The Naive Even Cartier Action Does Not Descend
Entry 502 proposes a Smith or minimal-presentation calculation over
Before that calculation, (x) must act on the complete exact cokernel. This is not automatic: Entry 449 already found multiplication commutators for the frozen exact image.
Using the complete frozen four-sector exact matrix, multiply every admissible exact-image column by (a^2), retaining only columns whose full support lies inside the cutoff. At
the enlarged image has one additional rank in every case:
Therefore
Consequence
The cyclic local-cohomology object (A_+\eta) of Entry 502 is valid in the conormal coefficient model, but the complete orbit cokernel is not yet an (A_+)-module. A Smith form over (A_+) is therefore undefined on the naive presentation.
The stable rank-one multiplication commutator is numerically compatible with the plus defect, but this is not an identification. The next comparison must construct a chain homotopy correcting multiplication by (a^2), or prove that no such correction exists. Only the corrected action can test whether the incidence generator (eta) supplies the invariant defect.
No new carrier datum is indicated. The obstruction is failure of a coefficient operation to descend through the exact differential.
Next falsifier
Compute
sector by sector and seek a source-derived homotopy through the retained gradient/Kodaira--Spencer complex. If it is nullhomotopic, repeat Entry 502's module presentation with that corrected action. If it is not, the even-incidence explanation of the plus defect fails in its present form.
Evidence
research/benincasa/marici-gm/src/bin/soft_axis_plus_defect_module.rs;- Entries 449, 473, 500, 501, and 502.