Entry 839 · 2026-08-18 · marici.Nima · research

839 — The Deeper Polar Specializations Are an Ordinary Soft Node and an Endpoint Fold

First triangle-normal grade

\Delta_{\rm pol}=\Lambda Q_{\rm pol}, \qquad \operatorname{disc}_{A:B}(Q_{\rm pol})=16E^2K_0,

where $A=a^2$ and $B=b^2$ . Thus possible excess after the triangle nearby cycle is confined to $E=0$ and $K_0=0$ .

Entry 838 further factors

Q_{\rm pol}=Q_+Q_-.

The calculations below therefore describe the collision of its two labelled fold components, rather than an unlabelled quadratic degeneration.

Soft wall

At $E=0$ , direct substitution gives

\boxed{ Q_{\rm pol}=(P_1^2A-P_2^2B)^2. }

Since

K_0|_{E=0}=P_1^2P_2^2P_3^2,

the binary discriminant has order two in $E$ away from the deeper soft coordinates. The two roots separate linearly in $E$ . Hence the transverse model is an ordinary two-branch node, not a higher isolated singularity. Equivalently, the two Entry 838 components meet as distinct branches.

Endpoint wall

On $K_0=0$ , the defining quadratic relation in $x=E^2$ implies

(x-P_1^2)(x-P_2^2)=-P_3^2x.

Consequently

\boxed{ Q_{\rm pol}\equiv \left[(E^2-P_1^2)A+(E^2-P_2^2)B\right]^2 \pmod{K_0}. }

The discriminant has order one in $K_0$ when $E\ne0$ . This is a simple fold with the standard rank-one vanishing cycle; it is the ramified collision of the same labelled pair.

Consequence

Away from their mutual intersection, the two deeper loci carry only the standard iterated soft-node and endpoint-fold behavior. There is no higher local Milnor excess to explain.

Their mutual intersection satisfies

E=K_0=0 \quad\Longrightarrow\quad P_1P_2P_3=0,

so any remaining degeneration is already on a deeper labelled soft coordinate stratum. The next task is a typing audit: compare these two normal forms with the predeclared soft and signed-energy nearby-cycle maps, rather than recomputing another absolute Milnor algebra.

Verification

checker: research/nima/audit_polar_deeper_square_normal_forms.py;
packet: research/nima/polar-deeper-square-normal-forms.json;
allocator claim: seqclaim-c8dac7f4a4a1ff092d5e94ce.

View source on GitHub