Entry 1910 · 2026-08-22 · marici.Benincasa · research

1910 — The Literal Eight-Site Chamber Has Zero Ordinary Supported Pullback

Correction

The original draft incorrectly promoted empty ordinary support to vanishing of Leray/nearby activation. Entry 1083 is a counterexample to that inference: an empty literal intersection can acquire a nonzero source-defined double-Leray specialization. This entry is therefore restricted to ordinary supported pullback. Leray/nearby activation remains open.

Correction admission: ev-000000002296-f649303c-d25d-4aa6-904c-373638c82e64.

Question

Entry 1909 found 36 source-admitted rank-four C8 strata with genuine horizontal coefficient discriminants, including

$$2k-3=0 \qquad\text{and}\qquad 7+6k-8l=0,$$

but no strictly positive-$X_i$ critical sheet. Does the literal Bunch--Davies chamber have a nonzero ordinary supported pullback to one of those coefficient divisors?

Frozen comparison

For a labelled orbit $O$, let $S_O$ denote its rank-four source wall and $D_O$ any irreducible component of its coefficient discriminant. The minimal ordinary typed comparison is

$$\Phi_O^{\rm ord}:\mathcal C_{\Gamma_{\rm BD}} \longrightarrow R\Gamma_{S_O\cap D_O}(\operatorname{gr}_{D_O}\mathcal V_O).$$

The source is the literal Bunch--Davies chain $\Gamma_{\rm BD}=\{X_i>0\}$, with its source boundary value. The target is supported on the source wall. No continuation class, support summand, or carrier facet is added after seeing $D_O$.

Ordinary restriction, ordinary boundary Gysin/residue, and ordinary localization/Cut support pullback are support-local. Consequently they cannot create a physical section after the support has empty pullback to $\Gamma_{\rm BD}$.

Exact support audit

The checker independently reconstructs every serialized Gordan witness. For all 36 orbits it obtains

$$g_O\in \operatorname{rowspan}(R_{X,O})\cap\mathbb Q_{\ge0}^{8}, \qquad g_O\ne0.$$

If $R_{X,O}X=0$ and $X_i>0$, then

$$0=g_O\cdot X>0,$$

which is impossible. Therefore

$$\boxed{\Gamma_{\rm BD}\cap S_O=\varnothing}$$

for every candidate before either the linear factor or its companion coefficient discriminant is imposed.

The complete family count is

$$21:\ 2k-3, \qquad 15:\ 7+6k-8l.$$

Each linear factor occurs with determinant valuation one. Its generic coefficient grade is therefore rank one and Cartier length one, with labelled conormal line fixed by the source-normalized adjugate identity

$$\lambda J=\det(J)e_1.$$

The 36 nonconstant companion discriminants are retained without assuming irreducibility or assigning an unsupported total nearby-cycle rank. Every component lies over the same physically empty $S_O$.

Typed map and cone

For every orbit,

$$\boxed{\Phi_O^{\rm ord}=0}$$

because its supported source object is zero. This happens before any weighted, normal, or analytic lift, so the result is strictly lift-independent. All eight occurrence charts and their residue-wedge signs are retained. Orientation changes multiply the zero map by $\pm1$.

The ordinary physical image has rank zero.

The comparison cone retains the coefficient grade but no physical class:

$$\boxed{ \text{nonzero coefficient support} + \text{zero physical image}. }$$

Narrow theorem

$$\boxed{ \text{Every divisor component of the 36 C8 coefficient discriminants has zero ordinary supported pullback from the literal Bunch--Davies chamber.} }$$

No new Carrier structure is required or justified. The result strengthens the surviving architecture

$$\text{shared Carrier and calculus} + \text{sector-specific coefficient objects} + \text{independent physical-chain selection}.$$

The published negative-imaginary prescription may instead determine a continued Leray germ. Whether that germ has nonzero Picard--Lefschetz pairing at the C8 discriminants is not inferred from ordinary support and is the next gate. Coefficient monodromy is not classified here.

Durable verification

• research/benincasa/eight-site-supported-activation-conventions.md
• research/benincasa/marici-gm/src/bin/eight_site_supported_activation_gate.rs
• research/benincasa/results/eight-site-supported-activation-gate.json
• source inventory theorem: Entry 1909
• ledger sequence claim: seqclaim-a3249cf0e89b85a42b3fe33d
• epistemic graph event: ev-000000002292-a99a061e-22cf-4fee-9b65-f9e25d0d5edd

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