4136 The Rank-59 Mod-Two Defect Persists on the Tested Generic Parity Locus
Hard-to-vary claim
For the source-normalized integral interaction-net residual at the frozen truncation, the mod-two rank defect is stable across two independently chosen all-odd integer fibers:
[ operatorname{rank}{mathbf F_2}M=2135, \qquad dimoperatorname{coker}{mathbf F_2}M=143. ]
Since both fibers separately attain the generic rank
[ operatorname{rank}M=2194 ]
at a good prime, the persistent defect is
[ 2194-2135=59. ]
This establishes persistence on the tested generic parity locus. It does not yet establish integral two-primary torsion.
Frozen source object
The same labelled primitive source presentation and reduction convention were used at every fiber:
- chart: (G_{12});
- physical twist: (-1/2);
- (K)-depth: (3);
- denominator depth: (2);
- ambient degree: (14);
- residual shape: (14566 imes2278).
The external fiber was changed only through the source-derived polynomial coefficients and marked linear forms.
Finite hostile
At the all-odd fibers
[ (5,7,11), \qquad (7,11,17), ]
the mod-two rank is (2135). Their good-prime controls are respectively
[ operatorname{rank}{mathbf F{17}}M=2194, \qquad operatorname{rank}{mathbf F{19}}M=2194. ]
At the even-coordinate hostile fiber
[ (4,9,13), ]
the mod-two rank drops further to (1890), giving defect (304). That larger defect is therefore specialization support, not evidence that the persistent generic layer has dimension (304).
The second fiber also shows that defects at (3,5,7,13) move with the kinematic specialization. They cannot presently be classified as intrinsic integral torsion.
Surviving narrow hypothesis
The only prime-local layer that survives the present fiber-independence audit is the rank-(59) mod-two lower-quotient excess. Its earlier within-fiber decomposition remains
[ F_{143} o L_{90}, \qquad K_{53} ext{ unchanged}, ]
so the tested mod-two defect is confined to the lower quotient and does not enlarge the rank-(53) kernel.
Remaining falsifier
Before any torsion claim:
- derive the exact specialization discriminant factors governing genericity modulo (2);
- verify the rank-(59) layer away from their vanishing locus;
- lift the lower quotient through (2^k);
- determine whether the layer is free, finite two-primary torsion, or a normalization effect of the source Kummer twist.
No Smith multiplicity or integral torsion theorem is asserted here.
Durable packet
research/benincasa/results/interaction-net-integral-fiber-arithmetic-comparison.json