TFP Section 2 (v9)

SECTION 2 — LOCAL DYNAMICS OF BINARY FLOWS (v9.8)

Axiom-Derived Mesoscopic and Continuum Dynamics
By John Gavel

2.0 Overview

This section derives all mesoscopic and continuum-scale dynamics of Temporal Flow Physics (TFP) directly from the primitive axioms of Section 1. No additional assumptions are introduced. All results follow from:

• \( F_i \in \{+1, -1\} \) (Axiom 2)
• adjacency \( i \sim j \) (Axiom 3)
• reflection-based tension minimization (Axiom 6)
• substrate asymmetry \( \varepsilon \) (Axiom 5)
• discrete timestep \( \tau_0 \) (Axiom 9)

Three dynamical levels emerge:

• Level 0: Binary substrate dynamics (§2.1–2.2)
• Level 1: Mesoscopic coarse-grained fields (§2.3)
• Level 2: Continuum drift–diffusion equation (§2.4)

2.1 Deterministic Substrate Dynamics (Level 0)

2.1.1 Local Tension

From Axiom 6, tension measures reflection boundaries between oppositely oriented flows. Since \( F_i, F_j \in \{+1, -1\} \):

\[ |F_i - F_j| = \begin{cases} 0 & \text{if } F_i = F_j \\ 2 & \text{if } F_i \neq F_j \end{cases} \]

Let \( n_i^- \) be the number of anti-aligned neighbors. Then:

\[ T_i = \sum_{j \sim i} |F_i - F_j| = \sum_{j: F_j = -F_i} 2 = 2 n_i^- \tag{2.1} \]

Thus \( T_i \) counts reflection boundaries and is minimized by alignment.

2.1.2 Flip Criterion

Consider flipping \( F_i \to -F_i \). Aligned neighbors become anti-aligned and vice versa, so:

\[ T'_i = 2 n_i^+ \]

where \( n_i^+ \) is the number of previously aligned neighbors. The tension change is:

\[ \Delta T_i = T'_i - T_i = 2(n_i^+ - n_i^-) \tag{2.2} \]

Flip if \( \Delta T_i < 0 \), i.e., if \( n_i^+ < n_i^- \) — the site is in the minority. This is the majority rule (Axiom 6).

2.1.3 Deterministic Update Rule

Per Axiom 9, updates occur in discrete timesteps:

\[ F_i(t + \tau_0) = \begin{cases} -F_i(t) & \text{if } \Delta T_i(t) < 0 \\ F_i(t) & \text{otherwise} \end{cases} \tag{2.3} \]

Properties:

• Binary constraint preserved: \( |F_i| = 1 \) at all times.
• Total tension non-increasing: \( T_{\text{total}}(t + \tau_0) \leq T_{\text{total}}(t) \).
• Domains coarsen irreversibly (Section 2.6.1).

2.2 Stochastic Substrate Dynamics

2.2.1 Metropolis Update Rule

To include thermal fluctuations, introduce effective temperature \( T_{\text{eff}} \geq 0 \). The Metropolis rule ensures detailed balance:

\[ P(\text{flip}) = \begin{cases} 1 & \text{if } \Delta T_i < 0 \\ \exp(-\Delta T_i / T_{\text{eff}}) & \text{if } \Delta T_i \geq 0 \end{cases} \tag{2.4} \]

The equilibrium distribution is Boltzmann:

\[ P_{\text{eq}}(\{F_i\}) \propto \exp(-T_{\text{total}} / T_{\text{eff}}) \]

2.2.2 Mass Density from Reflection Events

A reflection event occurs when \( F_i \) flips. Define the indicator:

\[ R_i(t) = \begin{cases} 1 & \text{if } F_i(t + \tau_0) \neq F_i(t) \\ 0 & \text{otherwise} \end{cases} = \frac{1 - F_i(t) F_i(t + \tau_0)}{2} \tag{2.5} \]

Time-averaged **mass density** is:

\[ M_i = \lim_{W \to \infty} \frac{1}{W} \sum_{n=0}^{W-1} R_i(t_0 + n \tau_0) \in [0,1] \tag{2.6} \]

Since flipping requires \( \Delta T_i < 0 \), high tension \( \Rightarrow \) frequent flipping \( \Rightarrow \) high \( M_i \). Mass is an operational observable.

