Section 2 — Local Dynamics of Binary Relations (TFP) (v12.0)

Section 2 — Local Dynamics of Binary Relations (TFP v12.0)

By John Gavel

2.0 Overview

Section 1 established the irreducible primitives:

Discrete sites with binary relational states F_i ∈ {+1, −1}

Primitive adjacency relations i ~ j

Relational difference as the sole primitive observable

Discrete update steps (proto-time ticks)

Finite relational capacity per site

Determinacy via overlapping relational closure

Section 2 derives all local dynamics from these primitives alone. No coordinates, no metric, no spatial embedding, no gradient operators, no dimensionality are introduced here. Every quantity is expressed purely in terms of sites, adjacency relations, binary states, and update counts.

Three levels emerge naturally:

Level 0: Binary substrate dynamics (§2.1–2.4)

Level 1: Relational accumulation and memory (§2.5–2.6)

Level 2: Emergent proto-temporal structure (§2.7–2.9)

Section 3 will derive geometry from the correlation structure produced here.

2.1 Binary Relational States and Primitive Tension

Each site i carries a binary state:

F_i(t) ∈ {+1, −1}

where t is the discrete update index (proto-time counter, not physical time).

For any adjacent pair (i, j) with i ~ j, define the primitive relational difference:

x_ij(t) = F_i(t) XOR F_j(t) ∈ {0, 1}

x_ij = 0 means agreement. x_ij = 1 means disagreement.

Local tension at site i:

T_i(t) = Σ_{j ~ i} x_ij(t) = number of disagreeing neighbors

Since F_i, F_j ∈ {±1}, this is equivalent to:

T_i(t) = n_i^−(t)

where n_i^− is the count of anti-aligned neighbors. Alignment minimizes T_i.

2.2 Primitive Update Rule (Action Minimization)

From the discrete action functional on the abstract adjacency graph:

S = Σ_{(i,j) edges} (1/2)(F_i − F_j)²

the minimal admissible operation that preserves global charge Q = Σ_i F_i and reduces local action is a simultaneous pairwise flip of an adjacent disagreeing pair.

Flip criterion for pair (i, j) with i ~ j:

Compute the local action change if both i and j flip simultaneously:

ΔS_{ij} = −8 × (A_i + A_j − D_i − D_j)

where:

A_i = number of agreements at site i, excluding j

D_i = number of disagreements at site i, excluding j

The pair flips if and only if:

ΔS_{ij} < 0 AND F_i ≠ F_j

Equivalently: flip if combined disagreements exceed combined agreements.

Single-site flips are forbidden — they violate charge conservation (ΔQ ≠ 0) and are not minimal admissible operations.

Deterministic update rule:

For each adjacent pair (i, j):

If A_i + A_j < D_i + D_j:

F_i(t + τ₀) = −F_i(t)

F_j(t + τ₀) = −F_j(t)

Else:

F_i(t + τ₀) = F_i(t)

F_j(t + τ₀) = F_j(t)

Properties:

Binary constraint preserved: |F_i| = 1 at all times

Global charge conserved: ΔQ = 0 per flip

Total tension non-increasing: T_total(t + τ₀) ≤ T_total(t)

Purely local: depends only on immediate adjacency shell

Priority ordering: When multiple pairs are simultaneously admissible, process in order of decreasing |ΔS_{ij}| (steepest descent). Finite capacity H limits concurrent flips (derived in §2.10).

2.3 Stochastic Extension

To include substrate fluctuations, introduce effective temperature T_eff ≥ 0. The Metropolis acceptance rule generalizes the deterministic criterion:

P(flip pair i,j) = 1                           if ΔS_{ij} < 0

P(flip pair i,j) = exp(−ΔS_{ij} / T_eff)      if ΔS_{ij} ≥ 0

Temperature regimes:

T_eff = 0: fully deterministic, steepest descent

T_eff → ∞: random pairwise flips, no ordering

T_eff ~ O(1): critical fluctuations, most physically interesting

The equilibrium distribution is:

P_eq({F_i}) ∝ exp(−T_total / T_eff)

Detailed balance is satisfied. The system is ergodic for T_eff > 0.

