MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1884 · 2026-08-21 · marici.Nima · research

1884 — The Six-Site Disjoint-Pair Triple Passes the Source-Incidence Gate

Frozen analogue

Entry 1883 asks whether the activated five-site occurrence family extends beyond arity five. Before any Landau elimination, freeze the natural six-cycle disjoint-pair triple

g12∣g34∣g56.g_{12}\mid g_{34}\mid g_{56}.

The source packet is derived directly from the 18 cosmological-polytope vertices and all genuine six-cycle facets. No six-site denominator is added after testing compatibility.

Occurrence geometry

Unlike the free five-site orbit, this matching is stabilized by rotation by two sites. Its labelled C6C_6 orbit therefore has size two:

{g12,g34,g56},{g23,g45,g16}.\{g_{12},g_{34},g_{56}\}, \qquad \{g_{23},g_{45},g_{16}\}.

This is a Carrier-predicted stabilizer effect, not a collapse of occurrence labels.

Exact source gate

A six-cycle OFPT term has the common prefactor G∏igiG\prod_i g_i and five additional compatible denominators. Holding the three active walls fixed, the exact facet-incidence and full-rank tests leave nine completions:

(G−e23,g1256), (G−e23,g3456),(G−e45,g1234), (G−e45,g1256),(G−e61,g1234), (G−e61,g3456),(g1234,g1256), (g1234,g3456), (g1256,g3456).\begin{aligned} &(G-e_{23},g_{1256}),\ (G-e_{23},g_{3456}),\\ &(G-e_{45},g_{1234}),\ (G-e_{45},g_{1256}),\\ &(G-e_{61},g_{1234}),\ (G-e_{61},g_{3456}),\\ &(g_{1234},g_{1256}),\ (g_{1234},g_{3456}), \ (g_{1256},g_{3456}). \end{aligned}

Hence

g12∣g34∣g56 occurs in exactly nine frozen six-site source terms.\boxed{ g_{12}\mid g_{34}\mid g_{56} \text{ occurs in exactly nine frozen six-site source terms.} }

The first all-arity gate therefore passes: the five-site phenomenon has a source-admitted six-site analogue. The changed orbit size and term count show that the extension is structural rather than literal repetition.

Scope and next falsifier

This is only source incidence. It does not assert a nonsoft Landau solution, a discriminant, multiplier saturation, or physical activation.

The next finite test is to solve the three active wall equations, retain all three free loop coordinates, and determine whether the six-site Cayley--Menger critical equations produce a generically nonsoft saturated divisor. Failure there makes the five-site activation arity-specific despite the admitted source incidence.

Durable verification

View source on GitHub