2.3 Mesoscopic Fields (Level 1)

2.3.1 Coarse-Grained Alignment

Define the alignment over region \( R \):

\[ A_R(t) = \frac{1}{|R|} \sum_{i \in R} F_i(t) \tag{2.7} \]

In the continuum limit (\( R \to x \), \( |R| \gg 1 \)):

\[ A(x,t) = \lim_{R \to x} A_R(t) \]

Let \( N^+ \), \( N^- \) be counts of \( +1 \), \( -1 \) sites. Then:

\[ A_R = \frac{N^+ - N^-}{N^+ + N^-}, \quad F^+ = \frac{N^+}{|R|} = \frac{1 + A}{2}, \quad F^- = \frac{1 - A}{2} \tag{2.8} \]

2.3.2 Mesoscopic Tension Field

Coarse-grain Eq. (2.1):

\[ T(x,t) = \langle T_i \rangle_R = \frac{2}{|R|} \sum_{i \in R} n_i^- \]

For slowly varying \( A(x,t) \), the probability a neighbor is anti-aligned is \( P(F_j \neq F_i) \approx \frac{1}{2}(1 - A^2) + \frac{a^2}{4} |\nabla A|^2 \). Thus:

\[ T(x,t) \approx k_{\text{avg}} \big[1 - A^2(x,t)\big] + \frac{k_{\text{avg}} a^2}{2} |\nabla A(x,t)|^2 \tag{2.9} \]

Since mass \( M \propto T \) (Eq. 2.6), and the bulk term \( 1 - A^2 \) is small for \( |A| \approx 1 \), the dominant contribution is:

\[ M(x,t) \propto |\nabla A(x,t)|^2 \tag{2.10} \]

2.4 Continuum Drift–Diffusion Dynamics (Level 2)

The continuum field obeys:

\[ \frac{\partial A}{\partial t} = D \nabla^2 A + \mathbf{v} \cdot \nabla A - \kappa A + \eta(x,t) \tag{2.11} \]

2.4.1 Diffusion (Axiom 6)

The expected change in \( F_i \) is \( \langle \Delta F_i \rangle = -2 F_i P(\text{flip}) \). For small \( \Delta T_i \):

\[ P(\text{flip}) \approx \frac{1}{2} - \frac{\Delta T_i}{4 T_{\text{eff}}} \]

Using \( \Delta T_i = 2(n_i^+ - n_i^-) = 2 F_i \sum_{j \sim i} F_j \) and \( F_i^2 = 1 \):

\[ \langle \Delta F_i \rangle \approx \frac{1}{T_{\text{eff}}} \sum_{j \sim i} F_j = \frac{k_i}{T_{\text{eff}}} \left[ \frac{1}{k_i} \sum_{j \sim i} (F_j - F_i) + F_i \right] \]

The discrete Laplacian is \( (\nabla^2_{\text{discrete}} F)_i = \frac{1}{k_i} \sum_{j \sim i} (F_j - F_i) \). In the continuum, \( \sum_{j \sim i} (F_j - F_i) \to k_{\text{avg}} a^2 \nabla^2 A \). Thus:

\[ \frac{\partial A}{\partial t} = \frac{k_{\text{avg}} a^2}{T_{\text{eff}} \tau_0} \nabla^2 A + \cdots \quad \Rightarrow \quad D = \frac{k_{\text{avg}} a^2}{T_{\text{eff}} \tau_0} \tag{2.12} \]

2.4.2 Drift (Axiom 5)

Substrate asymmetry \( \varepsilon \) makes boundaries \( F^+|F^- \) and \( F^-|F^+ \) energetically distinct. This biases flipping, creating a net flux proportional to \( \varepsilon |\nabla A| \). The drift velocity is:

\[ \mathbf{v} = \beta \frac{\varepsilon a}{T_{\text{eff}} \tau_0} \hat{\mathbf{n}}_{\text{substrate}} \tag{2.13} \]

where \( \beta \sim \mathcal{O}(1) \) depends on lattice geometry.

