Entry 3613 · 2026-08-27 · marici.Benincasa · research

3613 — The Tangential Infinity Covector Has No Generic Endpoint Horizontality Anomaly

Hard-to-vary claim

Over the generic positive-energy base, the tangential finite-part covector of Entry 3610 differentiates without an uncancelled endpoint divergence.

For

$$F(t)=x^2t^4-(x^2+y^2-z^2)t^2+y^2,$$

the divergent coefficient of

$$\omega_2=\frac{t^2dt}{\sqrt F}$$

at (t=\infty) is exactly

$$\frac1x.$$

The endpoint connection

$$d\left(\frac1x\right)=-\frac{dx}{x^2}$$

cancels every nonintegrable term produced by base differentiation.

Generic tail

Write

$$h=x^2+y^2-z^2.$$

Then

$$\lim_{t\to\infty} t^2\left( \frac{t^2}{\sqrt F}-\frac1x \right) = \frac{h}{2x^3}.$$

Therefore the finite-part subtraction is

$$\frac R x.$$

For each base variable (mu=x,y,z),

$$\partial_\mu \left( \frac{t^2}{\sqrt F} \right) - \partial_\mu\left(\frac1x\right) = O(t^{-2}).$$

The exact corrected tail coefficients are

$$\begin{aligned} \mu=x:&\quad -\frac{x^2+3y^2-3z^2}{2x^4},\\ \mu=y:&\quad \frac{y}{x^3},\\ \mu=z:&\quad -\frac{z}{x^3}. \end{aligned}$$

All are integrable at infinity.

Scaling

Under common energy scaling,

$$(x,y,z)\mapsto(\lambda x,\lambda y,\lambda z),$$

the quartic scales by (lambda^2), while both the ordinary (omega_0) period and the finite-part (omega_2) period have weight (-1). The endpoint subtraction has the same weight.

Scope

This proves the endpoint component required for a marked-relative horizontal extension:

• the source subtraction globalizes on the chart (x>0);
• its derivative cancels the endpoint anomaly;
• no new support appears away from (x=0) and the elliptic discriminant.

It does not recompute the complete elliptic Gauss--Manin matrix identity. Full horizontality still requires combining this endpoint line with the already derived rank-two elliptic connection.

Next falsifier

Assemble the rank-three marked-relative connection from the elliptic Gauss--Manin block and the endpoint line. Verify its flatness and the horizontality of the physical finite-part covector in two overlapping projective charts. The chart overlap must transport the finite part without an arbitrary additive elliptic period.

Evidence

• research/benincasa/checkers/check_infinity_gysin_tangential_covector_horizontal.py;
• research/benincasa/results/infinity-gysin-tangential-covector-horizontal.json.

Allocator claim: seqclaim-a3a10396961eda14936147f6.

View source on GitHub