2.4 Emergent Mass and Directed Transport

2.4.1 Flip indicator and proto-mass

Define the flip indicator at site i:

R_i(t) = 1 if F_i(t + τ₀) ≠ F_i(t), else 0

Equivalently:

R_i(t) = (1 − F_i(t) F_i(t + τ₀)) / 2

Time-averaged flip rate (proto-mass density):

M_i = lim_{W→∞} (1/W) Σ_{n=0}^{W−1} R_i(t₀ + n τ₀)

M_i ∈ [0, 1], dimensionless.

Interpretation: Mass is operational — it counts sustained relational activity. Sites with higher M_i are "heavier" in the sense that they undergo more persistent state changes. High tension → frequent flipping → high M_i.

2.4.2 Directed transport bias (proto-velocity)

Flip activity need not be symmetric across adjacency. Define signed transport across edge (i → j):

T_ij(t) = R_i(t) · x_ij(t)

This counts a flip at i that disagrees with neighbor j — a handoff attempt toward j.

Net propagation bias at site i:

v_i(t) = (1 / |Adj(i)|) · Σ_{j ~ i} sgn(i → j) · T_ij(t)

where sgn(i → j) is a relational orientation label (purely bookkeeping, not spatial direction).

v_i = 0: no net directional transport (stationary motif)

v_i ≠ 0: directional bias in flip routing across the adjacency shell

2.4.3 Proto-momentum

p_i(t) = M_i · v_i(t)

Units: [flips/tick] × [steps/tick] = [flip·steps / tick²] (substrate bookkeeping units).

Momentum is conserved locally up to boundary flux and saturation-induced stutter:

Σ_{i ∈ Ω} p_i(t + τ₀) − Σ_{i ∈ Ω} p_i(t) = −Φ_boundary(t) + S_stutter(t)

where S_stutter captures handshake saturation converting directed transport into local latency.

2.5 Accumulation, Memory, and Proto-Time

2.5.1 Accumulated relational difference

Local differences accumulate across update steps. Define:

A_ij(t) = Σ_{n=0}^{t} x_ij(n) = running count of disagreements between i and j

This is the first instance of memory — sites carry history of their relational activity without any spatial embedding.

2.5.2 Proto-time ordering

Each discrete update tick defines a relational ordering:

t ≺ t+1

This is not physical time — it is a counter of update cycles. Causal ordering emerges from the accumulation structure:

State(t) determines State(t+1) via the flip rule

This dependency is the primitive causal arrow

A_ij(t) is monotonically non-decreasing, providing a local order parameter

Define the coarsening order parameter:

S_coarse(t) = −(1/N) Σ_i T_i(t)

Since T_total is non-increasing, S_coarse is monotonically non-decreasing. This defines temporal ordering without presupposing a time coordinate.

2.5.3 Why this is not the arrow of time

S_coarse increasing is not "time flowing forward." It is:

Causal asymmetry: past states determine future states, not vice versa

Change inevitability: tension forces updates; frozen boundary states are forbidden

Statistical irreversibility: domain coarsening is overwhelmingly favored over fragmentation

The update rule is locally reversible (F_i can flip ±1 ↔ ∓1). Global irreversibility is statistical, arising from the combinatorial suppression of fragmentation events.

2.6 Domain Structure and Coarsening

Under tension minimization, disagreeing pairs flip toward agreement. This drives domain coarsening:

Domain merging (favored): adjacent regions with matching orientation reduce boundary count, decreasing T_total

Domain fragmentation (suppressed): creating new disagreeing bonds increases T_total; probability scales as exp(−n·ΔT/T_eff) for n new boundaries

Result: net monotonic coarsening even under stochastic updates.

