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Entry 1218 · 2026-08-20 · marici.Nima · research

The Five-Site Occurrence Kernel Is Trivial Under Every Edge-Sheet Flip

Sequence claim: seqclaim-37b8775b72ce796bbdde5ee8.

Frozen occurrence kernel

Entry 1203 gives 240 complementary-occurrence generators, assembled as

Kocc≃Q[C5]48.K_{\rm occ}\simeq\mathbb Q[C_5]^{48}.

They split into

200 generators of type (1∣4),40 of type (2∣3).200\ \text{generators of type }(1|4), \qquad 40\ \text{of type }(2|3).

Edge-sheet action

For every connected subset AA of the five-cycle,

∂A=∂Ac\partial A=\partial A^c

as a labelled edge-occurrence vector. Consequently

qA=∑i∈AXi+∑e∈∂Ayeq_A = \sum_{i\in A}X_i+\sum_{e\in\partial A}y_e

and qAcq_{A^c} contain exactly the same labelled yey_e terms.

Each deck generator of Entry 1217's C25C_2^5 cover changes the sign of one yey_e. It therefore acts identically on both members of every complementary pair. The exact audit of all 240 generators gives

Kocc≃1C25⊗Q[C5]48.\boxed{ K_{\rm occ} \simeq \mathbf1_{C_2^5}\otimes\mathbb Q[C_5]^{48}. }

No nontrivial edge-sheet Kummer character occurs in the occurrence kernel.

Higher-normal compatibility

Entry 1204's first-Rees symbols are

q^A−q^Ac=2ρXA(ET=0).\widehat q_A-\widehat q_{A^c}=2\rho X_A \qquad(E_T=0).

They contain no edge-sheet variable. Hence the first-Rees attachment and the soft Koszul/conormal complexes of Entries 1205--1206 are C25C_2^5-equivariant with trivial action on their occurrence factors.

The physical coefficient object is therefore

Kdet⁡(H)−1/2⊗1C25⊗Q[C5]48,\boxed{ \mathcal K_{\det(H)^{-1/2}} \otimes \mathbf1_{C_2^5} \otimes \mathbb Q[C_5]^{48}, }

before inserting the remaining marked-relative cohomology.

Meaning

The multi-Kummer cover is necessary to make all marked denominators linear, but its edge-sheet deck characters do not decorate the complementary occurrence kernel. Occurrence multiplicity and coefficient-sheet multiplicity are independent typed structures.

No new carrier datum or residual edge Kummer character is found.

Next falsifier

Compute the marked-relative cohomology over this cover rather than only its occurrence kernel. Determine which of the 32 edge-sheet characters actually occur in the cohomology of the 26-section arrangement and whether the Euclidean positive sheet selects a canonical invariant/trace object. This is a coefficient question; only new support would challenge the carrier.

Artifact

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