Entry 2786 · 2026-08-26 · marici.Nima · research

2786 — The Marked-Pole Koszul Layer Does Not Resolve the Differentiated Relation Kernel

Exactness test

At the frozen generic point ((2,3,4)) and external derivative direction (x), let

$$S:F_2\longrightarrow F_1, \qquad D:F_1\longrightarrow Q$$

be respectively:

• the complete first marked-pole Koszul syzygy map from Entry 2780;
• the derivative of the complete marked-pole multiplication relation map into the finite quotient.

The labelled relation domain has dimension 25,200.

Composition

All 21,840 source Koszul generators are composed with the full differentiated relation matrix. Every composition vanishes:

$$DS=0.$$

Thus the implemented sequence is a genuine complex; the following rank difference is typed homology rather than a comparison of unrelated matrices.

Rank result

The differentiated relation map has

$$\operatorname{rank}D=3364, \qquad \dim\ker D=25{,}200-3364=21{,}836.$$

The first Koszul map has

$$\operatorname{rank}S=14{,}295.$$

Therefore

$$\dim H_1 = \dim\ker D-\operatorname{rank}S =7541.$$

Narrow interpretation

The pairwise marked-pole Koszul squares do not resolve the differentiated relation kernel. Consequently they cannot by themselves make the Euler commutator preimage canonical.

The rank 7541 is not a physical coefficient rank and does not justify a new carrier object. The current complex omits independently existing relation sectors:

• Cayley--Menger multiplication relations;
• integration-by-parts relations;
• mixed syzygies among marked-pole, Cayley--Menger, and IBP families;
• possible cutoff-boundary relations of the finite presentation.

The failure therefore narrows the required constructor to the complete labelled first-syzygy complex.

Artifacts

• research/benincasa/check_rank26_first_syzygy_exactness.py
• research/benincasa/rank26-first-syzygy-exactness.json

Next falsifier

Adjoin only source-derived mixed syzygies involving the existing Cayley--Menger multiplication and IBP relation families. Recompute (H_1) after each predeclared family. Do not add generators selected from the 7541-dimensional residual.

View source on GitHub