A 128-Slope Smith Sweep Finds No Residual Exceptional Direction
Record
Status: bounded intrinsic-support sweep following Entries 421, 423, 426, and 433.
Hard-to-vary claim
If the exceptional master-image Fitting divisor has support away from the frozen strict transforms, then a broad test of the intrinsic minimum-over-minors Smith invariant should encounter a slope where either the generic rank 10 or generic valuation 66 fails.
A chosen maximal minor is inadmissible after Entry 433.
Frozen test
Use the ordinary exceptional chart [ E=t,\qquad X_2=ct ] and the intrinsic invariant [
u_{ m master}
delta_{r+10}([E M])-delta_r(E). ]
The computation used the prime field [ mathbf F_{2305843009213693951}, ] truncation (mathbf F[t]/(t^{12})), and 128 distinct deterministic pseudorandom slopes generated from seed (mathtt{0x9e3779b97f4a7c15}). The already classified directions [ c=0,\qquad c=1,\qquad c= frac12 ] were excluded. Eight slopes were independently repeated at exact-form degree 10 after the degree-8 sweep.
Result
All 128 degree-8 samples satisfy [ q=10,\qquad u_{ m master}=66. ]
There are no residual samples. All eight degree-10 cross-checks reproduce the same pair.
Thus [ \boxed{ ext{the bounded 128-slope intrinsic sweep finds no unclassified exceptional direction}.} ]
Interpretation
This result is invariant under the minor-chart criticism of Entry 433 because it uses determinantal minima, not one selected determinant. Together with the measured tangent excesses, it strengthens the candidate [ 5[c=1]+15[c= frac12]. ]
It does not derive that divisor. A finite pseudorandom sweep cannot exclude a sparse residual polynomial whose roots miss the sample.
Classification
- 128 generic sampled directions: existing exceptional coefficient geometry;
- residual sampled support: none;
- frozen strict-transform multiplicities: unchanged at 5 and 15;
- new carrier datum: none;
- saturated Fitting generator: still uncomputed.
Epistemic boundary
This is a bounded finite-field search sweep, not a symbolic primary decomposition and not a proof over characteristic zero. The pseudorandom seed, field, exclusions, truncation, and degree cross-check are frozen in the certificate.
Next falsifier
Derive an a priori degree bound for the saturated leading Fitting divisor on the exceptional (mathbf P^1), or construct the compatible exact-plus-master quotient determinant directly. Once the degree is bounded by 20, the already measured multiplicities 5 and 15 exhaust it and force [ (c-1)^5(2c-1)^{15} ] up to a unit. Without that degree bound, retain the expression only as a supported conjecture.
Evidence
- research/benincasa/dlog-smith-slope-scan-certificate.json
- research/benincasa/marici-gm/src/bin/marked_tangency_support.rs