A Chosen Maximal Minor Does Not Define the Exceptional Fitting Divisor
Record
Status: negative determinant-reconstruction test following Entries 421, 423, and 426.
Hard-to-vary claim
After dividing the master-image determinant by its generic exceptional order (t^{66}), a single fixed maximal-minor quotient chart should recover the intrinsic exceptional support polynomial. If true, its leading coefficient would have zeros only at the frozen strict-transform directions (c=1) and (c= frac12).
Frozen chart
Use [ E=t,\qquad X_2=ct. ] The intrinsic Smith difference is [
u_{ m master}
delta_{r+10}([E M])-delta_r(E), ] where (delta_k) is the minimum valuation of the (k)-minors, (E) is the exact-form block, and (M) is the twelve-master block.
For comparison, numerator and denominator maximal minors were selected at a generic base slope and then held fixed as functions of (c).
Result
The intrinsic Smith valuation remains [
u_{ m master}=66 ] at the generic tested slopes, including (c=5).
The fixed-minor quotient instead has valuation 67 at (c=5). Permuting the ambient row and column order before deterministic pivot selection did not remove this extra zero: the tested selections still landed in the same non-generating minor chart.
Therefore [ \boxed{ ext{a chosen maximal-minor ratio is not an intrinsic generator of the exceptional Fitting line}.} ]
The (c=5) zero is a minor-chart artifact, not coefficient support, because the minimum-over-all-minors Smith invariant does not jump there.
Surviving invariant statement
Entries 421 and 426 still give the valuation-divisor candidate [ 5[c=1]+15[c= frac12]. ] On the ordinary exceptional chart this has principal representative [ (c-1)^5(2c-1)^{15} ] up to a unit. This is supported by the exhaustive first-direction census and the measured tangent excesses, but it is not yet an explicitly derived global Fitting generator.
Classification
- (c=5): noncanonical minor-chart artifact;
- (c=1): frozen (E-X_2) strict-transform support, excess 5;
- (c= frac12): frozen (E-2X_2) strict-transform support, excess 15;
- new carrier datum: none;
- explicit exceptional Fitting generator: not yet derived.
Epistemic boundary
This result refutes the proposed single-minor reconstruction method, not the valuation-divisor candidate. Deterministic pivot permutations do not prove a gcd over all maximal minors. No zero of one chosen minor may be promoted to coefficient support without checking the determinantal minimum.
Next falsifier
Compute the leading (t^{66}) Fitting ideal itself: either obtain the gcd of the leading coefficients of sufficiently many certified maximal minors, or construct a compatible exact-plus-master quotient presentation whose determinant generates the saturated Fitting line. Test whether its exceptional divisor is exactly [ 5[c=1]+15[c= frac12]. ] Any surviving zero away from the frozen strict transforms is the first candidate for new coefficient support; it is not a carrier stratum unless independent incidence data require it.