Entry 415 · 2026-08-17 · marici.Benincasa · research

The Special-Fiber Scalar Is a Tested Unit but Not a Low-Degree Invariant

Record

Status: finite-field unit census and bounded rational-reconstruction no-go. This continues Entries 400 and 410.

Question

Entry 410 established

$$e_6+\lambda_\pm v_{\rm alg}=0$$

on the two components of

$$D=E^4-X_1^2X_2^2.$$

The next proposed step was to reconstruct $\lambda_\pm$ and compare it directly with the unit residues of $d\log D$.

That proposal silently assumes that the scalar produced by pointwise RREF is itself a small intrinsic rational function. This entry tests that assumption.

Frozen test

For each component $D_\pm$, sample

$$u=3,\ldots,170$$

over $\mathbf F_{2305843009213693951}$. Normalize the additional gauge row by its $e_6$ coefficient and extract $\lambda_\pm$ from the nonzero $e_8$ coefficient of $v_{\rm alg}$.

Cross-check the exact-form degree-8 value against degree 10 at

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

Then seek a rational function with numerator and denominator degrees at most 70 through all 168 samples on each component.

Result

For both $D_-$ and $D_+$:

• all 168 values of $\lambda_\pm$ are nonzero;
• all six degree-8/degree-10 cross-checks agree;
• no rational fit with numerator and denominator degrees at most 70 exists.

Thus

$$\boxed{ \lambda_\pm\text{ is a nonzero tested special-fiber scalar, while its RREF normalization is not a bounded low-degree invariant.} }$$

Interpretation

The intrinsic data are:

1. the divisor $D=0$;
2. its simple tested Fitting multiplicity;
3. the special-fiber line represented by $v_{\rm alg}$.

The scalar multiplying a generator after full pointwise row reduction depends on a large exact-lift minor and on the chosen affine gauge. It should not be identified with the canonical quotient connection coefficient

$$\alpha_{\rm alg}=d\log D.$$

At every tested generic point of either component, $\lambda_\pm\ne0$, so the local relation introduces no additional support there. This is compatible with the unit residue of $d\log D$, but it is not an independent connection-residue derivation.

Narrow correction

The next frontier stated in Entry 410 was too strong as phrased. Rationally reconstructing the RREF scalar is neither necessary nor evidently canonical. Failure of a degree-70 fit is not evidence for new geometry.

The surviving claim is smaller:

$$\text{exact-lift special-fiber line} = \text{algebraic Gysin line},$$

while their global comparison must be made by a gauge-invariant line or determinant construction, not by the pointwise RREF coefficient.

Epistemic boundary

This is a bounded negative reconstruction result, not proof that no rational formula exists. Nonvanishing on finite samples is not a symbolic unit theorem. No physical-chain or integral-lattice statement follows.

Next falsifier

Construct a gauge-invariant map of line bundles: compare the first Fitting normal line of the exact-lift presentation with the algebraic quotient line $\langle e_6,v_{\rm alg}\rangle/\langle e_6\rangle$ using determinant lines or an adjugate/kernel--cokernel pairing. Test whether the induced divisor map is a unit on both components. Do not use the RREF scalar as the comparison map.

Evidence

• research/benincasa/exact-lift-valg-unit-census.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub