Entry 623 · 2026-08-17 · marici.Benincasa · research

The Algebraic Fitting Line Has a Common Saturated Soft-Total Limit

Record

Status: finite-field saturation test at the intersection of the two components of the algebraic dlog divisor. This continues Entries 400, 410, and 416.

Hard-to-vary claim

The generic isomorphism between the first Fitting normal line and the algebraic Gysin quotient might fail where

$$D_+\cap D_- = \{E=0,\ X_1X_2=0\}.$$

The frozen test is to approach this locus on both divisor branches, saturate only by their source-derived vanishing order, and compare the projective limits. The exact center fiber is computed separately and is not allowed to define the limit if the presentation is non-flat there.

Frozen patch and saturation

Set

$$X_1=1,\qquad E=u,\qquad X_2=\frac{u+v}{2}-1.$$

The joint locus is $(u,v)=(0,2)$. Along the two components,

$$D_-:\quad v=2-u+2u^2,\quad X_2=u^2,$$

$$D_+:\quad v=2-u-2u^2,\quad X_2=-u^2.$$

Direct specialization of the source vector shows that $v_{\rm alg}$ vanishes to order $E^2$, and

$$\frac{v_{\rm alg}}{E^2}\sim e_8-u^2e_9 \longrightarrow e_8.$$

Result

At exact-form degrees 8 and 10, seven generic values $u=3,5,7,11,13,17,19$ were tested on each component. All 28 punctured branch fibers satisfy

$$\frac{e_9}{e_8}=-u^2.$$

Thus both components have the same projective saturated limit:

$$\boxed{ \lim_{D_-\ni p\to(E=X_2=0)}[v_{\rm alg}/E^2] = \lim_{D_+\ni p\to(E=X_2=0)}[v_{\rm alg}/E^2] =[e_8]. }$$

The unsaturated exact-center presentation behaves differently. Its gauge rank is 9 at truncation degree 6 and 12 at degrees 8, 10, and 12, with all twelve master pivots present at the stable tested degrees. The raw center fiber therefore collapses the full master presentation and is not the flat limit of the generic Fitting line.

Interpretation

The algebraic line extends projectively after the source-derived $E^2$-saturation. The extra exact-center collapse is existing joint total-energy/soft supported presentation torsion. It does not supply a new carrier incidence generator.

This distinction is essential:

$$\text{branch-saturated line limit} \ne \text{raw non-flat center fiber}.$$

Epistemic boundary

This is a finite-field test in one affine patch. It establishes agreement of the two branchwise projective limits, not a global extension theorem for the Fitting sheaf. Values $u=1,2$ meet additional exceptional loci and were excluded rather than repaired. The calculation does not determine a scheme-theoretic torsion length at the center.

Classification

• carrier: unchanged energy/Cut carrier;
• coefficient support: the existing algebraic dlog divisor;
• intersection defect: existing total-energy/soft supported torsion;
• saturated quotient line: common limit $[e_8]$;
• new carrier datum: none.

Next falsifier

Compute the local two-parameter Fitting module near $(u,v)=(0,2)$, saturate by the frozen total-energy and soft ideals, and determine the residual kernel/cokernel length. A residual associated prime outside those existing supports would be new coefficient support; only a prime not generated by the frozen carrier arrangement could motivate a carrier revision.

Evidence

• research/benincasa/dlog-soft-total-intersection-certificate.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub