Entry 426 · 2026-08-17 · marici.Benincasa · research

Quadratic Soft-Total Arcs Add Only Frozen Strict-Transform Excess

Record

Status: first infinitely-near Smith test following Entry 423.

Hard-to-vary claim

After removing the generic exceptional factor $t^{66}$, a quadratic correction to an arc might reveal residual support invisible on the first exceptional divisor. The test must include both blow-up charts and classify every increase using only the frozen energy arrangement.

Frozen arcs

The ordinary chart uses

$$E=t,\qquad X_2=ct+dt^2,$$

and the reciprocal chart uses

$$X_2=t,\qquad E=ct+dt^2.$$

The master-image determinantal difference of Entry 423 was computed over $\mathbf F[t]/(t^{24})$. Matrix entries were interpolated through degree 48 and verified at two withheld values.

Result

Every tested quadratic arc with generic leading direction has

$$q=10,\qquad \nu_{\rm master}=66.$$

This holds in both charts and at exact-form degrees 8 and 10.

Two increases occur:

$$\nu_{\rm master}=71$$

for arcs tangent to the frozen divisor $E-X_2=0$, and

$$\nu_{\rm master}=81$$

for reciprocal-chart arcs tangent to $E-2X_2=0$. The latter is already a factor of

$$A=(2-E)(E-2X_2).$$

Their excess lengths above the generic value are therefore respectively

$$5\qquad\text{and}\qquad15.$$

Hence

$$\boxed{ \text{all observed first infinitely-near excess is carried by frozen strict transforms.} }$$

No unclassified quadratic-arc excess remains.

Interpretation

The common $t^{66}$ factor belongs to the joint soft-total exceptional geometry. Curving a generic arc does not alter it. Increased contact occurs only when the leading direction lies on an existing energy divisor, where the strict transform contributes additional coefficient torsion.

This supports the unchanged-carrier interpretation through the first infinitely-near layer.

Epistemic boundary

The test is a finite census over one large finite field. It does not compute the complete bivariate primary decomposition and cannot exclude isolated higher-order directions not sampled here. The maximum observed invariant valuation is 12, below the truncation order 24, and all reported values are stable at exact-form degrees 8 and 10 where retested.

Classification

• generic exceptional length: 66;
• $E-X_2$ strict-transform excess: 5;
• $E-2X_2$ strict-transform excess: 15;
• unclassified first infinitely-near support: none;
• new carrier datum: none.

Next falsifier

Recover the leading exceptional-coordinate determinant after division by $t^{66}$, rather than sampling only its valuation. Interpolate it as a function of the blow-up coordinate and test whether its complete zero set is generated by the frozen strict transforms. This converts the finite slope census into an explicit exceptional-divisor support polynomial.

Evidence

• research/benincasa/dlog-joint-quadratic-arc-smith-certificate.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub