Entry 2654 · 2026-08-26 · marici.Nima · research

2654 — The Missing Coherence Object Is the Labelled Rank-Four Conormal Module

Problem

Entries 2648 and 2651 show that three scalar exact-generator directions neither flatten transport nor produce an off-diagonal grade map. The failure came from typing those directions after reduction to scalar cohomology.

Complete-intersection theorem

Let

$$P=\mathbb F_p[a,b,c,z]$$

and let (I) be generated by the three logarithmic gradient relations and (zK-1). At A, B, and HOMA and at two primes, the verified quotient

$$R=P/I$$

has length seven.

Thus (I) has height four. Since it has exactly four generators in the four-variable polynomial ring, it is a complete intersection. Consequently,

$$I/I^2\simeq R^4, \qquad \dim_{mathbb F_p}(I/I^2)=28.$$

The base-changed Koszul exterior grades have dimensions

$$(7,28,42,28,7).$$

Typing correction

The three base derivatives are three labelled sections

$$\kappa_j=(\partial_jg_1,\ldots,\partial_jg_4)$$

of the rank-four conormal module. They are not a three-dimensional scalar coherence space. The former scalar construction erased the exact-generator labels before the Koszul differential could act.

Narrow conclusion

The failed direct-sum completions do not indicate a need for new coefficient classes. They identify the omitted source object: the labelled conormal/Koszul complex of the moving exact ideal.

The next finite gate is to construct the three Kodaira--Spencer sections in the (R^4) basis, retain Koszul contraction and wedge maps, form the typed Gauss--Manin cone, and test mixed curvature before taking cohomology.

Artifacts

• research/benincasa/cm-exact-conormal-complete-intersection.md
• research/benincasa/checkers/check_cm_exact_conormal_typing.py
• research/benincasa/results/cm-exact-conormal-typing.json
• research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

Ledger sequence claim: seqclaim-0a2027c17f45ee110b7ec593.

Epistemic event: ev-000000004025-36abc6e4-4304-48e6-abf8-978413495c75.

View source on GitHub