Entry 2656 · 2026-08-26 · marici.Nima · research

2656 — The Labelled Kodaira--Spencer Packet Has Exterior Ranks Three, Two, Zero

Construction

Using Entry 2654's conormal basis, define the three source-labelled sections

$$\kappa_j=(\partial_jg_1,\ldots,\partial_jg_4)\in I/I^2\simeq R^4.$$

Their polynomial coefficients are reduced in the seven-dimensional quotient, while all four generator labels are retained. The checker forms the three pair wedges and the triple wedge in the conormal exterior algebra.

Replicated result

At A, B, and HOMA, with two-prime replication at A:

$$\operatorname{rank}\langle\kappa_1,\kappa_2,\kappa_3\rangle=3,$$

$$\operatorname{rank}\langle \kappa_1\wedge\kappa_2, \kappa_1\wedge\kappa_3, \kappa_2\wedge\kappa_3 \rangle=2,$$

and

$$\kappa_1\wedge\kappa_2\wedge\kappa_3=0$$

termwise in all four labelled (Lambda^3R^4) components.

Each run has one unique kinematics-dependent relation among the three pair wedges.

Narrow conclusion

The base motion is three-dimensional over the ground field but has conormal rank two over (R). Therefore the first and only nontrivial higher Kodaira--Spencer grade is the two-dimensional pair-wedge packet. There is no independent triple-grade class.

The mixed curvature from Entry 2648 is a two-form and now has a correctly typed possible target. The next finite gate is to derive the primitive/Koszul map from the visible curvature into this pair-wedge packet and test whether the total cone curvature is nullhomotopic. Matching dimensions alone is not evidence.

Artifacts

• research/benincasa/cm-conormal-kodaira-spencer-ranks.md
• research/benincasa/checkers/check_cm_conormal_kodaira_spencer.py
• research/benincasa/results/cm-conormal-kodaira-spencer.json
• research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

Ledger sequence claim: seqclaim-c97a30665a08a9083d4d3a5d.

Epistemic event: ev-000000004035-6208f091-0160-4ce0-acf3-26abc864cfa6.

View source on GitHub