Emergent 3D Geometry and Dimensionality in Temporal Flow Physics

Emergent 3D Geometry and Dimensionality in Temporal Flow Physics (TFP)

By John Gavel

Abstract

Starting from a minimal set of TFP primitives (discrete sites, primitive adjacency, binary site states, locality, ternary closure, and reversible local updates), this document gives a formal derivation showing how a 1-dimensional sequential causal engine (a fair 1D scheduler) together with accumulation/thresholding and determinacy conditions forces an emergent 3-dimensional relational geometry. Two independent derivations are provided:

1. Topological triangulation argument: the clique complex of the stabilized neighbor link forms a triangulated 2-sphere.
2. Spectral embedding argument: a 3-dimensional irreducible eigenspace of the adjacency of the neighbor link.

All prior clarifications are incorporated:

• Binary necessity is made precise (characteristic 2 is required for the linear-elimination mechanism underlying ternary closure).
• Determinacy conclusions are stated modulo the global flip symmetry.
• The claim “space is a projection of the 1D sequence” is rephrased as “space is the minimal embedding of the stabilized relational accumulation produced by the 1D sequence.”

1. Axioms / Primitive Assumptions (TFP)

Discrete Sites: \(V\) is a countable set of sites (vertices).

Primitive Adjacency: A symmetric adjacency relation \(\sim\) on \(V\). The undirected graph encodes adjacency.

Binary State: Each site carries a primitive binary state \(F_i \in \{+1,-1\}\).

Relational Difference: For each adjacent pair \(i \sim j\), define \(x_{ij} = \frac{1 - F_i F_j}{2} \in \{0,1\}\).

Finite Update / Locality: Updates at site \(i\) depend only on the \(x_{ij}\) for neighbors \(j\).

Ternary Closure / Determinacy: For any triangle \(i \sim j \sim k \sim i\), \(x_{ij} + x_{ik} + x_{jk} \equiv 1 \ (\text{mod } 2)\).

Primitive Reversibility (injective local update): There exists a local update rule \(F_i(t+1) = F_i(t) \oplus f(\Delta_i(t))\), where \(\Delta_i(t)\) depends on the \(x_{ij}\) for neighbors \(j\), and the map is injective.

2. Binary Necessity (Characteristic-2 Elimination)

Theorem 1 (Characteristic-2 Necessity for Algebraic Elimination):

Let the local closure mechanism be specified by ternary constraints of the form \(x_{ab} + x_{ac} + x_{bc} \equiv 1 \ (\text{mod } n)\).

The elimination strategy relies on substitution of equations for triples involving a center site, producing edge variables that appear twice in sums. Cancellation occurs only when \(1 + 1 \equiv 0 \ (\text{mod } 2)\).

Over any field with characteristic not 2, \(1 + 1 \neq 0\), so edge variables do not eliminate cleanly; determinacy-by-overlap fails or requires metric structure.

Conclusion: Characteristic 2 is necessary. The minimal finite field is \(\mathbb{F}_2\), forcing binary primitive states.

Corollary: Using ternary closure plus elimination implies algebra over \(\mathbb{F}_2\); binary primitives are structurally forced.

Clarification: This is a structural necessity for TFP’s closure/elimination mechanism, not a metaphysical claim.

3. Determinacy, Rank, and Global Symmetry

Let \(i\) be a site with neighbors \(N_1,\dots,N_K\). Let \(y_\ell = x_{i N_\ell}\). For adjacent neighbor pairs \(N_\ell \sim N_m\), the triangle constraint gives:

\(x_{N_\ell N_m} = 1 + y_\ell + y_m \ (\text{mod } 2)\)

Proposition (Local Determinacy Modulo Global Flip): If the triangle constraints among neighbors generate a linear system of rank \(K\) over \(\mathbb{F}_2\), then the \(y_\ell\) are uniquely determined up to a single global flip symmetry. Rank \(K+1\) gives absolute determinacy.

Remark: Achieving full rank requires sufficient linearly independent triangles among neighbors; the combinatorial/topological structure of the neighbor graph determines feasibility.

4. Sequential Processing, Accumulators, and Active-Edge Graphs

1D Sequential Scheduler: Selects one edge \(e(t) = (i,j)\) at each tick \(t\). Fairness: each edge is processed infinitely often.

