The Rank-Ten Exceptional Fitting Problem Is Not Projectively Flat at the Two Axes
Record
Status: projective-boundary falsification test following Entries 421, 423, 426, 433, and 437.
Hard-to-vary claim
The generic rank-ten master-image presentation on the first blow-up might extend unchanged to both boundary points of the exceptional (mathbf P^1_{[E:X_2]}). If so, its saturated leading Fitting divisor would admit a direct global degree count before any further boundary saturation.
The finite falsifier is the intrinsic determinantal difference [
u_{ m master}
delta_{r+q}([E M])-delta_r(E) ] at the two omitted projective directions, using both blow-up charts.
Frozen test
The ordinary chart is [ E=t,\qquad X_2=ct, ] whose point (c=0) is ([E:X_2]=[1:0]).
The reciprocal chart is [ X_2=t,\qquad E=ct, ] whose point (c=0) is ([E:X_2]=[0:1]).
The computation used (mathbf F_{2305843009213693951}[t]/(t^{12})) and was repeated at exact-form degrees 8 and 10. No support factor or rank correction was added after specialization.
Result
At the ordinary boundary, [ [E:X_2]=[1:0], \qquad (q, u_{ m master})=(7,55). ]
At the reciprocal boundary, [ [E:X_2]=[0:1], \qquad (q, u_{ m master})=(3,13). ]
Both results agree at exact-form degrees 8 and 10. They differ from the punctured generic pair [ (q, u_{ m master})=(10,66). ]
Therefore [ \boxed{ ext{the unsaturated rank-ten master-image presentation is not flat across either projective axis.} } ]
Interpretation
The two exceptional boundary points are not residual directions:
- ([1:0]) is the strict transform of the frozen soft divisor (X_2=0);
- ([0:1]) is the strict transform of the frozen total-energy divisor (E=0).
Thus the rank loss is supported on existing energy geometry. It supplies no new carrier incidence.
However, it invalidates the proposed shortcut from the punctured rank-ten calculation to a global degree bound on (mathbf P^1). The candidate [ 5[c=1]+15[c= frac12] ] lives on the rank-ten open locus only after the two source-derived boundary components have been saturated. A degree bound for its compactification cannot be read from the unsaturated presentation.
Classification
- (X_2=0) boundary: existing soft support;
- (E=0) boundary: existing total-energy support;
- punctured exceptional rank-ten locus: unchanged;
- residual projective boundary support: none;
- new carrier datum: none;
- naive unsaturated projective degree argument: refuted.
Epistemic boundary
This is a finite-field truncated Smith calculation, not a construction of the doubly saturated Fitting sheaf. The stable degree-8/10 agreement establishes the tested presentation behavior, but not its characteristic-zero flat extension. The result refutes only the unsaturated projective degree route; it does not refute the two-point divisor candidate on the punctured rank-ten locus.
Next falsifier
Construct the rank-ten master-image module after source-derived saturation by both boundary ideals ((E)) and ((X_2)). Compute its transition function between the ordinary and reciprocal charts. That transition—not the raw presentation—determines the compactified Fitting-line degree. Test whether it has degree 20. If it does, the measured multiplicities 5 and 15 exhaust the divisor; if it does not, the two-point expression remains non-exhaustive.
Evidence
- research/benincasa/dlog-smith-projective-infinity-certificate.json
- research/benincasa/marici-gm/src/bin/marked_tangency_support.rs