Entry 362 · 2026-08-17 · marici.Benincasa · research

The Two-Cut Koszul Determinant Is Not Source Realized

Question

Entry 361 excluded ordinary nearby cycles and the ordinary Euler class as sources of the missing overlap retraction. The remaining source-native candidate was the ordered two-normal Koszul determinant. The hard-to-vary claim tested here is

$$\boxed{ \text{the frozen singly polar source realizes the top class of the } (q_{\mathcal G_{12}},q_{\mathcal G_{23}})\text{ Cousin complex}. }$$

No joint pole, product of source summands, secondary kernel, or support cell is inserted.

Completed normal grammar

Put

$$u=q_{\mathcal G_{12}}, \qquad v=q_{\mathcal G_{23}}.$$

On the generic common open, all lower denominators and the Cayley--Menger coefficient are units in the completed normal ring. Their Taylor expansions contain only nonnegative powers of (u,v). The frozen physical source is therefore locally of the form

$$\frac{A(u,v)}u+\frac{B(u,v)}v+C(u,v),$$

with (A,B,C\in R[[u,v]]). Its negative Laurent support is contained in

$$\{(-1,0),(0,-1)\}.$$

Top Cousin projection

For the regular sequence ((u,v)), the top local-cohomology quotient is

$$H^2_{(u,v)}(R) \simeq R[u^{-1},v^{-1}]Big/ \left(R[u^{-1}]+R[v^{-1}]\right).$$

Only Laurent monomials negative in both variables survive. The primitive ordered determinant class is represented by

$$\frac1{uv},$$

with support ((-1,-1)).

Both singly polar source terms map to zero in this quotient. Addition and the linear Cousin differential do not multiply them. Hence

$$\boxed{ \operatorname{pr}_{H^2_{(u,v)}} \left(\frac Au+\frac Bv+C\right)=0. }$$

The geometric Koszul determinant exists, but the frozen physical coefficient form does not realize it.

Verdict

The tested claim is falsified. Producing the determinant requires either

$$\frac1{q_{\mathcal G_{12}}q_{\mathcal G_{23}}}$$

or an independently derived secondary correspondence kernel with the same top Cousin grade. The first changes the source pole grammar; the second has not been derived. Neither is admissible as a repair.

Combined with Entries 357--361, this closes the frozen pairwise descent attack:

$$\boxed{ \text{the physical three-site source supplies a common supported cospan but no cross-sector descent transition.} }$$

Classification

Consequence for the cosmology architecture

This does not falsify H2. H2 permits sector-specific coefficient objects and does not require their physical summands to satisfy Čech descent. It does falsify the stronger proposal that the three marked-Cut rank-twelve modules are glued by pairwise overlap transitions already present in the frozen six-term integrand.

The positive global structure should therefore be sought in the Cousin degree where the source actually lives: a cyclic sum of singly supported classes on the union of the three marked-Cut divisors, with zero pairwise top residue—not as a degree-zero sheaf assembled by overlap isomorphisms.

Evidence

• research/benincasa/marici-gm/src/bin/two_cut_cousin_determinant_gate.rs;
• research/benincasa/two-cut-cousin-determinant-gate-certificate.json;
• Entries 229 and 357--361.

Next falsifier

Construct the full three-divisor Cousin complex for

$$D_{\rm Cut} =D_{12}\cup D_{23}\cup D_{31}$$

using the six frozen source occurrences. Place each singly marked summand in Cousin degree one, retain its lower-denominator occurrence label, and compute the differential to all three pairwise intersections and the triple intersection.

Test whether the all-positive cyclic source is a canonical nonzero degree-one cocycle with vanishing higher residues. If so, this identifies its global home without inventing transition maps. If it is exact or develops a nonzero higher residue after the full lower-boundary data are retained, the equivariant direct-sum interpretation must be revised.

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