Bridge Paper: A Purity Bug Fix, a True H4-Symmetric Window, and a Cross-Method, Cross-Seed Replicated Occupancy Signal in the Elser-Sloane Quasicrystal Under WCM-PL

WCM-PL’s E8Sensor (240 roots) and Cell600Sensor (120 vertices, the binary icosahedral group) are, as polytopes, literallythe same geometric objects used to construct the Elser-Sloane quasicrystal via cut-and-project from the E8 lattice – butthe two fields use this shared geometry in opposite directions: quasicrystal theory projects a high-dimensional periodiclattice down to obtain aperiodic structure, while WCM-PL assigns low-dimensional data up onto a polytope’s vertexdirections to detect occupancy concentration. This paper reports three results. First, a genuine bug in a prior session’sE8-purity self-check is identified and fixed: the diagnosis at the time (generator redundancy, requiring a minimal 4-generator basis) was incorrect; the actual defect was the quadratic form used to verify lattice membership. The correctedformula, cross-checked against two independent literature sources, restores 100.00% even-integer norm-squared purity(up from 66-78.5% under the old formula) on both a confirmed-independent basis and the prior session’s own generatedpoint cloud. Second, the ball-window simplification used for the cut-and-project acceptance domain is replaced with thetrue H4-symmetric window (a scaled 600-cell, implemented via convex-hull containment). Third, an occupancy-concentration signal in the cut-and-project selection – structure beyond the trivial baseline expected from building pointsas sums of the sensor’s own vertex-direction generators – is cross-checked across three window-definition methods (thisdomain’s proposed analogue of minmax/rank/robust normalization) and five independent point-cloud seeds; all 15combinations agree in sign (mean z = 4.0-4.4, minimum z = 3.17). The finding is classified NUMERICAL: robust under thissession’s cross-method/cross-seed checks, but not yet DERIVABILITY-backed by a closed-form prediction, and obtainedwith a sensor (Cell600Sensor) that WCM-PL’s own same_sign_vertex_risk() audit rates CATASTROPHIC – a risk this paperflags explicitly rather than setting asid. ※The license for codes is AGPL-3.0-or-later