TFP Section 3 (v10)

SECTION 3 — EMERGENT SPATIAL STRUCTURE (v10.3)

Geometry, Distance, and Curvature Derived from Flow Correlations
By John Gavel

3.0 Overview

This section derives the emergent spatial manifold — distance, dimension, and curvature — from the binary flow dynamics of Section 2. No prior geometry is assumed. All structure arises from:

• Adjacency \( i \sim j \) (Axiom 3)
• Temporal flow \( F_i(t) \in \{+1, -1\} \) (Axiom 2)
• Time-lagged correlations \( C_{ij}(\tau) \)

The causal sequence is:

1. Stage 0: Metric-free dynamics on adjacency graph (Section 2)
2. Stage 1: Correlations define operational distance
3. Stage 2: Distance induces metric, dimension, curvature

Geometry is not input — it is output of the substrate.

3.1 Correlation Functions (Stage 1)

From the time series \( \{F_i(t)\} \) generated by the update rules (Section 2.1–2.2), define the temporal correlation:

\[ C_{ij}(\tau) = \langle F_i(t) F_j(t + \tau) \rangle_t \tag{3.1} \]

Since \( F_i^2 = 1 \) (Axiom 2), \( \langle F_i^2 \rangle = 1 \), so no normalization is needed. Properties follow directly:

• \( C_{ii}(0) = \langle F_i^2 \rangle = 1 \)
• \( |C_{ij}(\tau)| \leq \sqrt{\langle F_i^2 \rangle \langle F_j^2 \rangle} = 1 \) (Cauchy-Schwarz)
• \( C_{ij}(\tau) \to 0 \) as \( |i - j| \to \infty \) or \( |\tau| \to \infty \) (decorrelation)

Physical meaning: \( C_{ij}(\tau) \) quantifies how much \( F_i(t) \) predicts \( F_j(t+\tau) \).

3.1.1 Correlation Decay and Coherence Length

The coarse-grained field \( A(x,t) \) obeys the continuum equation (Section 2.4):

\[ \frac{\partial A}{\partial t} = D \nabla^2 A - \kappa A + \eta \tag{2.11} \]

In Fourier space, \( A(\mathbf{k},t) = \int A(x,t) e^{-i\mathbf{k}\cdot\mathbf{x}} d^d x \), this becomes:

\[ \frac{\partial A(\mathbf{k},t)}{\partial t} = -(D k^2 + \kappa) A(\mathbf{k},t) + \eta(\mathbf{k},t) \]

For stationary noise with \( \langle \eta(\mathbf{k},t) \eta(\mathbf{k}',t') \rangle = 2 D_{\text{noise}} \delta(\mathbf{k}+\mathbf{k}') \delta(t-t') \) (Eq 2.15), the steady-state two-point function is:

\[ \langle A(\mathbf{k}) A(\mathbf{k}') \rangle = \frac{2 D_{\text{noise}}}{D k^2 + \kappa} \delta(\mathbf{k} + \mathbf{k}') \tag{3.2} \]

In real space, for isotropic systems, the inverse Fourier transform in \( d \) dimensions yields exponential decay:

\[ C(r) = \langle A(\mathbf{x}) A(\mathbf{x} + \mathbf{r}) \rangle \propto \int \frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{k^2 + \kappa/D} \frac{d^d k}{(2\pi)^d} \propto e^{-r / L_c} \tag{3.3} \]

where the coherence length is:

\[ L_c = \sqrt{\frac{D}{\kappa}} \tag{3.4} \]

Substituting \( D = k_{\text{avg}} a^2 / (T_{\text{eff}} \tau_0) \) and \( \kappa = \mu_{\text{eff}}^2 / \tau_0 \) (Eqs 2.12, 2.14):

\[ L_c = a \sqrt{ \frac{k_{\text{avg}}}{T_{\text{eff}} \mu_{\text{eff}}^2} } \]

3.1.2 Causal Origin of Correlations

Although \( F_i(t) \in \{+1, -1\} \) has no intrinsic phase, correlations arise from constrained binary choices under tension minimization (Axiom 6). Consider a 1D chain where site \( i \) has neighbors \( i-1 \) and \( i+1 \):