Accumulator: For each edge \(e\), \(A_e(t+1) = A_e(t) + 1\) if \(x_{ij}(t) = 1\), otherwise 0.

Lemma 4.1 (Accumulation and Thresholding): If edge \(e\) has positive asymptotic mismatch frequency, \(A_e(t) \to \infty\). For separated long-run rates, there exist \(\tau\) and \(T_0\) such that for \(t > T_0\):

\(E_\tau(t) = \{ e : A_e(t) \ge \tau \}\)

Interpretation: Threshold graph stabilizes to the persistent relational subgraph produced by the 1D scheduler and local dynamics.

5. Neighbor Link Structure and Spherical Triangulation

Lemma 5.1 (Edge-Triangle Saturation from Determinacy): Each neighbor–neighbor edge must lie in at least two triangles (one including the center, one among neighbors) to achieve full rank.

Lemma 5.2 (Finite, Closed, Genus-0 Link): Finite capacity ensures the link has finitely many neighbors. Reversibility eliminates homological cycles that reduce rank. Conclusion: the neighbor link is a finite, closed triangulated 2-manifold with genus 0, i.e., a triangulated sphere \(S^2\).

Theorem 5.3 (K = 12): Using Euler’s formula for a finite, regular triangulated sphere:

V − E + F = 2, 3F = 2E, 2E = dV → d = 5 → K = 12

Lemma 5.4 (Full Activation by Stabilization): Fair 1D scheduling, mismatch accumulation, and reversibility force all edges of the triangulated link to be active in the long-time limit.

6. Topological Emergence

Lemma 6.1 (Euler Characteristic from Axioms): The stabilized neighbor link is a finite, closed, genus-0 triangulated 2-manifold. Using handshaking identities:

3F = 2E, 2E = dV, V − E + F = χ. Solving gives V*(1 − d/6) = χ → V = 12 / (6 − d).

Lemma 6.2 (Possible Degrees): d = 3 → V = 4 (tetrahedron), d = 4 → V = 6 (octahedron), d = 5 → V = 12 (icosahedron), d ≥ 6 → no finite solution.

Lemma 6.3 (Elimination of d < 5): Tetrahedron and octahedron fail determinacy or triangulation constraints.

Theorem 6.4 (Uniqueness of K = 12): Only d = 5 gives V = 12, full triangulation, rank = K − 1 = 11, tetrahedral subsets, and 3D embeddability.

Corollary (Handshake Capacity): Number of ordered neighbor comparisons: H = K(K − 1) = 12 × 11 = 132.

7. Spectral Confirmation: Adjacency Eigendecomposition

Let L_i be the neighbor link of site i, with adjacency matrix A.

Lemma 7.1 (Icosahedral Spectrum — Classical Fact): Eigenvalues: 5 (1), √5 (3), −√5 (3), −1 (5).

Theorem 7.2 (Spectral Embedding → 3D): Let v2, v3, v4 be an orthonormal basis of the √5 eigenspace. Define φ(j) = (v2(j), v3(j), v4(j)) for j = 1,..,12.

8. Final Theorem (3D Emergence)

Theorem 8.1 (Emergent 3D Geometry in TFP): Under TFP axioms with fair 1D scheduling, ternary closure over F_2, reversibility, and finite capacity:

• Determinacy → finite, closed triangulated neighbor link.
• Reversibility → genus 0 (sphere).
• Euler + regularity + non-degeneracy → K = 12.
• Stabilization + accumulation → all edges active.
• Clique complex → triangulated S^2.
• Minimal embedding dimension = 3.
• Spectral decomposition confirms 3D irreducible eigenspace.

Conclusion: 3-dimensional relational geometry emerges from 1D sequential reversible binary dynamics.

9. Clarifications and Notes

• Determinacy: Modulo global flip unless gauge fixed.
• K = 12: Derived from combinatorial, topological, and reversibility constraints.
• “Projection” language: Corrected — space is minimal embedding of the stabilized relational complex, not a linear projection.
• Reversibility: Ensures accumulators track relational differences faithfully.
• Ergodic technicalities: Stabilization can be formalized using standard mixing conditions but is orthogonal to structural logic.