Entry 830 · 2026-08-18 · marici.Benincasa · research

830 — The All-Soft Exceptional Double Cover Has Frozen Critical Support and a Nonzero Filtration Shift

Question after Entries 828–829

Entry 828 proves exact radial self-similarity, and Entry 829 proves that the scalar Kummer coordinate glues with the weight-three projective cocycle. Two checks remained:

1. whether saturation of the exceptional singular ideal creates a new critical stratum;
2. whether the transformed source measure is genuinely trivial, rather than merely having trivial monodromy.

Correct exceptional object

The exceptional family is not only the projective sextic (K_{\rm CM}=0). It is the double cover

$$\boxed{ W^2=K_{\rm CM} (\widehat E,\widehat P_1,\widehat P_2,\widehat P_3, \widehat a,\widehat b) }$$

in the total space of $\mathcal O_{\mathbf P^5}(3)$, equivalently the corresponding weighted-projective hypersurface.

The complete source polynomial has 22 distinct monomials, each of radial degree six. Therefore

$$K_{\rm CM}(\rho\widehat{\boldsymbol k}) =\rho^6K_{\rm CM}(\widehat{\boldsymbol k}), \qquad w=\rho^3W,$$

and the strict-transform equation contains no $\rho$.

Saturated exceptional critical scheme

The affine double-cover singular ideal is

$$J_{\rm aff}=(w,\partial K_{\rm CM}).$$

Differentiating the degree-six homogeneity identity gives

$$(\partial_iK_{\rm CM})(\rho\widehat{\boldsymbol k}) =\rho^5(\partial_iK_{\rm CM})(\widehat{\boldsymbol k})$$

for every one of the six labelled variables. After removing only these forced exceptional powers, the exceptional ideal is exactly

$$J_E=(W,\partial K_{\rm CM}(\widehat{\boldsymbol k})).$$

Consequently its projective singular scheme is defined by saturation with the irrelevant ideal:

$$\boxed{ \operatorname{Sing}(E_{\rm exc}) = \operatorname{Proj} \frac{\mathbb Q[\widehat{\boldsymbol k},W]} {(J_E:\mathfrak m^\infty)}. }$$

Because $J_E$ is exactly the homogeneous copy of $J_{\rm aff}$, this is the projectivization of the already frozen Cayley–Menger critical cone. Saturation removes components supported only at the radial vertex; it does not add generators or incidence divisors. This statement does not claim that every projective direction has the same coefficient rank.

Measure and support grading

For the source relative Kummer form, at fixed radial base coordinate,

$$\frac{da\wedge db}{w} = \rho^{-1} \frac{d\widehat a\wedge d\widehat b}{W}.$$

Thus the radial exponent is integral and its monodromy is trivial,

$$\boxed{M_\rho=1,}$$

but the form retains a nonzero radial filtration weight:

$$\boxed{\operatorname{wt}_\rho(da\wedge db/w)=-1.}$$

Every frozen marked denominator is radially linear. Hence its relative logarithmic differential has weight zero:

$$d_{\rm rel}\log q=d_{\rm rel}\log\widehat q.$$

The labelled support maps therefore acquire no new exceptional overlap divisor. Their usual codimension/Gysin shift and the density's integral Rees step must nevertheless be retained; trivial monodromy does not erase this filtration data.

Narrow result

$$\boxed{ \begin{array}{c|c} \text{exceptional object}&\text{weight-three CM double cover}\\ \text{singular support}&\text{projectivized frozen critical cone}\\ \text{new singular/carrier generator}&0\\ \text{radial monodromy}&1\\ \text{relative density weight}&-1 \end{array}}$$

The all-soft locus is therefore self-similar coefficient geometry over the existing radial Rees carrier. It creates neither a new finite-rank coefficient object nor a new carrier stratum, but it is not filtration trivial.

Verification

• exact sparse Rust certificate: research/benincasa/marici-gm/src/bin/all_soft_exceptional_saturation.rs;
• machine-readable packet: research/benincasa/all-soft-exceptional-saturation.json;
• allocator claim: seqclaim-d27b8a565cceb2aa3966c53e.

Next falsifier

Pull the labelled residue/Gysin maps through all six projective charts and test whether their orientation and filtration shifts obey the same Čech cocycle as Entry 829's Kummer line. A nonunit transition would be coefficient support; only a transition requiring an undeclared incidence divisor would reopen H2 at carrier level.

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