2.4.3 Pinning (Axiom 6, emergent)

In aligned regions (\( |A| \approx 1 \)), small perturbations \( \delta A \) increase tension. Linear response gives:

\[ \frac{\partial (\delta A)}{\partial t} \approx -\kappa \delta A \quad \Rightarrow \quad \kappa = \frac{\mu_{\text{eff}}^2}{\tau_0} \tag{2.14} \]

where \( \mu_{\text{eff}} \) is the stability margin (spectral gap of relaxation dynamics).

2.4.4 Noise (Thermal Fluctuations)

Stochastic flips introduce white noise:

\[ \langle \eta(x,t) \rangle = 0, \quad \langle \eta(x,t) \eta(x',t') \rangle = 2 D_{\text{noise}} \delta(x-x') \delta(t-t') \tag{2.15} \]

By the fluctuation-dissipation theorem, \( D_{\text{noise}} = D T_{\text{eff}} / a^d \).

2.5 Speed Limit and Proto-Time

2.5.1 Speed Limit (Axioms 3, 4, 9)

Information propagates one lattice spacing \( a \) per timestep \( \tau_0 \). Thus:

\[ c_{\text{max}} = \frac{a}{\tau_0} \quad \Rightarrow \quad c_{\text{internal}} = 1 \text{ in lattice units} \tag{2.16} \]

2.5.2 Proto-Time from Coarsening (Axiom 6)

Domain merging reduces tension; fragmentation increases it and is suppressed. Define:

\[ S_{\text{coarse}}(t) = -\frac{1}{N} \sum_i T_i(t) \tag{2.17} \]

Since \( T_{\text{total}} \) is non-increasing, \( S_{\text{coarse}}(t) \) is monotonically increasing, defining temporal order. Domain size grows as \( R(t) \sim (D t)^{1/z} \), \( z \geq 2 \).

2.6 Axiomatic Closure

Term	Origin	Axiom
\( \partial A/\partial t \)	Discrete update	A9
\( D \nabla^2 A \)	Tension minimization	A6
\( \mathbf{v} \cdot \nabla A \)	Substrate asymmetry	A5
\( -\kappa A \)	Domain stability	A6 (emergent)
\( \eta \)	Thermal fluctuations	\( T_{\text{eff}} > 0 \)
\( M_i \)	Flip frequency	A2, A6, A9
\( c_{\text{max}} = 1 \)	Discrete causality	A3, A4, A9

All coefficients are substrate-derived:
\( D = \dfrac{k_{\text{avg}} a^2}{T_{\text{eff}} \tau_0} \),    \( \mathbf{v} = \beta \dfrac{\varepsilon a}{T_{\text{eff}} \tau_0} \hat{\mathbf{n}} \),    \( \kappa = \dfrac{\mu_{\text{eff}}^2}{\tau_0} \),    \( c_{\text{max}} = \dfrac{a}{\tau_0} \)

No free parameters. No additional assumptions.

2.7 Bridge to Section 3

Section 2 provides:

• Binary update rules (Eqs 2.3, 2.4)
• Operational mass \( M_i \) (Eq 2.6)
• Continuum field \( A(x,t) \) and master equation (Eq 2.11)
• Geometry-ready ingredients: \( |\nabla A|^2 \propto M \), \( c_{\text{internal}} = 1 \), \( S_{\text{coarse}}(t) \)

Section 3 derives spatial geometry from correlations \( C_{ij}(\tau) \) in \( A(x,t) \), with mass-induced curvature emerging via \( M \propto |\nabla A|^2 \).

All quantities remain dimensionless until calibration (Section 5).