Sparse Law Mathematical Primer: Three-Factor Decomposition, Lyapunov Self-Averaging, and the Policy Distortion Factor
Councilman, J.
Abstract
v5.0 (March 2026): Revised terminology and epistemic framing throughout. Definitional identities relabeled as Propositions. Classification language updated. Mechanism descriptions qualified as proposed interpretations. License: CC-BY-4.0 (revised from the earlier AGPL-3.0-or-later statement). Scientific conclusions unchanged. v4.0 (March 2026): Revised for consistency with classification theorem v7.0. Demoted empirical results from Theorem to Observation/Conjecture. Corrected gamma normalization convention. Updated DOI cross-references to concept DOIs. Added self-DOI to author block. Qualitative conclusions unchanged. Complete mathematical reference for the sparse law of deep-space DTN communication. Derives the three-factor decomposition DR = S_T · exp(E[H]λ) · Φ from first principles. Key results: (1) Self-averaging of the oracle-path success predictor: ηOPSP = exp(E[H]λ) to < 0.4% across all 8 validation configurations, including high-disorder heliocentric cases (σ² = 0.80). (2) Wald CLT correction valid only when σ² < 0.5; diverges catastrophically above this boundary (+96% overshoot). (3) The policy distortion factor Φ spans [0.3, 8.6] across the 17,280-config orbital sweep and [0.6, 8593] when including CRAWDAD cluster-class traces. (4) Exact factorization confirmed to 1.11 × 10-16 across 154,000+ configurations (8 orbital targets + 4 CRAWDAD traces). Covers: CTMC efficiency model, OPSP bridge, k=1 gateway result, Lyapunov exponent, Wald extension, Braess phase surface, and the classification framework (γ order parameter separating trap and cluster structural classes).