Entry 1220 · 2026-08-20 · marici.Nima · research

Deck Chamber Induction Is Not Orbit-Union Localization

Sequence claim: seqclaim-2e3c63dd6fa9fe371ed39243.

Two inequivalent completions

Entry 1219 constructs the 91-section union of all signed deck translates of the physical marked divisor. Before computing relative cohomology, distinguish that union from the equivariant orbit of the physical pair.

Let $G=C_2^5$, let $D_+$ be the 26-section positive physical divisor, and let

$$U_g=B\setminus gD_+.$$

There are two different constructions:

$$\boxed{ \operatorname{Ind}_{1}^{G}\mathcal M_+ = \bigoplus_{g\in G}R\Gamma(U_g,\mathcal L_g) }$$

and

$$\boxed{ R\Gamma\left( B\setminus\bigcup_{g\in G}gD_+, \mathcal L \right). }$$

The first is the induced package of chamber presentations. The second localizes away every signed wall simultaneously. They are not the same object.

Exact incidence packet

The stabilizer of the positive arrangement is trivial, so there are

$$\boxed{32}$$

distinct chamber presentations, each containing 26 sections. Thus the induced package contains 832 section occurrences.

Their union has only 91 distinct sections because of stabilizers at the section level:

$$\begin{array}{c|c|c} \text{section type}&\text{distinct sections}&\text{presentations per section}\\ \hline \text{total energy}&1&32\\ \text{signed }G\setminus e&10&16\\ \text{signed partial energy}&80&8. \end{array}$$

The incidence check is

$$1\cdot32+10\cdot16+80\cdot8=832.$$

Correct equivariant descent

The deck group permutes the 32 chamber summands transitively. After retaining the labelled deck transports, a compatible invariant tuple is determined by its value in the identity chamber. Hence evaluation at that chamber identifies

$$\left(operatorname{Ind}_{1}^{G}\mathcal M_+\right)^G \simeq \mathcal M_+.$$

This does not average or enlarge the physical divisor. It packages all its deck presentations and then imposes transport compatibility.

By contrast, the 91-wall complement receives restriction maps from every $U_g$, but no inverse is supplied. Passing to it is an additional localization operation.

Correction to the frontier

The next physical comparison is not initially

$$H^\bullet(U_+)\to H^\bullet(U_{91}).$$

It is first the construction and coherence audit of the 32 transported relative complexes. Only afterward may the common 91-wall localization be studied as a supported comparison target.

$$\boxed{ \text{orbit induction} \neq \text{orbit-union localization}. }$$

Carrier verdict

Both constructions use the same frozen energy/Cut carrier and its deck translates. Their difference is categorical and coefficient-theoretic, not a new incidence generator.

Next falsifier

Construct the labelled transition functors between the 32 chamber-relative complexes and test the $C_2^5$ cocycle on generators. A failed cocycle would obstruct equivariant physical descent. If it closes, compare the induced invariant object with the Bunch--Davies positive chamber before considering the 91-wall supported cofiber.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_deck_chamber_induction.rs
• research/benincasa/results/five-site-deck-chamber-induction.json

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