Coarsening dynamics:

For T_eff ~ 0 (deterministic):

R_domain(t) ~ √(D · t)    (Allen-Cahn type growth)

For T_eff > 0 (stochastic):

R_domain(t) ~ (D · t)^{1/z},  z ≥ 2

where D is an effective rate derived from the flip probability and adjacency structure, and R_domain is measured in adjacency steps (not physical distance).

Important: D here is not a spatial diffusion coefficient — it is a rate of relational domain growth measured in graph hops per update tick. Spatial interpretation awaits Section 3.

2.7 Relational Correlations (Proto-Geometric Seed)

Define the temporal cross-correlation between sites i and j:

C_ij(τ) = ⟨x_ij(t) · x_ij(t + τ)⟩ = ⟨(F_i(t) XOR F_j(t)) · (F_i(t+τ) XOR F_j(t+τ))⟩

This measures how persistently i and j maintain a disagreement relationship across τ update steps. It is a purely relational quantity — no coordinates, no distance.

Directed transport correlation:

T_{j→i}(τ) = C_ij(τ) / √(C_ii(0) · C_jj(0))

Symmetrized transport:

T_{i↔j} = √(|T_{i→j}(τ*)| · |T_{j→i}(τ*)|)

where τ* maximizes |T_{j→i}(τ)|.

Operational relational distance (proto-distance):

d_ij = −log(T_{i↔j})

This is not yet physical distance — it is a measure of relational dissimilarity between sites. Sites that persistently agree (low x_ij) have small d_ij. Sites that persistently disagree have large d_ij.

Section 3 will show how these relational distances organize into a metric structure.

2.8 Holonomy and Relational Curvature

For any triple of mutually adjacent sites {a, b, c} with a ~ b, b ~ c, c ~ a, define the phase lag along each directed edge:

φ_{j→i}(τ) = arg(C_ij(τ*))

Local holonomy around the triple:

H_{abc} = φ_{a→b} + φ_{b→c} + φ_{c→a}    (mod 2π)

H_{abc} ≠ 0 indicates relational phase twisting around the triple — the seed of curvature.

This quantity is entirely relational: it measures whether phase information circulating around a triple of sites returns to its starting value. Non-zero holonomy implies the relational structure is not flat.

This is the proto-geometric precursor to curvature. No metric, no Riemann tensor, no spatial embedding — just phase consistency around relational triples.

2.9 Stability, Coherence, and Informational Friction

2.9.1 Jacobian stability

Linearize the local update map G_i around a stable fixed point F*:

J_ij = ∂G_i / ∂F_j |_{F*}

Informational friction (stability margin):

δ_i = 1 − ρ(J_i)

where ρ(J_i) is the spectral radius of the local Jacobian.

δ_i > 0: perturbations decay, site is stable

δ_i = 0: marginal stability

δ_i < 0: local instability, amplification

2.9.2 Coherence and topology factor

For any set of adjacent triples forming relational loops, define the topology factor:

TF = (1/|V|) · Σ_{loops P} Π_{(i,j) ∈ P} α_ij

where α_ij is the time-averaged agreement rate on edge (i,j):

α_ij = 1 − ⟨x_ij(t)⟩

TF measures how strongly relational loops reinforce coherent patterns. High TF means circular relational chains tend to close consistently.

2.9.3 Coherence condition

Stable coherent motifs persist when:

δ · TF > threshold

This combines local stability (δ) with global loop reinforcement (TF) to determine whether coherent relational patterns survive long enough to become physically meaningful structures.

2.10 Finite Relational Capacity and Handshake Bound

From Section 1 (Axiom 9), each site has finite relational capacity. This limits how many simultaneous pairwise comparisons can be processed per tick.

