§23.30 · Closing the Loop for Classes M and A: the Chain Is Free, and Why the Symmetric Crossing Is a Dead End

Abstract

Abstract§23.28 §7 flagged two related gaps for the vertex-figure classes M and A (hinge-sharing edge pairs whose tetrahedra do not share an edge): no discrete defect/holonomy loop had been defined for them, and no exponent formula existed for arbitrary defect multiplicities, since the §23.21 construction anchors both to a pair of tetrahedra T0a, T0b shared by e1 and e2 directly — which classes M and A lack by definition.A first attempt, reported here for completeness, searched for a genuinely new geometric object: a two-sided transport crossing between the two tetrahedra that e1and e2 DO each belong to (which share a face for class M, found in preliminary work on this note). Restricted to the unique transport thatfixes the hinge and exchanges the two bridge vertices — the only non-arbitrary choice — this crossing is well-defined and pair/hinge-independent, but it provably rotates axis(r′e2) onto axis(r′e1) exactly (cos γ 12′ = +1 to machine precision), collapsing the loop to a pure power of r′e1 and erasing the class-M information it was meant to capture (§2). The actual resolution needs no new object at all. Two facts, checked here across all 120 hinges with no exception, show that classes M and A already connect to the fully solved class-T machinery for free: (i) for a class-M pair, the two tetrahedra shared respectively by (e1 , ebridge) and by (ebridge, e2) already share a face — an ordinary wheel-step of the bridge edge’s own 5-cycle, not a new kind of step; (ii) for a class-A pair, the analogous 3-hop chain through 2 intermediate vertices is free in the same sense (§3). And the §23.21 §2.1–§2.2 exponent-transport identity, h·r′e1m·h−1 = ±(r′e2) m, holds for every valid conjugator h and every integer m — not just class T, but classes M and A too, unchanged (§4). Combining these: an ordinary §23.21-style bond defect on e1 gives k1 ; the SAME exponent-transportidentity that already built §23.21’s k2 formula for class T gives k2 for classes M and A too, with no modification; and Re(Lt) computed directly from the resulting quaternions matches the §23.28 closedform exactly. Verified on 1,056 cases spanning classes T, M and A, 4 hinges, and δ = 1..4, with maximum error 1.8×10−12 (§5). This closes both items of §23.28 §7: a discrete loop for M and A exists, built entirely from wheel-steps and the ordinary class-T crossing, chained through 1 or 2 intermediate vertices; and its defect-multiplicity exponents are exactly the §23.21 ones, since their derivation never used a shared tetrahedron between e1 and e2 in the first place — only that a conjugator h exists, which §23.28 §5(check D) already established for every class. Keywords600-cell · vertex figure · exponent transport conjugator h · wheel · discrete holonomy loop · classes M/A · exhaustive verification