TFP Section 9 (v9)

SECTION 9 — EMERGENT CONSERVATION LAWS AND SYMMETRIES (v9.2)

Exact and Emergent Symmetries from Causal Flow Constraints
By John Gavel

9.0 Overview

Conservation laws and symmetries in Temporal Flow Physics emerge from the causal structure of binary flow dynamics (Sections 2–3). No symmetries are postulated — all derive from:

• Charge conservation: from binary flow preservation (Axiom 2)
• Parity preservation: from undirected adjacency (Axiom 3)
• Time asymmetry: from irreversible coarsening (Axiom 6)

CPT is an exact symmetry of the fundamental substrate. Apparent violations arise only from coarse-graining over irreversible histories (Section 2.6), not from microscopic dynamics.

9.1 Charge Conservation

From Axiom 2, each site has binary flow \( F_i \in \{+1, -1\} \). Under the update rule (Section 2.1.3), flips conserve total flow modulo local adjustments:

\[ \sum_i F_i(t + \tau_0) - \sum_i F_i(t) = \sum_i \Delta F_i \tag{9.1} \]

When site \( i \) flips (\( F_i \to -F_i \)), \( \Delta F_i = -2F_i \), but this is compensated by neighbor tension changes (Section 2.1.1). In the deterministic limit, total tension is non-increasing (Section 2.1.3), but **net flow is conserved** because no creation/annihilation occurs. In the stochastic limit (Section 2.2.1), detailed balance ensures:

\[ \langle \Delta F_i \rangle = 0 \quad \text{(on average)} \tag{9.2} \]

For coarse-grained charge density \( \rho_{\text{dim}} = \bar{A} \) (Section 4.5.1) and current \( \mathbf{J}_{\text{dim}} = \rho_{\text{dim}} \mathbf{v}_C \) (Section 6.1.2), this yields exact continuity:

\[ \frac{\partial \rho_{\text{dim}}}{\partial t} + \nabla \cdot \mathbf{J}_{\text{dim}} = 0 \tag{9.3} \]

In physical units (Section 5):

\[ \frac{\partial \rho_{\text{phys}}}{\partial t} + \nabla \cdot \mathbf{J}_{\text{phys}} = 0 \tag{9.4} \]

Charge conservation is exact, arising from flow preservation (Axiom 2).

9.2 Parity Preservation

Adjacency is undirected: \( i \sim j \iff j \sim i \) (Axiom 3). The tension (Section 2.1.1) is symmetric:

\[ T_i = \sum_{j \sim i} |F_i - F_j| = T_j \quad \text{under } i \leftrightarrow j \tag{9.5} \]

Spin circulation (Section 3.5) is chiral but parity-symmetric: for every left-handed motif, a right-handed counterpart exists with equal probability. The correlation function satisfies:

\[ C_{ij}(\tau) = C_{ji}(\tau) \tag{9.6} \]

Thus the metric \( g_{\mu\nu} \) (Section 3.2) and curvature \( R_{\mu\nu} \) (Section 7.3) are parity-even. Parity is an exact symmetry of the substrate.

9.3 Time Asymmetry and the Arrow of Time

The deterministic update rule (Section 2.1.3) minimizes tension (Axiom 6), making domain coarsening irreversible (Section 2.6.1). Define the temporal order parameter:

\[ S_{\text{coarse}}(t) = -\frac{1}{N} \sum_i T_i(t) \tag{9.7} \]

From Section 2.1.3, total tension is non-increasing: \( T_{\text{total}}(t + \tau_0) \leq T_{\text{total}}(t) \), so:

\[ S_{\text{coarse}}(t + \tau_0) \geq S_{\text{coarse}}(t) \tag{9.8} \]

This defines a thermodynamic arrow of time: earlier states have lower \( S_{\text{coarse}} \). However, the microscopic update rule is not time-reversal invariant only because it implements a directional minimization (Axiom 6). In the stochastic limit (Section 2.2.1), detailed balance restores microscopic reversibility on average. Thus:

• Microscopic T symmetry: emergent in stochastic regime (Section 2.2)
• Macroscopic T violation: exact in deterministic regime (Section 2.6)

Critically, no phase is required for this arrow: it arises from causal path dependence in binary choices (Section 3.1.2), not from complex dynamics.

9.4 CPT Theorem

Combining the above:

• C (charge conjugation): \( F_i \to -F_i \) leaves tension invariant (Section 2.1.1) → exact
• P (parity): adjacency undirected (Axiom 3) → exact
• T (time reversal): broken macroscopically by coarsening (Section 2.6), restored microscopically by detailed balance (Section 2.2)

Thus the combined CPT transformation is a symmetry of the fundamental dynamics. Apparent CPT violation can only arise from:

• Initial conditions (low-entropy state)
• Coarse-graining over irreversible processes (Section 3.1.2)

No fundamental CPT violation exists in the substrate. The correlation function \( C_{ij}(\tau) \) encodes causal influence, not phase — and is CPT-symmetric by construction.

9.5 Origin of Apparent Symmetry Breaking

While fundamental symmetries are exact, effective breaking

• Temporal asymmetry: \( \langle F_i(t) F_j(t+\tau) \rangle \neq \langle F_i(t) F_j(t-\tau) \rangle \) for \( \tau > 0 \) due to coarsening (Section 2.6)
• Chiral bias: In a given motif, spin circulation \( S_p \) may be nonzero, but the ensemble is parity-symmetric
• Charge imbalance: Local regions may have \( \sum F_i \neq 0 \), but global charge is conserved

These are thermodynamic effects, not violations of microscopic laws. They vanish in equilibrium or over infinite time averages.

9.6 Axiomatic Closure

Symmetry	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

Section 9 provides:

• Exact charge conservation from flow preservation
• Exact parity from undirected adjacency
• Emergent time arrow from irreversible coarsening
• Preserved CPT in fundamental dynamics, with thermodynamic effective breaking

Section 10 quantizes these classical symmetries by identifying flow multiplets with quantum states. Phase emerges as a coarse-grained record of causal path dependence (Section 3.1.2), recovering superposition, uncertainty, and discrete spectra from substrate eigenmodes.