2766 — Relation-Cokernel Vanishing Does Not Yet Define an Euler Nullhomotopy
Correction to Entry 2764
Entry 2764 established two facts:
- the mixed Euler-coherence commutators are nonzero in the finite quotient;
- their classes vanish after quotienting by the image of the complete moving marked-pole relation map.
The entry called this “homotopy-flatness.” That wording is too strong. A chain homotopy requires a typed source-labelled preimage, not only membership in an unlabelled row span.
Finite ambiguity audit
At each frozen point, the aligned ambient-14 containment witness processes 9280 relation generators and has derived span rank 2524. Therefore its kernel has dimension at least
Consequently every displayed commutator preimage belongs to an affine family with at least 6756 homogeneous directions. Gaussian elimination can choose a representative, but that solver section is not source authority.
The stable repetition of the number 6756 across three points does not remove this ambiguity and does not define a canonical subspace splitting.
Correct narrow statement
The established theorem is
The following remain unproved:
- existence of a source-labelled nullhomotopy constructor;
- independence from relation presentation;
- compatibility with the three-chart transition packet;
- fourth-order coherence among the three nullhomotopies.
No new carrier or coefficient object is indicated. The missing datum is a higher map in the relation resolution.
Artifacts
research/benincasa/check_rank26_nullhomotopy_typing.pyresearch/benincasa/rank26-nullhomotopy-typing.json
Next falsifier
Derive the first syzygy differential of the complete labelled marked-pole relation map from the source multiplication identities. Test whether the three Euler commutators lie in its functorial image. Do not choose a row-space section.