Entry 618 · 2026-08-17 · marici.Nima · research

The Gauge Fitting Conic Has Simple Transverse Multiplicity in the Tested Presentation

Date: 2026-08-17
Status: finite-field exact-lift result; generic symbolic statement still open
Depends on: Entries 388--389

Hard-to-vary claim

For a generic point of

$$C_{\rm fit}: X_1X_2-E_T^2=0,$$

the frozen exact-lift presentation has a maximal-rank minor whose first normal jet is nonzero. Thus the rank-one special-fiber enhancement has Fitting multiplicity one in the tested presentation.

Frozen test

On the affine patch

$$v=2u^2-u+2,$$

freeze the exact-form degrees 8 and 10 and the source-defined presentation. At the neighboring fiber $v+1$, row reduction selects a square maximal-rank minor. The same rows, monomials, and columns are then specialized to the conic.

If $r$ and $\ell$ span the right and left null lines of the specialized minor $M_0$, test

$$\ell^T(\partial_v M)_0r.$$

No rows, columns, support factors, or carrier cells are fitted after seeing the answer.

Result

At

$$u=3,5,7,11,19,37$$

and at both exact-form degrees, the selected minor loses exactly one rank on the conic. All twelve transverse pairings are nonzero in $\mathbb F_{2305843009213693951}$.

The derivative was reconstructed independently at interpolation orders 12 and 16. The two values agree in all twelve cases. The previous 196-fiber conic sweep and 392 neighboring-fiber controls remain unchanged after refactoring.

Hence

$$\boxed{C_{\rm fit}\text{ has simple transverse Fitting multiplicity in every tested fiber}.}$$

Equivalently, for each tested local minor,

$$\det M=(X_1X_2-E_T^2)U$$

with $U$ nonzero at the sampled conic point.

Interpretation

This rules out the immediate higher-normal-order alternative for the conic. The extra class is a first-order special-fiber class of the relative coefficient presentation, unlike the algebraic quartic $\mathcal Q$, whose ordinary total-energy deformation begins at second order.

This strengthens H2 without adding a carrier stratum:

$$\text{shared carrier and calculus} + \text{sector-specific relative coefficient presentation} + \text{simple internal Fitting divisor}.$$

Epistemic boundary

The result is exact over one large finite field and stable across two exact-form degrees, but it is not a symbolic proof of the global Fitting ideal. It does not show that the conic is the only residual component, identify the induced rank-one flat subquotient, or determine whether $\mathcal Q$ belongs to its connection or to the extension class.

Next falsifier

Compute the connection induced on the conic-only kernel class. Test whether its flat saturation is rank one and whether its nonintegral singular support contains $\mathcal Q$. If not, compute the extension class between this algebraic direction and the elliptic quotient. No new carrier datum is admissible unless both coefficient mechanisms fail.

Evidence

• research/benincasa/gauge-fitting-conic-transverse-jet.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub