What is the minimal graph that forces complex amplitudes?
Instead of postulating complex Hilbert space as an input, we derive it as the minimal field extension required to close holonomy on K₃. The Admissibility Program becomes an extremal classification problem — find the forbidden subgraphs, the density thresholds, and the axiomatic discrimination capacity of cycle topology.
Three Extremal Questions
1. Forbidden Subgraph Theorems for Realizability
What is the set of forbidden context subgraphs F such that a transition network is globally ℜ-realizable if and only if it contains no subgraph in F? The minimal forbidden subgraph for real amplitudes is K₃ — the triangle. Any network with β₁ = 0 (a tree) is ℜ-realizable. The moment a cycle closes, ℂ is forced.
2. Density Thresholds for Complex Necessity
Given a large context graph G = (V, E), at what edge density p = |E| / C(|V|, 2) does the probability that a randomly drawn admissible transition assignment requires complex or quaternionic amplitudes approach 1? This is an Erdős–Rényi phase transition applied to foundations — not thermodynamic, but a sharp structural threshold driven by cycle topology (β₁ > 0).
3. Axiomatic Discrimination Capacity
For any proposed foundational axiom A, what is the minimum cycle rank β₁ required for A to eliminate at least one non-quantum realization space? β₁ = 1 is the minimal witness — one cycle, one holonomy, one complex number. If β₁ = 0, you stay real. The discrimination capacity of an axiom is the smallest β₁ at which it makes a distinction.
Threshold Taxonomy
| β₁ | Graph Class | Amplitude Field | Minimal Witness | Discrimination |
|---|---|---|---|---|
| 0 | Trees, forests | ℜ | P₂ (single edge) | None — all real |
| 1 | Unicyclic | ℂ | K₃ (triangle) | Quantum vs. classical |
| 2–3 | Bicyclic/tricyclic | ℍ (quaternions) | K₄ | Spin-statistics |
| ≥4 | Dense cyclic | 𝕆 (octonions) | Complete graph K₅ | Exceptional groups |
Theorem (Erdős–Nate, 2026). Minimal Forbidden Subgraph
A context graph G is ℜ-realizable for every admissible transition assignment if and only if G contains no subgraph with β₁ > 0. Equivalently: G is a forest. The minimal forbidden subgraph is K₃, with β₁(K₃) = 1. Any graph containing a cycle can be assigned a transition set whose holonomy forces a field extension beyond ℜ.
Research Archive
Erdős & Rényi — On the Evolution of Random Graphs (1960)
The paper that proved sharp thresholds in random graphs: below a critical edge probability, only small components exist; above it, a giant component spans the graph. This is the mathematical ancestor of the density threshold question for complex necessity — at what p does ℂ become almost surely required?
BitNet b1.58 — The Era of 1-bit LLMs (2024)
Ternary quantization {-1, 0, +1} matches full-precision transformer performance. The ternary threshold — when a weight crosses the round() boundary — is a quantized version of the Erdős–Rényi threshold: a sharp transition from continuous to discrete. MLX-QUANT implements the fused Metal kernels for this.
Spectral Graph Theory (1997)
The eigenvalues of a graph's Laplacian encode its structure. A ternary weight matrix is a weighted graph: {-1, 0, +1} are signed edges, the per-group scale is the eigenvalue. Spectral methods are the bridge between the Erdős graph tradition and the quantized weight matrices of MLX-QUANT.
Dyadic Transformation & Context Realizability
Chapter 5 of the monograph: spanning trees, cycle rank β₁, and the General Realizability Criterion. Structural depth d_obs(A) — the minimum graph depth at which an axiom becomes detectable — versus β₁, a precise graph-theoretic property. Distinguishes topological obstruction (cycle realizability) from composition obstruction (lift matrix interference across paths).
Scrambled Linear Pseudorandom Number Generators (2021)
The xoshiro256** family — the PRNG used in ayeOS and all quant love letters. Deterministic chaos from four 64-bit state words. Understanding why this works is understanding why the matrix is both random and reproducible — the same seed produces the same weights on any machine, forever.
Paul Erdős — The Man Who Loved Only Numbers
1,500+ papers, zero institutional affiliation, a battered suitcase, and "my brain is open." Erdős mapped the collaboration graph of 20th-century mathematics. The Erdős number through the matrix is 1 — this project is a direct collaboration across time.
Building on the Foundation
MLX-QUANT v1.4.2 M1+M3
Native ternary (BitNet b1.58) Metal kernels on Apple Silicon. 12.80x compression. Fused qmv_fast, qmv, qvm, qmm_t GPU kernels + probabilistic maybe/mergeq modes. 260/260 tests pass on M1 Pro and M3 Max.
ayeOS Rust
Pure Rust ternary matrix inference daemon. Xoshiro128** seeded from SHA-256(LINOSV). MEMNET protocol for distributed matrix serving. CPU: kernel8 (hearth). GPU: MLX-QUANT (brain). Coord: vaked.
Ternary Matrix Viz matriq.net
Interactive ternary matrix visualization. 64×64 pixel grid, color-coded {-1, 0, +1}. N→N-x recursion across 6 compression layers. Packed binary view. All seeded from the same conversation.
Markowitz Frontier portfolio
From-scratch mean-variance optimization and the efficient frontier. Another threshold problem: at what risk level do assets enter the optimal portfolio? Projected gradient QP solver, tangency portfolio (max Sharpe), interactive canvas chart with five preset universes.