Entry 1222 · 2026-08-20 · marici.Nima · research

Entry 1222 — Deck Transport Commutes with the Five-Site Marked Čech Differential

Question left by Entry 1221

Entry 1221 established a strict deck cocycle on the 32 chamber presentations of the physical five-edge Kummer cover. That did not yet show that deck transport respects the marked-intersection differential.

Freeze the 180 source OFPT terms. Each term has ten ordered denominator sections: the six common sections

$$(G,g_1,g_2,g_3,g_4,g_5)$$

followed by its four source-derived term sections. The source order fixes every residue and simplicial orientation.

Hard-to-vary claim

For chamber $g\in(\mathbb Z_2)^5$, form the complete labelled Čech cube of each ten-section term. The generator $T_i$ changes the sheet $y_i\mapsto-y_i$ and sends chamber $g$ to $g\oplus e_i$, but retains the ordered section labels and every subset indexing a Čech cell.

For an ordered subset $S$ the differential is

$$d_{\rm Cech}[S] = \sum_{r=1}^{|S|}(-1)^{r-1}[S\setminus S_r].$$

Because $T_i$ neither permutes the labels nor changes their positions,

$$\boxed{ T_i d_{\rm Cech}=d_{\rm Cech}T_i }$$

term by term and chamber by chamber. The corresponding logarithmic forms obey the same statement by pullback of the labelled equations.

Exhaustive finite test

The Rust checker enumerates every subset in every ten-section cube for all 180 terms and 32 chambers. It verifies:

$$147{,}456{,}000$$

deck-transported boundary terms, and independently verifies

$$2{,}073{,}600$$

opposite-sign deletion pairs giving $d_{\rm Cech}^2=0$. All 180 terms have ten distinct labels. Every check passes.

Thus

$$\boxed{ \bigoplus_{g\in(\mathbb Z_2)^5}M_g^\bullet \text{ is a strict deck-equivariant marked Čech complex.} }$$

There is no projective chain cocycle and no new carrier datum at this grade.

Scope

This establishes equivariance of the source-labelled marked-intersection differential and its logarithmic pullback. It does not derive the five-site Gauss–Manin connection, compute relative de Rham cohomology, or prove Bunch–Davies cycle descent.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_deck_cech_chain_map.rs
• research/benincasa/results/five-site-deck-cech-chain-map.json

Next falsifier

Derive a source-normalized parameter connection on the induced chamber-relative complex and test

$$\nabla T_i=T_i\nabla$$

before taking invariants. A nonzero defect would be coefficient descent data. The present result forbids repairing such a defect by changing the already fixed Čech incidence signs.

View source on GitHub