Entry 2664 · 2026-08-26 · marici.Nima · research

2664 — The Ordinary Quotient Koszul Differential Cannot Kill the Curvature Obstruction

Frozen complex

For

$$I=(5K_a,5K_b,5K_c,zK-1)\subset P$$

and $R=P/I$, retain the four source-labelled generators and their ordinary Koszul complex. Its differential is

$$d(e_i)=g_i, \qquad d(e_i\wedge e_j)=g_i e_j-g_j e_i.$$

Exact quotient audit

At every replicated point and prime, reduction by the same Gröbner basis gives

$$g_1=g_2=g_3=g_4=0\quad\text{in }R.$$

Consequently every ordinary quotient Koszul differential vanishes. In exterior degrees $0$ through $4$, its ranks are

$$(0,0,0,0).$$

Narrow result

The ordinary quotient Koszul complex cannot supply a boundary for Entry 2662's nonzero rank-four obstruction. The candidate coherence must retain variation of the labelled generators before quotient specialization: a differentiated Koszul, Spencer, Atiyah, or equivalent Gauss–Manin cell.

This does not prove that such a differentiated cell exists. It removes the raw Koszul differential from the candidate list and prevents a zero quotient differential from being reinterpreted as the required homotopy.

Updated acceptance equation

Any surviving differentiated construction must produce a labelled homotopy $H_c$ satisfying

$$D H_c+H_cD = c_{12}F_{12}+c_{13}F_{13}+c_{23}F_{23},$$

where $D$ retains both the Koszul differential and its base variation before passing to $R$.

Artifacts

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

Next falsifier

Construct the first differentiated Koszul differential directly from $\partial_j g_i$, including the principal generator $zK-1$, and test its mixed square against the full-rank obstruction. No free homotopy matrix may be introduced.

View source on GitHub