The Soft-Axis Exact-Form Tail Has Persistent Quartic Cokernel
Record
Status: graded-symbol falsification test following Entry 440.
Hard-to-vary claim
The cutoff-dependent exact-form tail at (X_2=0) might become contractible in sufficiently high polynomial degree, with canonically trivial determinant. If so, the leading symbol map should have full target rank in all sufficiently high homogeneous shells.
Frozen symbol
At the joint soft-total center, [ E=X_2=0, ] the leading fiber data in the integration variables are [ K_4=a^4,\qquad L_{1,mathrm{top}}=b,\qquad L_{2,mathrm{top}}=a. ]
For all four exact sectors [ (s_a,s_b)in{(1,1),(1,0),(0,1),(0,0)}, ] take the highest homogeneous part of the two exact-form columns generated by each monomial (f=a^ib^j). Combine the sectors that land in each fixed target-degree shell.
No completion or support summand is added.
Result
Target degrees 5 through 40 were computed. The deficiencies begin [ 4,4,4,4,4,4,4,5 ] through degree 12. From degree 13 through degree 40, every shell has deficiency exactly four: [ operatorname{rank}sigma_d=d-3 \qquad ext{in a target shell of dimension }d+1. ]
Structural inspection explains the persistent four. Since [ K_4=a^4, ] every leading exact-form image is divisible by (a^4). Hence the four classes [ 1,\qquad a,\qquad a^2,\qquad a^3 ] times the appropriate power of (b) cannot lie in the image.
The computed rank reaches the full dimension of the ideal ((a^4)) in every tested shell (dge13). Thus the tested eventual symbol cokernel is [ \boxed{ operatorname{coker}sigma_{mathrm{tail}} simeq mathbf F[a,b]/(a^4) } ] at the level of graded shell dimensions and canonical representatives.
Degree 12 has one additional finite resonance. It does not alter the persistent four-dimensional tail.
Interpretation
The tail is not contractible. The continuing cutoff dependence in Entries 439--440 is therefore not merely numerical failure to reach a stable range: each higher cutoff adds genuine graded soft-boundary coefficient classes.
Deleting this cokernel solely to force a finite determinant would be a prohibited fitted support repair. It may be removed only if the source problem independently imposes a relative boundary condition whose cone kills (mathbf F[a,b]/(a^4)). Otherwise it must remain part of the coefficient object.
The quartic cokernel is generated by degeneration of the existing Cayley--Menger fiber polynomial (K_4). It is coefficient-boundary geometry, not a new energy/Cut carrier incidence.
Classification
- persistent tail: (mathbf F[a,b]/(a^4)) in the tested graded symbol;
- degree-12 extra deficiency: finite coefficient resonance;
- tail contractibility conjecture: refuted;
- existing carrier: unchanged;
- new carrier datum: none;
- finite soft-axis determinant line: not defined without a source-derived relative boundary condition.
Epistemic boundary
Divisibility by (a^4) is symbolic in the frozen leading geometry. Equality of the image with the full ideal ((a^4)) is verified shellwise over one large finite field through degree 40, not proved for every degree or in characteristic zero. The statement identifies the associated graded tail, not the complete filtered exact-form complex.
Next falsifier
Return to the frozen physical relative integration problem and determine whether its integration-domain boundary supplies a canonical relative condition whose associated-graded cone kills (mathbf F[a,b]/(a^4)). If yes, construct that cone before recomputing the soft-axis determinant. If no such source boundary exists, abandon the finite compactified Fitting-line conjecture and retain an infinite graded soft-boundary coefficient module.
Evidence
- research/benincasa/soft-axis-exact-symbol-certificate.json
- research/benincasa/marici-gm/src/bin/marked_tangency_support.rs