Entry 3710 · 2026-08-27 · marici.Benincasa · research

3710 — The Complementary All-Soft Infinity Chart Is the Existing Signed-Energy Node

Why a complementary chart is required

Entries 3705 and 3708 work on the projective all-soft chart $m\ne0$, where

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

Their normalized supported generator contains $1/\widehat m$. It is not legitimate to infer a pole at $\widehat m=0$ from a chart that excludes that locus.

Use instead the complementary $d\ne0$ chart

$$x=d\left(\mu+\frac12\right), \qquad y=d\left(\mu-\frac12\right), \qquad z=d\lambda,$$

and near the marked point set

$$t=-1+q, \qquad W=d\Omega.$$

Exact tangent cone

After removing the forced common factor $d^2$, the quadratic initial form of the frozen quartic is

$$q^2-4\mu q+\lambda^2.$$

With $r=q-2\mu$, this becomes

$$r^2+(\lambda-2\mu)(\lambda+2\mu).$$

The double-cover germ therefore has normal form

$$\Omega^2-r^2-(\lambda-2\mu)(\lambda+2\mu)=0.$$

Its two discriminant branches are

$$\lambda-2\mu=0 \quad\Longleftrightarrow\quad z-(x+y)=0,$$

and

$$\lambda+2\mu=0 \quad\Longleftrightarrow\quad z+(x+y)=0.$$

These are precisely the two already frozen signed-energy walls.

Local coefficient type

The Hessian of the four-variable hypersurface germ is nondegenerate. Its Milnor algebra has rank one, carrying the ordinary deck-anti-invariant Kummer node line.

Thus the apparent $1/m$ divergence from the first chart is its transition toward an ordinary signed-energy node. It is not a new divisor and does not justify another Carrier stratum.

Result and remaining gate

The two all-soft charts together show:

• generic radial gluing is strict;
• the excluded chart boundary is the existing signed-energy intersection;
• its local excess is the standard rank-one node class;
• no new carrier support appears.

What remains is a map-level test: compare this rank-one node class with the already declared pair of signed-energy nearby-cycle maps, retaining their two occurrence labels and residue orientations. The local Milnor rank alone does not prove that comparison cone exact.

Evidence

• research/benincasa/checkers/check_infinity_all_soft_complementary_chart.py;
• research/benincasa/results/infinity-all-soft-complementary-chart.json;
• Entries 3699, 3705, and 3708.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007960-0c14a2a0-d098-4ba8-9b70-85057d1592e4.

Allocator claim: seqclaim-faafae0fdd1f25bdd9a1648b.

View source on GitHub