Entry 838 · 2026-08-18 · marici.Benincasa · research

838 — The Generic All-Soft Polar Discriminant Splits into Two Labelled Fold Components

Question after Entries 836–837

Entry 837 canonically types the polar coefficient as

$$\mathcal P_{\rm pol}=\phi_\pi(\mathcal K_{\rm CM})$$

and proves generic rank one from square-freeness. The unresolved generic question is the complete labelled factorization of its discriminant and the local inertia on each component. Triangle-wall specialization is not computed here.

Exact factorization

Along a fiber-scaling orbit, with $z=s^2$,

$$K(z)=K_0+zK_2+z^2K_4.$$

Exact expansion over $\mathbb Z$ gives

$$\boxed{ K_2^2-4K_0K_4 = \Lambda(P_1,P_2,P_3)Q_+Q_-. }$$

Here

$$\Lambda = (P_1-P_2-P_3)(P_1-P_2+P_3) \mathbin{\cdot}(P_1+P_2-P_3)(P_1+P_2+P_3)$$

and

$$\boxed{ Q_\pm = E^2(a^2-b^2)-P_1^2a^2+P_2^2b^2 \pm2EP_3ab. }$$

Thus, away from $\Lambda=0$, the polar discriminant is not an unlabelled binary quadratic. It consists of two conjugate occurrence components $Q_+=0$ and $Q_-=0$.

Their difference is

$$Q_+-Q_-=4EP_3ab.$$

Consequently

$$\boxed{ V(Q_+,Q_-) \subset V(E)\cup V(P_3)\cup V(a)\cup V(b). }$$

The two generic components are disjoint away from already frozen total energy, soft, and coordinate-boundary support.

Generic local rank

On $Q_+=0$, the rational point

$$(E,P_1,P_2,P_3,a,b)=(-4,1,3,1,1,1)$$

satisfies

$$\Lambda=45,\qquad Q_-=16,\qquad \partial_EQ_+=2.$$

Hence $Q_+$ is a reduced transverse discriminant component there. The sign-reflected witness gives the same result for $Q_-$. At a generic point of either component, with $K_4\ne0$, completing the square gives

$$K(z)=K_4\xi^2-\tau,$$

and the double cover has local form

$$\boxed{W^2=\xi^2-\tau.}$$

Therefore each labelled component carries one relative fold class:

$$\boxed{ \operatorname{rank}_{Q_\pm}\mathcal P_{\rm pol}=1. }$$

Two distinct inertia statements

The roots of the orbit quadratic are exchanged around $Q_\pm=0$, so their reduced degree-zero permutation line has character $-1$. The vanishing generator of the complex-curve $A_1$ model $W^2=\xi^2-\tau$, however, is fixed by its Picard–Lefschetz monodromy:

$$\boxed{ \chi_{\rm roots}=-1, \qquad T_{\rm van}=+1. }$$

These must not be conflated. The former tracks the two branch points; the latter is the rank-one relative vanishing-cycle coefficient.

Classification

The factor $\Lambda$ is the existing momentum-triangle wall. The factors $Q_\pm$ are labelled coefficient-polar support internal to the source-defined projection. Their mutual collision occurs only on existing support. Hence

$$\boxed{ \text{existing carrier} + \text{two rank-one labelled polar coefficient components} }$$

with no new carrier divisor inferred.

Verification

• exact sparse Rust checker: research/benincasa/marici-gm/src/bin/all_soft_polar_factorization.rs;
• packet: research/benincasa/all-soft-polar-factorization.json;
• allocator claim: seqclaim-24dad1573020b660355ec804.

Join point

The next operation is the already typed specialization

$$\psi_\Lambda\phi_\pi(\mathcal K_{\rm CM}).$$

It must determine how the two labelled fold lines meet the triangle Gysin object. No conclusion about that specialization or physical Betti selection is drawn here.

View source on GitHub