Section 2 — Local Dynamics of Binary Relations (TFP) (v11.1)

Section 2 — Local Dynamics of Binary Relations (TFP) (v11.1)

By John Gavel

2.0 Overview

This section derives the local dynamics of relational differences directly from the axioms of Section 1.
No additional assumptions are introduced; all behavior emerges from:

• Binary relational states (Axiom 2)
• Primitive adjacency (Axiom 3)
• Relational closure (Axiom 7)
• Discrete updates (Axioms 8–9)

We focus on the proto-temporal dynamics: the minimal processes that allow differences to propagate, interact, and accumulate.

2.1 Primitive Update Rule (Level 0)

Axiomatically Derived Rule:

Each site \(i\) updates its state based solely on the states of its adjacent sites.
Differences propagate via comparison, not physical motion.

Formally:

Let \[ F_i(t) \in \{+1, -1\} \] and let \( \text{Adj}(i) \) be the set of adjacent sites to \(i\).
Define the local difference count:

\[ \Delta_i(t) = \sum_{j \in \text{Adj}(i)} \big(F_i(t) \oplus F_j(t)\big) \]

Update Rule:

\[ F_i(t + \tau_0) = F_i(t) \oplus f(\Delta_i(t)) \]

Where \(f\) is a binary function that determines whether to flip \(F_i\) based on local relational differences.

Logical Meaning:

• This is the proto-time step: a discrete relational tick.
• \(\Delta_i\) propagates relational information.
• The system remains binary and discrete; nothing yet resembles vectors or velocity.

2.2 Accumulation and Memory (Emergent Proto-Time)

Local differences accumulate in overlapping neighborhoods.
Let \(A_{ij}(t)\) represent the accumulated difference between sites \(i\) and \(j\):

\[ A_{ij}(t+1) = A_{ij}(t) + \big(F_i(t) \oplus F_j(t)\big) \]

Logical Role:

• Provides the first instance of memory of past differences.
• Serves as the mechanism for proto-time: ordering relational events without assuming historical time.

2.3 Local Deterministic and Stochastic Updates

Deterministic flips:

\[ F_i(t + \tau_0) = \begin{cases} F_i(t) \oplus 1 & \text{if } \Delta_i(t) > \text{threshold} \\ F_i(t) & \text{otherwise} \end{cases} \]

Stochastic updates:

Flip probability: \[ P(\text{flip}) = g(\Delta_i(t), T_\text{eff}) \] where \(g\) maps accumulated differences to a flip probability; introduces variability and emergent “noise” while preserving locality.

Interpretation:

• Flipping events encode relational activity.
• Repeated activity creates measurable markers (proto-mass) without assuming pre-existing space.

2.4 Emergent Mass and Stability

Define a flip indicator: \[ R_i(t) = \begin{cases} 1 & \text{if } F_i(t + \tau_0) \neq F_i(t) \\ 0 & \text{otherwise} \end{cases} \]

Time-averaged mass density:

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

Interpretation:

• Mass is operational: it counts relational activity.
• Sites with more frequent changes are “heavier” in relational significance.

2.5 Proto-Geometry Ingredients

Differences propagate along adjacency, creating correlations across the network: \[ C_{ij}(\tau) = \langle F_i(t) \oplus F_j(t + \tau) \rangle \]

Proto-time ordering emerges from sequential accumulation of differences.
No pre-existing temporal axis is assumed.
Updates define a local temporal sequence.
These correlations will form the foundation for distance in Section 3.

2.6 Emergent Principles

• Directional Representations are Emergent: Differences propagate bidirectionally; orientation emerges only after closure. Vectors, flow, or spatial directionality are descriptions of correlated activity, not primitives.
• Closure as Local-to-Global Determinacy: Overlapping updates across triples allow relational closure. Once enough overlaps exist, global structure can be inferred from local difference dynamics.
• Proto-Time: Each discrete update is a tick of relational change, not historical time. Accumulation of flips defines an emergent sequence of events.

2.7 Bridge to Section 3

Section 3 will derive:

• Emergent distance from correlations between sites
• Spatial geometry as a consequence of relational closure
• Mass-induced curvature emerging from accumulated differences

All constructs remain derived from Section 1 primitives.