Section 9 — Emergent Conservation Laws and Symmetries (TFP, v11.1)

Section 9 — Emergent Conservation Laws and Symmetries (TFP, v11.1)

Exact and Emergent Symmetries from Causal Flow Constraints

9.0 Overview

Conservation laws and symmetries in TFP emerge directly from causal structure, binary flow rules, and finite substrate constraints (Sections 1–3). No symmetries are postulated; all arise naturally:

• Charge conservation ← binary flow preservation (Axiom 2)
• Parity (P) ← undirected adjacency (Axiom 3)
• Time asymmetry (T) ← irreversible coarsening (Axiom 6)
• CPT symmetry is exact at the substrate level. Apparent violations arise only from coarse-graining over irreversible update histories.

9.1 Charge Conservation

Each site \(i\) has binary flow \(F_i \in \{+1,-1\}\) (Axiom 2). Updates flip nodes while preserving total flow modulo local tension redistribution: \[ \sum_i F_i(t + \tau_0) - \sum_i F_i(t) = \sum_i \Delta F_i \] Flip of a site \(i\): \(F_i \to -F_i \Rightarrow \Delta F_i = -2 F_i\), compensated by neighbors through tension redistribution.

Deterministic limit: total tension non-increasing, net flow conserved.
Stochastic limit: detailed balance ensures \(\langle \Delta F_i \rangle = 0\).

Coarse-grained fields: \[ \rho_\text{dim} = \text{average}(A), \quad J_\text{dim} = \rho_\text{dim} \cdot v_C \] Continuity equation: \[ \frac{\partial \rho_\text{dim}}{\partial t} + \nabla \cdot J_\text{dim} = 0 \] Physical units mapping: \[ \frac{\partial \rho_\text{phys}}{\partial t} + \nabla \cdot J_\text{phys} = 0 \] Interpretation: Charge conservation is exact, derived from binary flow preservation.

9.1.1 Recursive Flow and Directional Identity

Sequence of flips \(F_i^+ \to F_j^-\) defines local chiral state. H = 132 ensures handshake balance over forward/return cycles. Incomplete sequences ("half-handshake") are suppressed by the tension functional. Quantization emerges naturally: each unit of charge corresponds to a complete binary handshake cycle.

9.2 Parity Preservation

Adjacency is undirected: \(i \sim j \iff j \sim i\) (Axiom 3). Local tension is symmetric under site exchange: \[ T_i = \sum_{j \sim i} |F_i - F_j| = T_j \text{ under } i \leftrightarrow j \] Spin circulation may be chiral at the motif level, but the ensemble is parity-symmetric. Correlation function: \[ C_{ij}(\tau) = C_{ji}(\tau) \Rightarrow g_{\mu\nu}, R_{\mu\nu} \text{ parity-even} \] Interpretation: Parity (P) is exact at the substrate level.

9.3 Time Asymmetry and Arrow of Time

Deterministic update rules minimize tension (Axiom 6), making domain coarsening irreversible. Temporal coarse-graining order parameter: \[ S_\text{coarse}(t) = - \frac{1}{N} \sum_i T_i(t) \] Total tension: \[ T_\text{total}(t + \tau_0) \le T_\text{total}(t) \Rightarrow S_\text{coarse}(t + \tau_0) \ge S_\text{coarse}(t) \] Macroscopic arrow of time emerges from causal, irreversible coarsening. Microscopic reversibility restores detailed balance; T symmetry is statistical.

9.4 CPT Theorem

• C (charge conjugation): \(F_i \to -F_i\) preserves tension → exact
• P (parity): undirected adjacency → exact
• T (time reversal): broken macroscopically, restored microscopically

Combined CPT is exact in the substrate. Apparent CPT violations arise from:

• Low-entropy initial conditions
• Coarse-graining over irreversible updates

Correlation functions \(C_{ij}(\tau)\) encode causal influence, not phase, maintaining CPT symmetry.

9.5 Origin of Apparent Symmetry Breaking

Fundamental symmetries are exact; effective breaking is thermodynamic:

• Temporal asymmetry: \(\langle F_i(t) F_j(t+\tau) \rangle \ne \langle F_i(t) F_j(t-\tau) \rangle\) due to coarsening
• Chiral bias: individual motif circulation \(S_p \ne 0\), ensemble symmetric
• Local charge imbalance: \(\sum_i F_i \ne 0\) locally, global charge conserved

Motifs trapped in recursive loops (11/12 handshake deficit) produce frozen thermodynamic bias → explains matter-antimatter asymmetry.

9.5.1 Thermodynamic Floor and \(k_B\)

Substrate handshake efficiency: \[ \eta = \frac{H}{K^2} = \frac{132}{144} = \frac{11}{12} \] Residual fraction: \[ \delta = 1 - \eta = \frac{1}{12} \] Define Boltzmann constant operationally: \[ k_B \approx \frac{E_c \cdot \delta}{T_c} = \frac{E_c}{12 \, T_c} \] Example: \(E_c \approx 0.217 \, \text{GeV}\), \(T_c \approx 5.6 \times 10^{-25} \, \text{s} \Rightarrow k_B \approx 1.38 \times 10^{-23} \, \text{J/K}\)

Interpretation: Thermodynamic entropy measures unsynchronized bit-flips; \(k_B\) arises naturally from the 132-bit substrate.

9.6 Axiomatic Closure

Symmetry / Conservation	Substrate Origin	Axiom
Charge Conservation	Binary flow preservation	A2
Parity (P)	Undirected adjacency	A3
Time Asymmetry (T)	Irreversible coarsening	A6
CPT	Combination of C, P, T	A2, A3, A6

9.7 Bridge to Section 10 — Quantization

Conservation laws are flow constraints. Section 10 derives discrete spectra from these constraints. A single handshake (F⁺ → F⁻) defines Planck constant \(h\) as energy per bit-flip cycle. Heisenberg uncertainty emerges naturally as address jitter from finite 132-bit handshake routing. This links classical field conservation (Sections 8–9) to the quantum of action, completing the bridge from coarse-grained observables to discrete spectra.