The Heliocentric Sparse Law: Distance, Link Physics, and the Geometry–Efficiency Orthogonality
Councilman, J.
Abstract
v4.0 (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. v3.0 (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. Extension of the sparse law to heliocentric relay architectures across the solar system. An eleven-experiment campaign covering synodic phase sweeps, SEP-angle transitions, phase diagram collapse, multi-body universality (Venus, Mars, Jupiter), and CGR recovery analysis. The campaign spans 3,459 configurations across three target bodies (Venus, Mars, Jupiter), four constellation sizes, six load levels, twelve link-margin settings, and distances from 0.27 to 5.95 AU. The factorization holds at machine epsilon across all configurations; the two-dimensional phase diagram DR(d, pref) collapses onto a single effective variable peff = pref/d² (R² = 0.96), and the crossover exponent α ≈ 0.65 is stable across constellation sizes. Mars remains highly active despite its 1.45 AU distance, with diversity gain dominating at far distances and the dead-end tax dominating dense inner-planet constellations.