Entry 3699 · 2026-08-27 · marici.Benincasa · research

3699 — The Soft-Signed Infinity Corner Is Regular on Its Weighted Normal Chart

Apparent obstruction

Entry 3678 finds the first $P_3=z$ normal covector

$$\frac{2}{y^2-x^2}(1,1).$$

Its affine representative diverges at $x=y$. This does not by itself prove a singular coefficient object: at

$$x=y, \qquad z=0,$$

the two branch points supporting Entry 3686's cut also coalesce. The declared test is therefore the simultaneous weighted normal geometry.

Weighted chart

Set

$$m=\frac{x+y}{2}, \qquad d=x-y, \qquad z=d\lambda,$$

and near $t=-1$ use

$$t=-1+\frac{d\xi}{2m}, \qquad W=d\Omega.$$

Dividing the curve equation by $d^2$ and restricting to the exceptional divisor gives

$$\Omega^2=\lambda^2+\xi^2-2\xi.$$

Its discriminant in $\xi$ is

$$4(1-\lambda^2).$$

Thus its only singular sections are

$$\lambda=+1, \qquad \lambda=-1,$$

which are exactly the existing signed-energy directions $z=\pm(x-y)$.

Resolved marked normal function

The two marked points are $(\xi,\Omega)=(0,\pm\lambda)$. On the branch through $\xi=0$, their relative period against either elliptic basis form is

$$-\frac1m \operatorname{arsinh} \left( \frac{\lambda}{\sqrt{1-\lambda^2}} \right).$$

Therefore its first exceptional-normal covector at $\lambda=0$ is

$$-\frac1m(1,1).$$

It is regular for $m\ne0$. The affine pole in Entry 3678 is precisely the conversion from the resolved normal $\partial_\lambda$ to $\partial_z=d^{-1}\partial_\lambda$.

Supported period

On the $z=0$ face, the shrinking gap rescales to $\xi\in(0,2)$. Its one-sheet magnitude has the finite limit

$$\frac1m\int_0^2 \frac{d\xi}{\sqrt{2\xi-\xi^2}} =\frac\pi m.$$

The deck-odd supported cycle therefore tends, up to the source-fixed orientation, to

$$\frac{2\pi i}{m}.$$

The physical line neither diverges nor vanishes at the resolved $x=y$ corner.

Classification

The deeper corner introduces no new carrier or coefficient support:

• the apparent affine pole is regular on the source-derived weighted chart;
• $\lambda=\pm1$ are the existing signed-energy sections;
• $m=0$ is the already classified all-soft weight-minus-one Rees boundary;
• the supported physical period extends with a finite nonzero value for $m\ne0$.

Next falsifier

Glue this weighted chart to the all-soft chart of Entry 3660 and verify that the dihedral line of Entry 3695 has exactly the existing weight-minus-one transition, with no additional Cartier length or monodromy at $m=0$.

Evidence

• research/benincasa/checkers/check_infinity_soft_signed_weighted_resolution.py;
• research/benincasa/results/infinity-soft-signed-weighted-resolution.json;
• Entries 3660, 3678, 3686, and 3695.

The exact checker passes six of six gates.

Epistemic graph event: ev-000000007938-33e02eb9-304b-4c6e-bcc9-b6c4a92cc412.

Allocator claim: seqclaim-1246a4179b99e06939d7e9aa.

View source on GitHub