Section 3 — Emergent Spatial Structure and Proto-Time (TFP, v11.1)

Section 3 — Emergent Spatial Structure and Proto-Time (TFP, v11.1)

The Causal–Relational Manifold
By John Gavel

3.0 Overview

In this section, we derive the dimensionless structure of space and proto-time directly from the first principles of TFP:

• Binary relational states (Axiom 2)
• Primitive adjacency (Axiom 3)
• Relational closure and ternary accumulation (Axiom 7)
• Discrete updates (Axioms 8–9)
• Finite relational capacity (Axiom 10)

No geometric primitives are assumed. Distance, dimensionality, coherence, and propagation all emerge as logical consequences of the substrate’s discrete relational rules. We first derive the key quantities in a principled manner, then discuss what they represent conceptually.

3.1 Relational Capacity and Neighbor Coordination

Derivation:

Each site has a finite capacity, denoted \(K\), for simultaneous adjacency interactions. The total number of directed neighbor pairs it can manage in a single tick is:

\[ H = K \times (K - 1) \]

Interpretation:

\(K\) represents the coordination shell of a site. The substrate’s capacity to resolve differences is bounded; this finite capacity directly limits the system’s ability to propagate differences simultaneously across multiple neighbors. \(H\) is therefore a dimensionless bandwidth for relational interactions.

3.2 Operational Distance

Derivation:

Let \(d_{ij}\) be the minimal number of adjacency steps (hops) required to connect site \(i\) to site \(j\) along allowed causal paths:

\[ d_{ij} = \min_{\text{paths } P \text{ from } i \text{ to } j} |P| \] where \(|P|\) counts edges along \(P\). By construction:

• Reflexivity: \(d_{ii} = 0\)
• Symmetry: \(d_{ij} = d_{ji}\)
• Triangle inequality: \(d_{ik} \le d_{ij} + d_{jk}\)

Interpretation: \(d_{ij}\) defines a purely topological, dimensionless distance in the relational network. No metric assumptions are made; this is the first emergent notion of proximity in the substrate.

3.3 Temporal Sequencing and Proto-Time

Derivation:

Discrete updates (Axioms 8–9) impose a sequence of relational ticks. When a site cannot resolve all adjacency relations within a single tick due to finite capacity, unresolved differences are carried forward, forming reflections:

Reflection indicator: \(R_n \in \{+1, -1\}\), where \(R_n = 1\) if a reflection occurs at tick \(n\)
Cumulative proto-time:

\[ S_n = \sum_{i=0}^{n} R_i \]

Interpretation:

• \(S_n\) represents the integrated history of unresolved relations, forming a proto-temporal ordering.
• Proto-time is bidirectional and relational; the arrow of historical time emerges from the fact that differences accumulate and propagate through these sequential ticks.
• Causality is relational: sequences of accumulated comparisons create the perceived flow from past to future.

3.4 Phase and Coherence

Derivation:

Correlation between sites \(i\) and \(j\) over \(\Delta t\) ticks:

\[ C_{ij}(\Delta t) = \langle F_i(t) \oplus F_j(t + \Delta t) \rangle \]

Phase differences emerge naturally from discrete update lags along adjacency chains:

\[ \theta_{ij} = \omega \times d_{ij} \times \tau_0 \]

Linearizing local dynamics gives the Jacobian of updates:

\[ \delta F_i(t + \tau_0) = \sum_j J_{ij} \delta F_j(t) \]

Eigenvalues \(|\lambda_j|\) determine persistence of relational patterns:

• \(|\lambda_j| < 1\) → decay
• \(|\lambda_j| = 1\) → coherent propagation

Interpretation: Phase is a bookkeeping device encoding relative timing and correlation of difference propagation. Coherent sequences of differences across the network allow emergent directional structure, but orientation is not primitive—it arises from relational closure and overlap.

3.5 Saturation and Folding

Derivation:

Finite capacity \(K\) imposes a limit on simultaneous difference resolutions. Let \(R_\text{geom}\) be the number of relational states required to fully satisfy local closure. If \(R_\text{geom} > H\), then a residual deficit occurs:

\[ \delta = (R_\text{geom} - H) \times f(H_\text{total}) \] where \(H_\text{total} = H + \text{combinatorial overhead}\), and \(f\) scales the deficit dimensionlessly.

Interpretation: \(\delta\) represents irreducible stutter: unavoidable leftover relational differences per tick. It is not noise; it is the mechanical consequence of finite relational capacity. Accumulation of \(\delta\) over time contributes to operational mass and curvature in later sections.

3.6 Emergent Dimensionality

Derivation:

Each spatial dimension requires a fixed number of local constraints (forward, backward, exclusion):

\[ \text{Constraints per dimension} = 4 \]

For total neighbor capacity \(K\):

\[ d_\text{max} = \lfloor K / 4 \rfloor \]

Interpretation: The substrate’s coordination directly determines \(D = 3\) for the emergent relational manifold. Dimensionality is locked by local saturation; under- or over-constrained arrangements are unstable.

3.7 Summary of Derived Quantities

Quantity	Derived From	Meaning
K	Axiom 10	Neighbor coordination capacity
H	K × (K − 1)	Max directed relations per tick
d_ij	Adjacency graph	Operational distance (dimensionless)
S_n	Cumulative reflections	Proto-time sequence
θ_ij	Discrete lag along path	Phase for correlation bookkeeping
δ	R_geom − H	Folding tax; residual stutter
D	K / 4	Emergent dimensionality

All quantities are dimensionless, principled, and logically derived from the first principles in Sections 1–2. Historical time, phase, distance, and dimensionality emerge naturally, without introducing geometric or temporal primitives.

3.8 Bridge to Section 4

With these derivations, the substrate now has:

• A discrete relational manifold with measurable dimensionless distance
• Proto-time defined by accumulated reflections
• Phase coherence for correlated propagation
• Residual stutter \(\delta\) from finite relational capacity

Section 4 will show how stable motifs emerge in this manifold, and how mass, charge, and curvature are operationally defined from these purely relational and discrete dynamics.