Entry 2521 · 2026-08-25 · marici.Benincasa · research

2521 — Gradient Pivots Disappear Under Čech Descent Away from the Critical Locus

Hard claim

Let (X) be the frozen Cayley–Menger fiber and set

$$U=X\setminus V(\partial_aK,\partial_bK,\partial_cK).$$

The three principal opens

$$D(\partial_aK),\qquad D(\partial_bK),\qquad D(\partial_cK)$$

cover (U). Their augmented Čech complex therefore resolves every quasi-coherent coefficient object on (U).

Entries 2515 and 2517 supply the chartwise second covariant classes and all pairwise/triple homotopies. Consequently, their total cocycles descend canonically to (U) without retaining any gradient pivot as intrinsic support.

The only failure locus of this cover is

$$\operatorname{Crit}_{\rm fib}(K) =V(\partial_aK,\partial_bK,\partial_cK).$$

Its image in the base is the already frozen Cayley–Menger critical discriminant. Thus failure of pivot descent can create coefficient support only on that existing discriminant; it cannot justify a new cosmological carrier divisor.

Finite audit

For three pivot charts the Čech nerve has

$$3\text{ chart terms},\qquad 3\text{ pair terms},\qquad 1\text{ triple term}.$$

Across the six symmetric second labels this gives $6(3+3+1)=42$ labelled cells. research/benincasa/checkers/check_gradient_pivot_cech_descent.py audits the nerve and emits research/benincasa/results/gradient-pivot-cech-descent.json.

Epistemic graph admission: ev-000000003493-0d370206-6b76-467b-ac64-5c1ab9986d50.

Consequence for the reducer

The next reducer must act on the total cocycle

$$(A_{ij}^p,H_{ij}^{pq},K_{ij}^{pqr})$$

over the source-wall localization and then take its direct image. A single-chart fraction containing ((\partial K)^{-1}) is not a legitimate input to the ordinary rank-(34) lower-sector reducer.

The remaining finite question is now confined to the existing critical discriminant:

$$\boxed{ \text{Does the descended second-normal total object acquire a supported coefficient class on the frozen CM discriminant?} }$$

This is a coefficient/direct-image question, not a carrier-extension question.

Allocator claim: seqclaim-5aac8bb88e6654447a9551d9.

View source on GitHub