• If \( F_{i-1} = +1 \) and \( F_{i+1} = -1 \), site \( i \) cannot align with both
• Whichever state it chooses breaks local symmetry and propagates influence
• This creates path-dependent histories: the state at \( j \) depends on the sequence of tension-minimizing choices along paths from \( i \) to \( j \)

The correlation \( C_{ij}(\tau) \) quantifies this causal entanglement. The time lag \( \tau \) that maximizes \( |C_{ij}(\tau)| \) encodes the minimum update steps for influence to propagate — a discrete analog of signal delay.

The maximum propagation speed is (Section 2.5.1):

\[ c = \frac{a}{\tau_0} \tag{3.4a} \]

This enforces a strict bound between spatial separation \( d_{ij} \) and causal delay \( \tau_{i \to j} \):

\[ \tau_{i \to j} \leq d_{ij} \cdot \frac{\tau_0}{a} \tag{3.4b} \]

Thus, correlations cannot propagate faster than one lattice spacing per update step. In the continuum limit, this discrete causal bound converges to Lorentz-invariant propagation, with phase accumulation \( H_p \) (Section 3.4.2) emerging only at the coarse-grained level and remaining consistent with relativistic causality. No phase exists at the substrate level, but these path-dependent constraints generate the proto-geometric structure that later supports phase in coarse-grained fields (Section 10).

3.2 Operational Distance

Define distance from maximum predictive power (information-theoretic metric):

\[ d_{ij} = -\ln \left( \max_{\tau} |C_{ij}(\tau)| \right) \tag{3.5} \]

For systems with short memory (Section 2.5.2), the peak occurs at \( \tau = 0 \), so:

\[ d_{ij} \approx -\ln |C_{ij}(0)| \tag{3.6} \]

3.2.1 Metric Properties

From \( 0 < |C_{ij}| \leq 1 \):

• \( d_{ij} \geq 0 \) (non-negativity)
• \( d_{ii} = -\ln 1 = 0 \) (identity)
• \( d_{ij} = d_{ji} \) because \( C_{ij}(\tau) = \langle F_i(t) F_j(t+\tau) \rangle = \langle F_j(t-\tau) F_i(t) \rangle = C_{ji}(-\tau) \), so \( \max_\tau |C_{ij}(\tau)| = \max_\tau |C_{ji}(\tau)| \)

For smooth fields \( A(x,t) \), \( C_{ik} \gtrsim C_{ij} C_{jk} \) by conditional independence, so:

\[ \ln C_{ik} \gtrsim \ln C_{ij} + \ln C_{jk} \quad \Rightarrow \quad d_{ik} \lesssim d_{ij} + d_{jk} \tag{3.7} \]

Thus \( d_{ij} \) is a statistical metric.

3.2.2 Connection to Physical Distance

In the exponential regime (Eq 3.3), \( C_{ij}(0) \approx e^{-|\mathbf{x}_i - \mathbf{x}_j| / L_c} \), so:

\[ d_{ij} = \frac{|\mathbf{x}_i - \mathbf{x}_j|}{L_c} \tag{3.8} \]

Operational distance is geometric distance scaled by \( L_c \).

3.3 Emergent Dimension

3.3.1 Combinatorial Dimension

From adjacency (Axiom 3), the coordination number is \( k_i = |\{ j : j \sim i \}| \). In a regular \( d \)-dimensional lattice, \( k \approx 2d \). Solving \( k = 2^{d_{\text{eff}}} \) gives:

\[ d_{\text{eff}}^{\text{(coord)}}(i) = \frac{\log k_i}{\log 2} \tag{3.9} \]

3.3.2 Correlation-Based Dimension

The number of sites within correlation length \( L_c \) scales as \( N_c \propto L_c^{d_{\text{eff}}} \). Taking logarithms:

\[ d_{\text{eff}}^{\text{(corr)}} = \frac{\log N_c}{\log L_c} \tag{3.10} \]

Consistency \( d_{\text{eff}}^{\text{(coord)}} \approx d_{\text{eff}}^{\text{(corr)}} \) validates the dimension.

