TFP Section 17 (v8)

Section 17 — Quantum-Relativistic Simulation Framework for Emergent Gauge Couplings in Temporal Flow Physics (v8.4)

17.0 Core Objective

Defines a computational and theoretical framework for simulating discrete causal evolution of temporal flow multiplets \(\Psi_i\) in TFP. Focuses on quantum-relativistic behavior: ΔF cluster dynamics, coherence \(C(l)\), residual misalignment \(\beta(l)\), loop holonomy, and local informational friction \(\delta(l)\). Cosmology and large-scale expansion are handled in Section 20.

17.1 Characteristic Units (TFP Calibration)

All variables use TFP characteristic units consistent with Sections 1–5, 15, and 16, explicitly linked to gravitational calibration constant \(C_G\):

• Characteristic Length: \(L_c =\) cluster coherence scale
• Characteristic Time: \(T_c =\) temporal coherence scale
• Characteristic Energy: \(E_c =\) energy per ΔF cluster
• Characteristic Speed: \(c_\text{char} = L_c / T_c\)
• Characteristic Mass: \(M_c = E_c \cdot T_c^2 / L_c^2\)
• Characteristic Action: \(\hbar_c = E_c \cdot T_c\)

Gravitational effective coupling (calibrated, from Sections 15–16):

\(G_\text{eff}(l) = C_G \, C_\text{EM}^* \, \frac{\delta_n(l)}{D_n(l)} \frac{L_c^3}{M_c T_c^2}\)

Where \(\delta_n(l)\) and \(D_n(l)\) are defined in Section 15.6.

Electromagnetic coupling:

\(\alpha_\text{EM} \approx \langle F_\text{topo} \cdot (\delta_n / D_n) \cdot (L_c / T_c) \rangle_\text{clusters} \cdot \kappa_\text{EM}\)

\(F_\text{topo}\) = topological reinforcement factor
\(\kappa_\text{EM}\) = emergent scaling from cluster statistics and stochastic π-resonant kicks

17.2 Temporal Flow Multiplets

Each node \(i\) carries an n-component complex temporal flow multiplet:

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

• Phase encodes gauge degrees of freedom
• Amplitude encodes coupling to gravitational/matter sector
• Bidirectional evolution:

  \(\Psi_i(t + \Delta t) = \Psi_i(t) + \Delta t \left[ -\frac{dV}{d\Psi_i} + C_2 \sum_j w_{ij} (\Psi_j - \Psi_i) + \eta_i^+ - \eta_i^- + \sum_p H_p(i,j) \right]\)
  

• \(V(\Psi_i)\) = local stabilizing potential (Section 16.1)
• \(w_{ij}\) = causal neighbor weight based on \(d_\text{eff}(i,j)\)
• \(\eta_i^\pm\) = stochastic noise mapped to \(\delta_n(l)\) and \(D_n(l)\)
• \(H_p(i,j)\) = holonomy contribution from minimal loops

17.3 Cluster-Level Coherence Extraction

C(l)^2 = (1 / N_l^2) sum_{i,j in cluster} cos^2(theta_i - theta_j) * w_ij
beta(l) = 1 - C(l)^2
C(l)^2 ≈ C_0^2 * (l_0 / l)^d, d = effective spatial dimension

Coherence directly feeds into \(\delta_n(l)\) and phase-alignment stability. ΔF cluster properties (ω_i, R_ij, TF_i) define node-level dynamics.

17.4 Emergent Gauge Coupling Computation

alpha_a(l) ≈ average over coherent clusters of Re[ Psi_i* · U_ij^a · Psi_j · H_p(i,j) · w_ij ]

Where \(U_{ij}^a = \exp(i \theta_{ij}^a)\) (Section 16.4). Friction correction:

Phi_delta(l) = kappa * (delta_n(l)/D_n(l)) * alpha_a * topology_factor(l)
d(alpha_a)/d(log l) = alpha_a^2 * F_a(l) - Phi_delta(l)

17.5 Recursive Osculation Depth with Noise Mapping

d(D_n)/d(log l) = -Gamma_decay(l) * D_n(l) + Gamma_form(l) * [1 - f(delta_n(l), topology_factor(l))]

Links directly to gravitational coupling via \(\delta_n / D_n\). Gamma_decay and Gamma_form derive from stochastic node evolution and phase alignment.

17.6 Classical Domain Formation

Domains form when:

• \(\delta_n(l)\) exceeds threshold → damping of microscopic fluctuations
• topology_factor(l) large → stable modular substructures

Domains produce emergent locality and stable effective couplings \(\alpha_a(l)\) without external collapse rules.

17.7 Emergent Gauge Unification

At high coherence scale l*:
D_1(l*) ≈ D_2(l*) ≈ D_3(l*) = D_unified
alpha_1(l*) ≈ alpha_2(l*) ≈ alpha_3(l*) = alpha_U

Emergent from balanced coherence formation, informational friction, and network topology. First-principles TFP mechanism for gauge unification, including \(\delta_n / D_n\) and \(C_G\).

17.8 Simulation Parameters and Implementation

Evolves discrete ΔF clusters with bidirectional \(\Psi_i(t)\), stochastic noise, topology effects, and π-resonant kicks.

17.8.1 Key Parameters

• ΔF cluster density and coherence seeds
• Phase noise threshold \(\epsilon_\text{phase}(l)\)
• Temporal locality \(\tau_l\)
• Recursive weighting coefficients \(w_n\)
• Informational friction baseline \(\delta_0\) and calibration \(\kappa\)
• Topological Reinforcement Factor \(F_\text{topo}\) via persistent homology (1-cycles, loop density, motif saturation)

17.8.2 Bidirectional Update Rules

Psi_i(t + Δt) = Psi_i(t) + Δt * [ -∂V/∂Psi_i + C2 Σ_j w_ij (Psi_j - Psi_i) + η_i^+ - η_i^- + Σ_p H_p(i,j) ]

17.8.3 π-Resonant Stochastic Kicks

kick_prob(l) = π * < δ_n(l)/D_n(l) * F_topo(l) * (L_c / T_c) >_clusters
converges to π × 10^-3 ≈ 0.0031415

17.8.4 Gauge Coupling Update

alpha_a(l + Δ log l) = alpha_a(l) + [ alpha_a^2 * F_a(l) - Phi_delta(l) ] * Δ log l
Phi_delta(l) = kappa * (delta_n(l)/D_n(l)) * alpha_a * topology_factor(l)

17.8.5 Recursive Depth Update

D_n(l + Δ log l) = D_n(l) + [ -Gamma_decay(D_n, η_i) + Gamma_form * (1 - f(delta_n(l), topology_factor(l))) ] * Δ log l

17.8.6 Holonomy Integration

Loop holonomies \(H_p(i,j)\) included in \(F_a(l)\) feedback; π-phase flips applied probabilistically via kick_prob.

17.9 Summary

• ΔF clusters with bidirectional \(\Psi_i\) evolve under causal, phase-aligned dynamics
• Cluster coherence \(C(l)\), residual misalignment \(\beta(l)\), and \(\delta_n(l)\) regulate emergent physics
• Loop holonomy and causal neighbor weighting explicitly incorporated
• \(\alpha_a(l)\) emerges, runs, and unifies naturally
• \(D_n(l)\) governs classical domain formation
• \(G_\text{eff}(l) \propto C_G \cdot (\delta_n / D_n)\)
• Cosmology and black hole scale effects handled in Section 20