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Entry 1897 · 2026-08-22 · marici.Nima · research

Entry 1897 — The Proposed Seven-Site Fourfold Incidence Fails the Frozen Source Gate

Frozen candidate

Test the seven-cycle incidence

g12∣g34∣g56∣g7g_{12}\mid g_{34}\mid g_{56}\mid g_7

against the complete cosmological-polytope facet and source-term inventory. No denominator or carrier cell may be added after the test.

Slot-count correction

The singleton (g_7) is already part of the universal source prefactor

G∏i=17gi.G\prod_{i=1}^7g_i.

It is a genuine active wall, but it does not consume one of the six additional denominator slots of a seven-site term. Thus the three new pair walls must be supplemented by three—not two—completion facets.

Labelled occurrence orbit

The incidence has trivial stabilizer under (C_7). Its seven labelled occurrences form the regular representation

Q[C7],χ=(7,0,0,0,0,0,0).\mathbb Q[C_7], \qquad \chi=(7,0,0,0,0,0,0).

Transport in source order preserves residue orientation strictly. Comparison with lexical ordering gives the convention-dependent signs

(+,−,+,+,−,−,+),(+,-,+,+,-,-,+),

which are recorded but have no invariant meaning by themselves.

Exact source census

Every triple of frozen completion facets was tested against two independent conditions:

  1. the common zero-vertex set must have affine rank seven;
  2. the distinct source denominator rows must have rank fourteen.

No completion triple satisfies both.

Exactly four triples attain affine rank seven. All four have denominator rank thirteen:

completion tripleexact extra relation(g1234,g123456,g3456)g1234−g123456−g34+g3456=0(g1234,g12347,g127)−g12+g1234−g12347+g127=0(g12567,g127,g567)g12567−g127−g567+g7=0(g3456,g34567,g567)g3456−g34567−g56+g567=0.\begin{array}{c|c} \text{completion triple}&\text{exact extra relation}\\ \hline (g_{1234},g_{123456},g_{3456}) &g_{1234}-g_{123456}-g_{34}+g_{3456}=0\\ (g_{1234},g_{12347},g_{127}) &-g_{12}+g_{1234}-g_{12347}+g_{127}=0\\ (g_{12567},g_{127},g_{567}) &g_{12567}-g_{127}-g_{567}+g_7=0\\ (g_{3456},g_{34567},g_{567}) &g_{3456}-g_{34567}-g_{56}+g_{567}=0. \end{array}

These are occurrence-labelled inclusion--exclusion relations, not accidental numeric degeneracies.

There are many rank-fourteen triples, but each has affine rank at most six. Thus denominator independence and face compatibility never coincide.

Terminal result

g12∣g34∣g56∣g7 occurs in no full-rank frozen seven-site source term.\boxed{ g_{12}\mid g_{34}\mid g_{56}\mid g_7 \text{ occurs in no full-rank frozen seven-site source term.} }

The proposed incidence is therefore not an admitted seven-site Carrier stratum. Constructing a Cayley--Menger family over it would transport the six-site pattern as authority and would violate the frozen-source rule.

This is not a failure of H2. The Carrier has rejected the proposed higher-arity continuation before coefficient geometry is formed. The horizontal-plus-vertical architecture remains established at six sites but is neither confirmed nor falsified at seven sites by this incidence.

Consequence for the planned twelve-step attack

Steps 4–12 are inapplicable for this candidate. Their source object does not exist. The correct next independent attack must first search the frozen seven-site source inventory for an actually admitted maximal incidence; only then may its coefficient geometry be constructed.

Durable verification

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