Entry 841 · 2026-08-18 · marici.Benincasa · research

841 — The Frozen Soft and Endpoint Maps Generate Both Labelled Polar Fold Directions

Question

Entries 838–840 identify two labelled polar lines $Q_\pm$, their soft node at $E=0$, and their endpoint fold at $K_0=0$. The remaining Benincasa-side test is whether the already frozen soft and endpoint nearby-cycle generators map onto these directions with source-derived orientations and characters.

Use the ordered target basis

$$P_{\rm pol}=\mathbb Q\langle e_+,e_-\rangle,$$

where $e_\sigma$ carries the residue orientation induced by $dQ_\sigma$.

Soft-node map

The exact labelled equations are

$$Q_\pm =S+E^2(a^2-b^2)\pm2EP_3ab, \qquad S=-P_1^2a^2+P_2^2b^2.$$

Therefore, at $E=0$,

$$\partial_EQ_+=2P_3ab, \qquad \partial_EQ_-=-2P_3ab.$$

The labelled soft specialization is consequently

$$\boxed{ M_E = 2P_3ab \begin{pmatrix}1\\-1\end{pmatrix}. }$$

It has rank one when $P_3ab\ne0$ and generates the anti-diagonal node class. Its rank becomes zero precisely on the already frozen soft or coordinate boundaries

$$P_3ab=0.$$

No replacement map is fitted there; those fibers belong to the existing iterated soft–coordinate complexes.

Endpoint-fold map

On the chart $b\ne0$, set $r=a/b$ and

$$c=E^2-P_1^2.$$

Then

$$\frac{Q_\sigma}{b^2} =cr^2+\sigma\,2EP_3r+(P_2^2-E^2).$$

Define

$$\xi_\sigma=2cr+\sigma\,2EP_3.$$

Exact completion of the square gives

$$\boxed{ 4c\frac{Q_\sigma}{b^2} = \xi_\sigma^2-\frac{4K_0}{P_3^2}. }$$

Both labelled folds inherit the same $r$-orientation because

$$\partial_r\xi_+=\partial_r\xi_-=2c.$$

Hence the frozen endpoint generator maps as

$$\boxed{ M_{K_0} = \begin{pmatrix}1\\1\end{pmatrix}. }$$

This normalization is valid on the generic endpoint chart

$$P_3(E^2-P_1^2)\ne0.$$

Its failure locus is again an existing soft/signed-energy intersection; the matrix is not extended across it by choosing a new denominator.

Characters and orientations

Under the labelled fiber reflection $a\mapsto-a$, the two equations are exchanged. The residue orientation contributes the Jacobian sign, so

$$e_+\longmapsto-e_-, \qquad e_-\longmapsto-e_+.$$

It follows that

$$\boxed{ \chi_{\rm occ}(M_E)=+1, \qquad \chi_{\rm occ}(M_{K_0})=-1. }$$

Both target lines remain anti-invariant under the Kummer deck involution $W\mapsto-W$. The occurrence reflection and Kummer deck character are distinct operations and are retained separately.

Rank test

In the ordered source basis

$$(\text{soft normal},\text{endpoint fold}),$$

the combined map is

$$M_{\rm comb} = \begin{pmatrix} 2P_3ab&1\\ -2P_3ab&1 \end{pmatrix}, \qquad \boxed{\det M_{\rm comb}=4P_3ab.}$$

Thus the two existing source classes generate both labelled polar directions generically:

$$\boxed{\operatorname{rank}M_{\rm comb}=2.}$$

On $P_3ab=0$, the rank drops to one because the first normal soft map vanishes. This is exactly the deeper support already reserved for the iterated soft/coordinate audit.

Narrow conclusion

Away from existing deeper support, no polar coefficient excess remains at the matrix level:

$$\boxed{ \operatorname{coker} (M_E\oplus M_{K_0})=0. }$$

This is not yet the global Beck–Chevalley theorem. It supplies the exact labelled matrices for comparison with Nima's functorial $\psi_E\psi_\Lambda\phi_\pi$ and $\psi_{K_0}\psi_\Lambda\phi_\pi$ squares.

Verification

• exact Symbolica checker: research/benincasa/marici-gm/src/bin/polar_soft_endpoint_maps.rs;
• packet: research/benincasa/polar-soft-endpoint-maps.json;
• allocator claim: seqclaim-7c3d4255f2123845685c44f7.

Next gate

Insert these matrices into the canonical iterated localization diagram. Only cohomology of its supported comparison cone can represent a residual coefficient class. Do not infer one from the controlled rank drop on $P_3ab=0$.

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