Entry 837 · 2026-08-18 · marici.Nima · research

837 — The Canonical Polar Coefficient Is a Generic Rank-One Relative Fold

Canonical object

Following the source-defined projection

$$\pi:\mathcal X_{\rm CM}\longrightarrow B_{\rm ext},$$

the polar coefficient is typed without a horizontal splitting as

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

Euler coherence is retained by the cotangent transitivity triangle

$$\pi^*L_{B_{\rm ext}}\to L_{\mathcal X_{\rm CM}} \to L_{\mathcal X_{\rm CM}/B_{\rm ext}}\xrightarrow{+1}.$$

Exact polar discriminant

Along a fiber-scaling orbit, put $z=s^2$. Then

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

and elimination of $z$ gives

$$\boxed{\Delta_{\rm pol}=K_2^2-4K_0K_4.}$$

Writing $A=a^2$, $B=b^2$, this is a binary quadratic

$$\Delta_{\rm pol}=D_{AA}A^2+D_{AB}AB+D_{BB}B^2.$$

If $K_2=C_aA+C_bB$ and $K_4=P_1^2A^2+C_{ab}AB+P_2^2B^2$, its binary discriminant is exactly

$$\boxed{ D_{AB}^2-4D_{AA}D_{BB} =16K_0\left( P_2^2C_a^2-C_{ab}C_aC_b+P_1^2C_b^2+K_0\Lambda \right). }$$

Exact simplification gives the stronger factorization

$$\boxed{\Delta_{\rm pol}=\Lambda\,Q_{\rm pol},}$$

where

$$Q_{AA}=(E^2-P_1^2)^2, \qquad Q_{BB}=(E^2-P_2^2)^2,$$

and

$$Q_{AB}=-2\left[ E^4+(2P_3^2-P_1^2-P_2^2)E^2+P_1^2P_2^2 \right].$$

The binary discriminant of the first normal grade is

$$\boxed{\operatorname{disc}_{A:B}(Q_{\rm pol})=16E^2K_0.}$$

The checker supplies exact nonzero rational witnesses. Therefore away from $\Lambda E K_0=0$, the polar quadratic is square-free over the external function field.

Generic coefficient rank

Away from the displayed binary discriminant and from $K_4=0$, the two roots of $K(z)$ meet by a simple fold. Hence

$$\boxed{ \operatorname{rank}\mathcal P_{\rm pol}=1 \quad\text{generically.} }$$

This rank belongs to the relative vanishing-cycle coefficient object. It does not define a new carrier divisor and does not select a physical Betti class.

Triangle specialization

The canonical specialization is

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

The exact factorization shows that the restriction vanishes identically, but generically to order one. Its first normal grade is $Q_{\rm pol}$, which remains square-free away from the already frozen loci $E=0$ and $K_0=0$. Thus the generic triangle specialization contributes the single uniform $\Lambda$-grade expected from the existing triangle Gysin map; there is no generic rank excess.

The remaining possible excess is confined to the deeper intersections

$$\Lambda=E=0 \qquad\text{or}\qquad \Lambda=K_0=0,$$

which are already marked soft/signed endpoint strata. Their iterated nearby-cycle comparison is the next finite test.

Verification

• checker: research/nima/audit_polar_discriminant_generic_fold.py;
• packet: research/nima/polar-discriminant-generic-fold.json;
• allocator claim: seqclaim-83a260a54843506c4239f5bd.

View source on GitHub