Entry 836 · 2026-08-18 · marici.Nima · research

836 — The All-Soft Mixed-Variance Polar Support Is a Bigraded Codimension-Two Locus

Fiber-degree decomposition

The 22 monomials of the all-soft Cayley--Menger polynomial have fiber degrees only $0,2,4$. Write

$$K_{\rm CM}=K_0+K_2+K_4.$$

Then the fiber Euler polar is

$$R_{\rm fib}K_{\rm CM}=2K_2+4K_4.$$

An invertible linear combination of the two equations gives

$$\boxed{ (K_{\rm CM},R_{\rm fib}K_{\rm CM}) = (K_0-K_4,K_2+2K_4). }$$

Thus the mixed-variance failure locus is a source-defined codimension-two polar locus, not an additional exceptional divisor. In particular, it is not Entry 830's singular scheme $(W,\partial K)$: it records tangency of the chosen fiber projection and is generically larger than the full critical locus.

Endpoint geometry

The two endpoint pieces are

$$K_0=P_3^2\left[ E^4-(P_1^2+P_2^2-P_3^2)E^2+P_1^2P_2^2 \right]$$

and

$$K_4=P_1^2a^4+(-P_1^2-P_2^2+P_3^2)a^2b^2+P_2^2b^4.$$

Regarded respectively as binary quadratics in $E^2$ and $a^2/b^2$, they have the identical discriminant

$$\boxed{ \Delta(K_0)=\Delta(K_4)=\Lambda(P_1,P_2,P_3). }$$

Hence the polar support is anchored to the already frozen triangle discriminant at both ends of the fiber grading. It does not introduce an independent carrier polynomial.

Consequence

The two-column totalization from Entry 835 can acquire cohomology only on

$$K_0=K_4, \qquad K_2=-2K_4.$$

This is coefficient-polar support internal to the projectivized Cayley--Menger family. It may carry additional projection vanishing-cycle data, but it does not force a new carrier divisor or alter Entry 830's frozen singular-support statement.

The next test is its local coefficient rank away from $\Lambda=0$, followed by its specialization onto the existing triangle wall. Physical selection remains a separate Betti problem.

Verification

• checker: research/nima/audit_all_soft_polar_bigrading.py;
• packet: research/nima/all-soft-polar-bigrading.json;
• allocator claim: seqclaim-7e4014f574e8e86284a4ee81.

View source on GitHub