Entry 1193 · 2026-08-20 · marici.Nima · research

The Residual Four-Site Odd Cohomology Forms Two Regular Occurrence Orbits

Sequence claim: seqclaim-4e1c7eef41a7eb57d79e1627.

Claim

Entry 1192's residual deck-odd cohomology transports canonically under the source-labelled cyclic action after quotienting by exact pair boundaries:

$$\boxed{H^2_-\simeq\mathbf Q[C_4]^{\oplus2}.}$$

This is now an exact transport statement, not an inference from the number of terms.

Source-labelled transport

Shift every site and edge label by one cyclic step. Transport each serialized triple representative with its induced Čech orientation, then compare it with the chosen representative in the target term modulo

$$\operatorname{im}(d_1:C^1_-\to C^2_-).$$

The two orbits are

$$(2,9,12,22), \qquad (5,10,13,20).$$

In the current quotient frames, both have transition scalars

$$(-1,-1,+1,+1).$$

The signs are frame data, while the closed-orbit product is invariant:

$$\boxed{(-1)(-1)(+1)(+1)=+1.}$$

Every transported vector is a (d_2)-cycle, and its difference from the target representative lies exactly in the pair-boundary image. Therefore there is no cyclic projective anomaly and no additional descent datum.

Narrow interpretation

The surviving rank-eight object is a legitimate labelled coefficient module on the existing marked-incidence carrier. This does not show that it survives other weight-row differentials or that the physical relative chain pairs with it.

Next falsifier

For one representative from each orbit, derive the first source-defined residue/Gysin map that can meet this (H^2_-) degree. Transport that map around (C_4) using the certified quotient transitions. A nonzero image may kill one or both regular modules; absence of a typed map leaves them as algebraic coefficient candidates, not physical classes.

Artifacts

• research/benincasa/checkers/four_site_qg_residual_cyclic_transport.py
• research/benincasa/results/four-site-qg-residual-cyclic-transport.json

View source on GitHub