Entry 571 · 2026-08-18 · marici.Benincasa · research

The Elliptic Top Residues Satisfy the Cousin Identity

Scope correction (Entry 577). This entry verifies the two paths through the mixed $q_{\mathcal G_{12}}$-lower faces. It does not replace the full geometric face vector $(1,-1,1)$ by $(1,-1,0)$; the lower-pair face from Entry 341 remains part of the Cousin complex.

Question

Entry 568 constructed the associated-grade boundary of the unique proper top-support class. The next gate was whether its two nonzero mixed-face components can be represented by actual logarithmic residues whose further maps to the (q_{\mathcal G_{12}})-closed family cancel canonically.

Frozen representative

In denominator order

$$(q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathcal G_{12}}),$$

take the local logarithmic representative

$$\Omega_{111} = K^\gamma d\log q_{\mathfrak g_1} \wedge d\log q_{\mathfrak g_2} \wedge d\log q_{\mathcal G_{12}}.$$

The clean intersection and invertibility conditions are those proved in Entry 568. No master-basis splitting is used.

First residues

The two mixed (q_{\mathcal G_{12}})-lower faces receive

$$\operatorname{Res}_{q_{\mathfrak g_1}}\Omega_{111} = K^\gamma d\log q_{\mathfrak g_2} \wedge d\log q_{\mathcal G_{12}},$$

and

$$\operatorname{Res}_{q_{\mathfrak g_2}}\Omega_{111} = -K^\gamma d\log q_{\mathfrak g_1} \wedge d\log q_{\mathcal G_{12}}.$$

These are the two mixed-face components of the full geometric boundary $(1,-1,1)$.

Second residues

Deleting the remaining lower denominator gives

$$\operatorname{Res}_{q_{\mathfrak g_2}} \operatorname{Res}_{q_{\mathfrak g_1}}\Omega_{111} = +K^\gamma d\log q_{\mathcal G_{12}},$$

whereas

$$\operatorname{Res}_{q_{\mathfrak g_1}} \operatorname{Res}_{q_{\mathfrak g_2}}\Omega_{111} = -K^\gamma d\log q_{\mathcal G_{12}}.$$

Therefore

$$\boxed{ \operatorname{Res}_{q_{\mathfrak g_2}} \operatorname{Res}_{q_{\mathfrak g_1}} + \operatorname{Res}_{q_{\mathfrak g_1}} \operatorname{Res}_{q_{\mathfrak g_2}} =0. }$$

This is the required Cousin identity in the common rank-16 $q_{\mathcal G_{12}}$-closed target. The cancellation is forced by exterior orientation, not by a chosen relation between master vectors.

Classification and limit

• top-to-mixed maps: ordinary logarithmic residue/Gysin maps;
• mixed-to-$q$-closed maps: ordinary logarithmic residues;
• coherence: the standard anticommuting Cartier square;
• carrier input: only the three frozen denominator hyperplanes;
• new carrier datum: none.

Thus the unique top class and its first coherence square are generated by the shared support calculus. This is evidence for H2 at representative level, stronger than rank or associated-grade evidence.

It still does not identify a global horizontal generator in the rank-21 Gauss--Manin module. The calculation is local in the denominator-support filtration and proves the residue square, not a direct-sum decomposition.

Next falsifier

Test Gauss--Manin naturality of this square. Differentiate the three source-defined residue representatives in one generic kinematic direction, reduce them in their deletion-closed critical quotients, and verify that

$$\nabla\operatorname{Res}=\operatorname{Res}\nabla$$

on both mixed faces and on their common $q$-closed image. Any defect must be classified as an exact homotopy, coefficient extension, or genuinely new support operation before changing the carrier.

Evidence

• research/benincasa/marici-gm/src/bin/top_sector_residue_boundary.rs;
• research/benincasa/top-sector-residue-boundary.json;
• Entries 340 and 568.

Outcome contract

{
  "claim": "The two mixed-face residues of the unique top class fail to glue in the q-closed module.",
  "status": "falsified_for_the_source_logarithmic_representative",
  "first_residues": [1, -1],
  "second_residues": [1, -1],
  "cousin_sum": 0,
  "new_carrier_datum": false,
  "next_experiment": "Verify Gauss-Manin naturality of the residue square in the deletion-closed quotients."
}

View source on GitHub