§23.20 · The §23.14 Residual Is a Genuine Population Feature, Not a Sampling Artifact — Exhaustive Verification and a Ruled-Out Candidate

Abstract

AbstractTT-DAV §23.14 found that its gauge-independent defect-insertion outer-loop holonomy, evaluated on a sample of 300 edge-pair configurations, is exactly determined in its SUPPORT by the §23.10 (k1,k2) transport class — class (1,4) configurations produce only {0°,120°,144°}, class (2,3) only {72°,144°} — butnot in its exact value: within class (1,4) the sample split 80:100:20 across the three angles, and within class (2,3) it split 50:50, with no identified explanation, and no relation found to the §23.11 6-vs-10 dichotomy.This note first reproduces §23.14's construction independently and matches its reported sample distributions exactly. It then computes the FULL population of valid configurations exhaustively — all 3,906 distinct (e1,e2,T0a,T0b) quadruples arising from all 600 tetrahedra of the 600-cell, not a 300-configuration sample — and finds proportions matching the original sample within ordinary sampling noise: class (1,4) splits 11.50%/40.15%/48.35% across {0°,120°,144°} (population) versus 10.0%/40.0%/50.0% (sample); class (2,3) splits 52.99%/47.01% across {72°,144°} (population) versus 50.0%/50.0% (sample). This confirms the residual is a stable, genuine feature of the 600-cell's geometry,not an artifact of which 60 tetrahedra the original sample happened to draw.The note then tests the candidate mechanism §23.14 §8 proposed — the specific pair of "outer" verticescompleting each tetrahedron beyond the two edges' own vertices — against the full population, and finds it insufficient: grouping all 3,906 configurations by (transport class, ordered outer-vertex pair) gives2,713 distinct groups, of which 459 mix more than one resulting angle. The candidate does not determine the outcome. The mechanism responsible for the residual remains unidentified and open.Keywords600-cell · binary icosahedral group 2I · discrete Volterra construction · gauge independence · transport class · exhaustive population verification · sampling artifact1. Background§23.14 built a gauge-independent recipe for inserting a holonomy defect into an outer loop composed of two edges e1,e2 sharing a vertex, continued through two shared tetrahedra T0a,T0b of their respective face-adjacency 5-cycles. On a random sample of 300 such configurations (60 tetrahedra, seed11), the resulting defected holonomy angle was found to lie in a set exactly determined by the §23.10 transport class of (e1,e2) — {0°,120°,144°} for class (1,4), {72°,144°} for class (2,3) — but the specific value within that set varied: 80:100:20 and 50:50 respectively. §23.14 §8 proposed, as a candidate explanation, the specific pair of vertices completing each tetrahedron beyond e1 e2's own four ∪vertices, and left both the explanation and the question of whether the sample was representative open.