3.4 Curvature

3.4.1 Coordination Defects

In flat \( d \)-space, expected coordination is \( k_{\text{exp}} = 2 d_{\text{eff}} \). Define the defect:

\[ \Delta k_i = k_i - k_{\text{exp}} \tag{3.11} \]

Interpretation:

• \( \Delta k_i > 0 \): excess neighbors → positive curvature (spherical)
• \( \Delta k_i = 0 \): Euclidean
• \( \Delta k_i < 0 \): deficit → negative curvature (hyperbolic)

3.4.2 Holonomy from Temporal Lags

The peak correlation lag (Section 2.5.3) defines edge phase:

\[ \tau_{i \to j} = \arg\max_{\tau} C_{ij}(\tau) \tag{3.12} \]

For a closed loop \( p = (i \to j \to k \to i) \), the holonomy is the total phase accumulation:

\[ H_p = \tau_{i \to j} + \tau_{j \to k} + \tau_{k \to i} \tag{3.13} \]

In flat space, transport is path-independent, so \( H_p \approx 0 \). Nonzero \( H_p \) signals curvature.

Averaging over loops containing site \( i \):

\[ R(i) \propto \frac{1}{\langle A_p \rangle} \sum_{p \ni i} H_p \tag{3.14} \]

where \( \langle A_p \rangle \) is the mean loop area (computed from \( d_{ij} \)).

3.4.3 Curvature from Tension Laplacian

From Section 2.3.2, tension \( T(x) \propto |\nabla A|^2 \). The discrete Laplacian on a graph with coordination defects differs from the flat-space Laplacian by curvature terms:

\[ (\nabla^2_{\text{discrete}} f)_i = \frac{1}{k_i} \sum_{j \sim i} (f_j - f_i) = \nabla^2_{\text{flat}} f + R_{\text{Ricci}} f + \cdots \tag{3.15} \]

Applying this to \( T(x) \) and solving for curvature:

\[ R_{\text{Ricci}}(x) \approx \frac{\nabla^2 T(x)}{T(x)} \tag{3.16} \]

Since \( M \propto T \) (Section 2.2.2), curvature is sourced by mass density gradients.

3.5 Spin from Flow Circulation

Define local spin as a topological winding number of binary flow:

\[ S_p = \sum_{(u,v) \in \partial p} F_u F_v \cdot \mathrm{sgn}(u \to v) \tag{3.18} \]

This integer-valued quantity counts the net chiral circulation around plaquette \( p \). It is nonzero only for motifs with \( d \ge 2 \) (see Section 4.3.3), which support asymmetric update paths.

In the continuum limit, \( S_p \) corresponds to vorticity \( \nabla \times \mathbf{F} \), but on the discrete substrate it is quantized by construction. Physical spin \( \pm \hbar / 2 \) emerges after calibration to the action scale \( \hbar_c \) (see Section 10.6.1).

3.6 Axiomatic Closure

Geometric Object	Origin	Axiom
Distance \( d_{ij} \)	Max correlation \( C_{ij}(\tau) \)	A2, A3, A9
Dimension \( d_{\text{eff}} \)	Coordination \( k_i \) or \( N_c(L_c) \)	A3
Curvature \( R \)	Coordination defects, holonomy, \( \nabla^2 T \)	A2, A3, A6
Spin \( s_i \)	Flow circulation on loops	A2, A3

All geometric quantities are computed from substrate data alone. No embedding space is assumed.

3.7 Bridge to Section 4 — Particle Structure

Section 3 provides:

• Operational distance \( d_{ij} \) → emergent manifold
• Coherence length \( L_c = \sqrt{D/\kappa} \) → physical scale
• Curvature \( R \propto \nabla^2 M \) → geometric response to mass
• Spin \( s_i \) → chiral flow modes

Section 4 identifies stable flow motifs (particles) as localized, self-reinforcing patterns in this geometry. Their recursion depth, holonomy divisors, and coherence budgets \( \Omega \) determine mass and charge — all without free parameters.