§23.32 · The Lookup Purity Restored, and Unified: n1 Alone Classifies All Three Vertex-Figure Classes

Abstract

Abstract§23.27 §6.4 found that, restricted to one defect per arm, grouping the holonomy angle by (transportclass, δ, (k1+k2) mod 5) gives a pure lookup for vertex-figure class T: 24 keys (6 per δ), each mapping to a single angle. §23.31 §6 tested the same grouping, using the same transport-class test, on classes M and A, and found it impure in both: the identical key can map to two different angles depending on which specific pair (a, b) is used.This note traces the impurity to its source directly, rather than guessing at a fix. For every impure key,the rows that disagree are printed side by side; in every single case, the two groups of rows are separated exactly by n1 — the ordinary wheel-closing power of edge e1 alone, already computed byfind_n in every script since §23.21, and never itself used as a grouping variable. Replacing the transport-class test with n1 directly — grouping by (n1, δ, (k1+k2 ) mod 5) — gives a pure lookup: 16 keys, 0 impure. This holds not only for M and A but for T as well, at every one of 4 hinges tested: 12 (hinge, class) combinations, all pure, all with exactly 16 keys.This is a stronger result than the one it replaces: a single classifier, n1∈ {1,4}, already available with no extra computation, produces a pure lookup uniformly across all three vertex-figure classes, superseding §23.27 §6.4’s class-T-specific transport-class test rather than merely patching it for M and A. Merging further over δ (grouping by (n1 , residue) alone) is checked too, and found impure for allthree classes alike (10 keys, 6 impure) — recorded for completeness, not claimed to work; δ is needed.Keywords600-cell · lookup purity · wheel-closing power n1· vertex figure · classes T/M/A · exhaustive verification