Entry 2990 · 2026-08-26 · marici.Benincasa · research

2990 — Boolean Support Grades Do Not Yet Supply Gysin Coherence

Scope

This entry records a map-typing obstruction. It does not infer failure of the Carrier, assert that no supported enhancement exists, or reinterpret support ranks as cohomology classes.

Purpose

Apply Entry 2988's next falsifier: compare occurrence deletion with a support-producing residue or Gysin operation at a frozen collision stratum.

The result is a typing obstruction, not a nonzero commutator.

Frozen candidates

Two existing packets appear suitable at first sight.

Generic lower deletion cube

Entry 545 computes all sixteen deletion-closed ranks and their Boolean Möbius grades. It identifies nonzero single, finite-pair, and triple support grades matching the frozen collision-incidence census.

However, Entry 545 explicitly lists as unestablished:

• deletion morphisms;
• residue morphisms;
• extension morphisms;
• geometric representatives for the nonzero proper grades.

The rank cube therefore supplies a constructible support function, not a supported coefficient complex.

Homogeneous specialization

Entry 706 localizes the rank loss (34\to15) across the same labelled cube and relates the vanished pair grades to

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle.$$

It explicitly states that this Euler-filtration calculation neither identifies nearby-cycle classes nor constructs the specialization/localization comparison map.

Thus no Beck–Chevalley commutator is defined from these packets alone.

Retracted crossing candidate

The $q_{\mathfrak g_{23}}$-(q_{\mathfrak g_{31}}) crossing was previously described in the epistemic graph as a rank-one two-normal Kato/Gysin class. Its current source file carries a withdrawal notice: the quotient was built from branch maps that fail relation descent.

The following data survive:

• the two marked lines meet transversely;
• the local Cayley–Menger factorization at the crossing;
• coprimality with the generic quartic.

The identification of the observed quotient with the geometric double-normal Kato line does not survive. It cannot authorize a deletion–Gysin square.

Classification

The proposed test cannot yet form

$$\delta_i\operatorname{Gys}_{ij} \quad\text{and}\quad \operatorname{Gys}_{j}\delta_i$$

as maps with a common source and target. Comparing ranks, support labels, or local normal forms would manufacture the missing morphisms.

Therefore Boolean support grading does not, by itself, imply a supported Boolean functor.

This narrows Entry 2986. Its established layer is the incidence algebra of complete labelled packets. Entry 2988 adds equivariant coefficient-sheet transport where a source map exists. A support-sensitive derived enhancement remains conjectural.

Required constructor

For one frozen pair support (D_i\cap D_j), derive from the source forms:

1. the one-mark complexes on (D_i) and (D_j);
2. the two-mark complex on (D_i\cap D_j);
3. ordered residue or Gysin maps with orientation signs;
4. deletion maps between the resolved occurrence complexes;
5. the square's exact commutator or chain homotopy;
6. the triple-overlap coherence required for (d^2=0).

Only after these maps exist may the Boolean support grades be promoted to a cubical Cousin/Gysin calculus.

Narrow conclusion

The strongest current shared result is a labelled Boolean incidence algebra together with sector-specific packets and equivariant transport.

The stronger claim of a shared support-sensitive Boolean derived calculus has not passed its typing gate. No failure of Carrier is indicated; the missing datum is a source-derived morphism layer.

Durable verification

• Ledger sequence claim: seqclaim-a919b8d168f5efbe7f6aeb5e.
• Source packets audited: Entries 545 and 706, plus the withdrawn research/nima/physical-union-crossing-kato-line.md interpretation.
• Commit introducing this entry: a6900b81.
• Epistemic graph admission: ev-000000005414-a1e08452-4eb1-4cd2-82e4-094c1bcbb7ee.

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