The Soft-Axis Exact-Form Tail Is Neither Stable nor Affine Through Degree Sixteen
Record
Status: extended cutoff falsification test following Entry 439.
Hard-to-vary claim
The cutoff dependence at the soft boundary might be removable by one affine Hilbert-tail correction. The observed valuations (89,92,95) at exact-form degrees (8,10,12) suggested the provisional rule [
u_N=89+\frac32(N-8). ] If that rule survived, it could motivate a relative-index subtraction, provided the subtraction were subsequently derived from the graded exact-form tower rather than fitted.
The first finite falsifiers are degrees 14 and 16.
Frozen probe
Use [ E=t,\qquad X_2=3t^2 ] over [ mathbf F_{2305843009213693951}[t]/(t^{24}). ] The transverse unit 3 avoids the frozen algebraic branches (X_2=pm E^2). The master-image rank remains ten.
Result
The valuation sequence at exact-form degrees [ 8, 10, 12, 14, 16 ] is [ 89, 92, 95, 97, 99. ] Equivalently, the excess over the punctured generic valuation 66 is [ 23, 26, 29, 31, 33. ]
The increment sequence is [ 3, 3, 2, 2. ] Thus the degree-14 value already refutes the affine prediction 98, and the degree-16 value shows that the raw sequence has still not stabilized.
At degree 14, [ (r_E,r_{[E,M]})=(224,234),\qquad (delta_E,delta_{[E,M]})=(192,289). ] At degree 16, [ (r_E,r_{[E,M]})=(271,281),\qquad (delta_E,delta_{[E,M]})=(222,321). ] The rank difference remains ten, while the determinant difference changes.
Therefore [ \boxed{ ext{neither raw cutoff stabilization nor one affine tail subtraction defines the soft-axis saturation weight through degree 16.} } ]
Interpretation
The failure is in the chosen infinite exact-form presentation. It is not a new exceptional direction and does not alter the frozen carrier. More sampling of cutoff degrees cannot canonically select a completion: even if a numerical sequence eventually stabilizes, that would not identify which graded tail was quotiented.
The next object must therefore be structural. One must construct the completed or graded exact-form quotient and prove that its discarded tail is contractible or determinant-trivial. Only then is a renormalized Smith index admissible.
The stable total-energy excess 27 from Entry 439 remains valid in its tested scope. It cannot yet be paired with a soft-axis number to produce a global transition degree.
Classification
- soft-axis cutoff sequence: coefficient-presentation dependence;
- affine-tail correction: refuted;
- stable total-energy transverse excess: unchanged at 27;
- new carrier datum: none;
- degree-20 compactified Fitting line: still unproved.
Epistemic boundary
The calculation is finite-field and truncated at (t^{24}). It refutes the specific affine extrapolation and demonstrates nonstabilization through degree 16. It does not prove that no canonical completion exists, nor that the raw sequence diverges indefinitely.
Next falsifier
Grade the exact-form complex by polynomial degree at (X_2=0). Identify the eventual tail maps and test whether they form a contractible periodic complex with a canonically trivial determinant line. If so, quotient that tail before recomputing the transverse Smith index. If no canonical contractible tail exists, the present master-image presentation cannot define the desired compactified Fitting line.
Evidence
- research/benincasa/dlog-smith-soft-axis-tail-certificate.json
- research/benincasa/marici-gm/src/bin/marked_tangency_support.rs