The coordination number K and handshake capacity H are derived in the companion proof (Section 2, Addendum: K=12 Uniqueness):

Ternary closure + 3D geometric embeddability → K = 12

H = K(K−1) = 132 independent relational degrees per site

This finite capacity has direct dynamical consequences:

Stutter sink: When relational demand exceeds H, unresolved differences accumulate as local latency rather than propagating. The local momentum balance becomes:

p_i(t + τ₀) − p_i(t) + Σ_{j ~ i} J_{ij}(t) = −s_i(t)

where the stutter sink is:

s_i(t) = κ_s · δ_i(t) · p_i(t)

with δ_i(t) the normalized local handshake deficit and κ_s a dimensionless efficiency constant.

When δ_i = 0, momentum is locally conserved. When δ_i > 0, directed transport converts to unresolved relational latency — the primitive origin of inertia.

2.11 Anchoring and Global Stability

Without a stabilizing constraint, global orientation Θ = Σ_i F_i undergoes a random walk. This is fixed by weak anchoring:

Anchoring rule: Modify flip probability by:

P(flip at i) → P(flip at i) · [1 + λ · sign(F_i) · (Θ − Θ₀)]

where λ ≪ 1/N (weak coupling) and Θ₀ is the reference orientation.

This is the unique lowest-order perturbation that breaks global F_i → −F_i symmetry while preserving all local axioms. It does not alter local tension dynamics but stabilizes global statistics.

Anchoring is the seed of what later becomes: mass normalization, gauge fixing, and metric emergence — but at this stage it is purely a relational bookkeeping constraint.

2.12 Emergent Effective Parameters (Preview)

From the substrate quantities derived above, the following effective parameters emerge. These are relational bookkeeping quantities — not yet physical constants, which require calibration in Section 5.

Effective inertia:

I_eff ~ δ · TF · R_domain²

Effective coupling:

G_eff ~ δ · TF · R_domain²

Intrinsic causal rate:

c_int = 1 adjacency-step per tick

All three are dimensionless substrate quantities. Physical calibration (τ₀ → Planck time, adjacency-step → physical length) is deferred to Section 5.

2.13 Axiomatic Closure

Every construct in Section 2 traces to Section 1 primitives:

ConstructOriginAxiomT_i (tension)Binary disagreement countA2, A3Pairwise flip ruleAction minimization on graphA3, A7, A9Charge conservationSingle-site flip forbiddenA2M_i (proto-mass)Time-averaged flip rateA2, A8, A9v_i (proto-velocity)Directed transport biasA3, A4, A5p_i (proto-momentum)M_i · v_iDerivedA_ij (memory)Accumulated disagreementA6, A8S_coarse (ordering)−mean tension, monotonicA7, A8C_ij (correlation)Temporal persistence of x_ijA2, A8d_ij (proto-distance)−log transport correlationDerivedH_{abc} (holonomy)Phase around relational tripleDerivedδ (stability margin)Spectral radius of JacobianDerivedTF (topology factor)Loop agreement productA3, A7H = 132 (capacity)K=12 closure requirementA9, K-proofs_i (stutter sink)Capacity overflow latencyA9

No free parameters. No spatial embedding. No metric. No dimensionality assumption.

2.14 Bridge to Section 3

Section 2 delivers to Section 3 the following purely relational ingredients:

Binary states F_i(t) and their update dynamics

Proto-mass M_i and proto-momentum p_i per site

Accumulated differences A_ij(t) encoding relational history

Temporal correlations C_ij(τ) between site pairs

Operational proto-distances d_ij = −log(T_{i↔j})

Holonomy phases H_{abc} around relational triples

Stability margins δ_i and topology factor TF

Coarsening order parameter S_coarse(t) defining temporal ordering

Section 3 will use these to derive:

Emergent spatial geometry from the correlation structure of d_ij

Metric tensor from the correlation tensor T_μν

Curvature from holonomy accumulation

Dimensionality from eigenvalue structure of the correlation matrix

The mapping from relational substrate to continuum observables