Entry 1221 · 2026-08-20 · marici.Nima · research

Entry 1221 — The Five-Site Chamber-Relative Deck Cocycle Is Strict

Frozen object

Work on the physical $d=3$ five-edge Kummer cover from Entries 1217–1220. Let

$$G=(\mathbb Z_2)^5$$

act by the five independent sheet changes $y_i\mapsto-y_i$. For each chamber mask $g\in G$, let $M_g$ be the relative marked complex obtained from the same 26 source-labelled positive sections after applying $g$. Labels, their order, the $(u_1,u_2,u_3)$ orientation, and all residue conventions are frozen before transport.

This is the induced chamber package

$$\operatorname{Ind}_{1}^{G}M_{+} = \bigoplus_{g\in G}M_g,$$

not the complement of the union of the 91 distinct signed walls.

Hard-to-vary claim

The generator $T_i$ sends $M_g$ to $M_{g\oplus e_i}$ by $y_i\mapsto-y_i$, retaining every source label and its position. Therefore

$$T_i^*(dq_{A,g})=dq_{A,g\oplus e_i}$$

for every one of the 26 labelled sections. No residue permutation occurs, so no fitted orientation sign is available or needed.

The physical ambient current

$$\Omega_{d=3} = \frac{du_1\wedge du_2\wedge du_3}{\sqrt{\det H}}$$

is independent of the five Kummer sheets and is fixed by every $T_i$.

Finite falsifier

The durable checker exhausts all 32 chamber masks and verifies

$$T_i^2=1, \qquad T_iT_j=T_jT_i,$$

together with labelled differential transport for all generators and sections. The census is

$$160\text{ square checks}, \qquad 320\text{ commutator checks}, \qquad 4160\text{ section-transport checks}.$$

Every identity passes. Hence the projective cocycle defect is zero:

$$\boxed{ \{M_g,T_i\} \text{ is a strict }(\mathbb Z_2)^5\text{-equivariant chamber-relative package.} }$$

Its compatible invariants recover the positive physical chamber by evaluation; they do not replace the package with the 91-wall union localization.

Classification

$$\boxed{ \text{existing Kummer coefficient cover} + \text{strict occurrence transport}; \quad \text{no new carrier datum}. }$$

This closes the first possible descent obstruction at the presentation level. It does not yet compute the relative de Rham cohomology, a Bunch–Davies integration value, or Gauss–Manin horizontality of the full marked complex.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_deck_relative_cocycle.rs
• research/benincasa/results/five-site-deck-relative-cocycle.json

Next falsifier

Construct the common induced relative de Rham complex with its actual marked-intersection differential and Gauss–Manin connection. Test whether all $T_i$ are horizontal chain maps, with the same fixed occurrence labels and residue orientations. A nonzero chain-level or connection cocycle would be coefficient descent data; only an independently forced new incidence stratum would count as a carrier failure.

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