow Physics (TFP) substrate and to extract emergent quantum, gauge, and gravitational behavior from first principles.

No new axioms are introduced. All quantities defined here are coarse-grained observables derived from:

• Section 1 (axioms)
• Section 2 (binary dynamics and proto-time)
• Section 3 (emergent geometry)
• Sections 12–16 (coherence, holonomy, gravity, unification)

This section bridges ontology to validation. Cosmology is deferred to Section 20.

17.1 DeltaF Clusters: Derived Dynamical Units

From Axiom 2, node states are binary:

\(F_i(t) \in \{+1, -1\}\)

Local relational difference is defined by XOR:

\(\Delta F_{ij}(t) = F_i(t) \oplus F_j(t)\)

Persistent relational differences generate stable motifs when decoherence is suppressed (Sections 15.5–15.7).

Definition: A DeltaF cluster is a connected set of nodes satisfying:

• Phase coherence condition: \(C(l) \ge C_{\text{thresh}}\)
• Adjacency condition: Nodes separated by no more than \(l_{\text{max}}\) adjacency steps

Cluster dynamics:

• Clusters merge if overlapping nodes satisfy both conditions
• Clusters split if coherence drops below threshold

Clusters are informational constructs. Spatial interpretation emerges only after closure (Section 3).

17.2 Characteristic Units (Emergent Calibration)

Characteristic units are not fundamental constants. They are normalization scales extracted from steady-state cluster statistics under finite substrate capacity (Section 16.11).

Define:

• Characteristic length: \(L_c\)
• Characteristic time: \(T_c\)
• Characteristic energy: \(E_c\)

Derived quantities:

• Characteristic action: \(\hbar_c = E_c \cdot T_c\)
• Characteristic speed: \(c_{\text{char}} = L_c / T_c\)
• Characteristic mass: \(M_c = \frac{E_c T_c^2}{L_c^2}\)

17.3 Emergent Gravitational Coupling

From Section 16.3, gravity arises from curvature in causal correlations induced by accumulated relational activity.

Define:

• \(\delta_n(l) =\) local informational friction
• \(D_n(l) =\) n-fold recursive stability (Section 17.8)

Scale-dependent gravitational coupling:

\[ G_{\text{dim}}(l) = \frac{L_c^3}{M_c T_c^2} \cdot \frac{\delta_n(l)}{D_n(l)} \cdot \kappa_G \]

If \(\delta_n / D_n \to 1\), then \(G_{\text{dim}} \to G_{\text{observed}}\). Gravity emerges as a classical correlation phenomenon; no quantization is assumed.

17.4 Emergent Electromagnetic Coupling

Axiomatic basis: closed causal loops generate holonomy (Section 16.4). Gauge structure arises from loop misalignment, not force mediation.

Definition:

\[ \alpha_{\text{em}}(l) \propto \langle H_p \cdot C_\theta(l)^2 \cdot (\delta_n(l)/D_n(l)) \cdot F_{\text{topo}} \cdot \kappa_{\text{EM}} \rangle \]

Where:

• \(H_p =\) loop holonomy
• \(C_\theta(l) =\) phase coherence
• \(F_{\text{topo}} =\) topological reinforcement (persistent homology)
• \(\delta_n / D_n =\) informational friction

Holonomy is the primary source; coherence modulates stability.

17.5 Temporal Flow Multiplets (Coarse-Grained Form)

Binary node histories admit statistical compression:

\(\Psi_i(t) = [ F_i^{(1)}(t), F_i^{(2)}(t), \dots, F_i^{(n)}(t) ]\)

Statistical evolution equation:

\[ \Psi_i(t + \Delta t) = \Psi_i(t) + \Delta t \cdot \Big[ -\frac{dV}{d\Psi_i} + C_2 \sum_j w_{ij} (\Psi_j - \Psi_i) + \eta_i \Big] \]

Noise constraint: \(|\eta_i| \le \Omega \cdot \Psi\)

17.6 Cluster-Level Coherence

For cluster of size \(N_l\):

\[ C(l)^2 = \frac{1}{N_l^2} \sum_{i,j} \cos^2(\theta_i - \theta_j) \cdot w_{ij}, \quad \beta(l) = 1 - C(l)^2 \]

Recursive scaling:

\[ C(l)^2 \approx C_0^2 \left( \frac{l_0}{l} \right)^d \]

17.7 Emergent Gauge Coupling Computation

For gauge sector \(a\):

\[ \alpha_a(l) \approx \langle \text{Re} [ \Psi_i^* \cdot U_{ij}(a) \cdot \Psi_j \cdot H_p(i,j) \cdot w_{ij} ] \rangle \]

Where \(U_{ij}(a) = \exp(i \theta_{ij}(a))\)

Scale evolution:

\[ \frac{d \alpha_a}{d \log l} = \alpha_a^2 F_a(l) - \Phi_\delta(l), \quad \Phi_\delta(l) = \kappa \cdot \frac{\delta_n(l)}{D_n(l)} \cdot \alpha_a \cdot \text{topology\_factor}(l) \]

17.8 Recursive Osculation Depth

\(D_n(l)\) = n-fold phase-locked recursion stability

Evolution equation:

\[ \frac{d D_n}{d \log l} = -\Gamma_{\text{decay}}(l) \cdot D_n + \Gamma_{\text{form}}(l) \cdot [1 - f(\delta_n(l), \text{topology\_factor}(l))] \]

17.9 Classical Domain Formation

Classical domains emerge when \(\delta_n(l)\) exceeds fluctuation threshold and topology_factor stabilizes recursion.

17.10 Emergent Gauge Unification

At coherence equilibrium scale \(l_\star\):

\(D_1(l_\star) \approx D_2(l_\star) \approx D_3(l_\star) = D_{\text{unified}}\)
\(\alpha_1(l_\star) \approx \alpha_2(l_\star) \approx \alpha_3(l_\star) = \alpha_U\)

17.11 Simulation Control Parameters (Non-Physical)

• Initial cluster density
• Phase noise threshold \(\epsilon_{\text{phase}}\)
• Temporal locality \(\tau_l\)
• Recursive weights \(w_n\)
• Informational friction baseline \(\delta_0\)
• Holonomy extraction window

No parameter tunes physical outcomes arbitrarily.

17.12–17.14 Icosahedral Charge, Mass, and Scorecards

All charge and mass relations arise from topological occupation of the icosahedral substrate. Charge is phase-locked occupancy, not additive flow. See Sections 12–16 for derivations.

17.15 Finite Hardware Tension (132-Bit Constraint)

Define hardware tension factor:

\(\tau_{\text{HW}} = \theta_{\text{experimental}} / \theta_{\text{ideal}}\)

Deviations encode:

• Finite bit capacity
• Phase-volume compression
• Stochastic update noise

17.16 The Top Quark — Double Saturation Limit

Mass relation:

\(M_{\text{Top}} \approx M_H + M_Z \cdot \Psi\)

17.17 W and Z Bosons — Latency Thresholds

Z boson: Pure volumetric obstruction
W boson: Obstruction plus parity-alignment cost

SECTION 17 SUMMARY

Section 17 provides a complete, first-principles simulation framework for extracting quantum, relativistic, and gauge behavior from the TFP substrate. No fields, forces, or symmetries are assumed. All observed physics arises from:

• Binary difference
• Finite adjacency
• Recursive coherence
• Holonomy
• Finite computational capacity