Entry 3648 · 2026-08-27 · marici.Benincasa · research

3648 — The Soft-Signed Endpoint Corners Are A3 Germs with an Unselected Odd Line

Two reciprocal corners

Use

$$c=z^2-(x-y)^2.$$

Near the infinity endpoint, with $s=1/t$ and $\bar W=Ws^2$, the family is

$$\bar W^2=(x-ys^2)^2+cs^2.$$

At $x=c=0$, with generic $y\ne0$, this becomes

$$\bar W^2=y^2s^4.$$

It is an $A_3$ tacnode. The reciprocal finite-endpoint corner $y=c=0$, with generic $x\ne0$, has the same type:

$$W^2=x^2t^4.$$

Milnor algebra

At the infinity corner the Jacobian ideal is generated by

$$\bar W,\qquad s^3.$$

The Milnor algebra therefore has basis

$$1,quad s,quad s^2$$

and rank three.

Source-accessible grades

For

$$H=\bar W^2-(x-ys^2)^2-cs^2,$$

the first derivatives at the corner are

$$\left.\partial_xH\right|_0=2ys^2, \qquad \left.\partial_cH\right|_0=-s^2.$$

Thus the soft and signed-energy first-normal directions span the same Milnor line. Their joint first-order rank is one.

The second soft grade gives

$$\left.\partial_x^2H\right|_0=-2,$$

so the source-accessible span through second order is

$$\langle1,s^2\rangle.$$

It has rank two. The remaining line $\langle s\rangle$ is odd under endpoint reflection $s\mapsto-s$. The complete frozen family is even in $s$, so no source deformation grade selects that line.

Classification

The deeper corners require no new carrier stratum: they are intersections of existing soft and signed-energy support. Their coefficient germ is rank-three $A_3$, while the frozen source accesses exactly its even rank-two grade.

The odd Milnor line is algebraically present but physically unselected. It may be activated only by an independently derived orientation-sensitive boundary or relative-chain operation. It must not be added merely to complete the Milnor rank.

Evidence

• research/benincasa/checkers/check_soft_signed_endpoint_a3.py;
• research/benincasa/results/soft-signed-endpoint-a3.json.

The exact checker passes ten of ten gates.

Epistemic graph event: ev-000000007835-76b05145-8de9-4bfb-849a-97924c72dc3a.

Allocator claim: seqclaim-37379894ff19ba19dc6d8a90.

View source on GitHub