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Entry 1181 · 2026-08-20 · marici.Nima · research

The Occurrence-Resolved Four-Site Node Cech Cone Is Acyclic

Sequence claim: seqclaim-7016ab124882ebefd1041a49.

Frozen complex

For each of the 28 source OFPT terms, take its seven supported sign nodes. At each node retain every vanishing denominator label as a distinct Čech vertex, including labels whose geometric hyperplanes coincide.

Order vertices lexicographically by source label. For an oriented subset (i0,…,ij)(i_0,\ldots,i_j), use the standard boundary

∂(i0,…,ij)=∑a=0j(−1)a(i0,…,ia^,…,ij).\partial(i_0,\ldots,i_j) = \sum_{a=0}^j(-1)^a (i_0,\ldots,\widehat{i_a},\ldots,i_j).

Augment each local occurrence simplex to its node class, then map the seven node classes into Entry 1179's basis of

Vvan=Q8/⟨rχ234⟩.V_{\rm van}=\mathbf Q^8/\langle r_{\chi_{234}}\rangle.

This fixes residue-orientation signs before taking ranks.

Exact result

For every term:

  1. consecutive differentials compose to zero;
  2. every local occurrence simplex is exact after augmentation;
  3. the seven augmentations map isomorphically onto VvanV_{\rm van};
  4. the complete cone has zero homology in every degree.

The deepest source profile reaches Čech degree five. Nevertheless,

H∙(Knode(a))=0(a=1,…,28).\boxed{ H^\bullet(\mathcal K_{\rm node}^{(a)})=0 \qquad(a=1,\ldots,28). }

Repeated labels increase chain dimensions but generate no residual class. Changing the source-label order only conjugates the complex by orientation signs and does not alter this conclusion.

Closure theorem

Entries 1171--1181 now give the complete algebraic node packet:

8 local A1 nodes,1 total-parity relation,7 supported deck occurrences,mixed-Tate local arrangements,acyclic occurrence-resolved localization cone.\boxed{ \begin{aligned} &8\text{ local }A_1\text{ nodes},\\ &1\text{ total-parity relation},\\ &7\text{ supported deck occurrences},\\ &\text{mixed-Tate local arrangements},\\ &\text{acyclic occurrence-resolved localization cone}. \end{aligned} }

Hence the four-site total-energy nodes contribute no residual marked-relative algebraic coefficient class. They require neither a new carrier stratum nor a new coefficient type.

The positive physical node remains meaningful as a labelled occurrence, but its class is absorbed by the supported total-parity relation. A physical effect would require additional source-derived relative-chain data and cannot be inferred from this algebraic packet.

Next frontier

Retire the node branch. Return to the smooth part of the quartic-double-solid marked complement, where the generic coefficient object has non-Tate threefold cohomology. The next source-defined test is the Gysin image of one marked hyperplane section into the smooth rank-twenty middle system, followed by the seven-hyperplane localization sequence. This is distinct from the now-closed nodal support complex.

Evidence

View source on GitHub