Theory of Optical Phase-Memory Correlators (TOPMC): An Axiomatic Design Theory for Refractive Identity, Causal Integrity, and Quantum-Limited Readout

Abstract

We present the Theory of Optical Phase-Memory Correlators (TOPMC), an axiomatic design theory for storing and authenticating identity and causal history in weak refractive phase patterns, and for reading those patterns under classical matched-filter and optional quantum-optimal (Helstrom) measurements. TOPMC does not claim a new microscopic material law; rather, it composes classical coding geometry, matched-filter detection, causal support constraints, and quantum state discrimination into a single operational calculus with falsifiable predictions. Under five axioms—weak-phase linearity (A1), code-addressable writing (A2), a noisy score operator (A3), causal support for ledger reads (A4), and an optional coherent-state upgrade (A5)—we prove eight theorems that constitute the theoretical discoveries of the program: orthogonality separation (T1), an explicit false-accept law FA(τ)=Q((τ−½)/σ) (T2), a capacity–dimension bound for orthogonal libraries (T3), a hierarchical forge exponent Adv≤FAD (T4), causal integrity of past-restricted correlators (T5), a Helstrom dominance threshold η* separating quantum-advantage and classical regimes (T6), retention–fade duality (T7), and a security-calibration map λ↦τ (T8). Three phenomenon hypotheses (H1–H3) state the empirical content required to promote TOPMC from an in-model theory to a physical discovery, each equipped with an explicit falsifier and a minimum experiment. Default design numbers at code length M=32 include σ≈0.02, matched-filter SNR ρ≈25, FA≈6.2×10−3 at τ=0.55, Shannon capacity C1≈149 bits/read, hierarchy depth D*=5 for λ_path≳32 bits, retention horizon t*(f*=0.90)≈1217 days, and Helstrom crossover η*∼6×10−6. We emphasize epistemic honesty: emulation and theoremhood do not entail nature until Bridge protocol TOPMC-Bridge-1 advances past H1. The practical meaning of TOPMC is a design envelope—how to choose codes, thresholds, hierarchy depth, and readout modality against stated adversaries—rather than a claim that silica (or any dielectric) already implements the theory.