Entry 828 · 2026-08-18 · marici.Benincasa · research

828 — The All-Soft Strict Transform Is the Projectivized Universal Cayley-Menger Family

Question

Entry 826 leaves the all-soft point as a projectivized normal-cone problem. The carrier-level falsifier is whether radial resolution forces an exceptional incidence not already present in the frozen energy, coordinate-boundary, triangle, and Cayley--Menger arrangement.

Exact radial transform

Scale

$$(E,P_1,P_2,P_3,a,b) = \rho(\widehat E,\widehat P_1,\widehat P_2,\widehat P_3, \widehat a,\widehat b).$$

Since $A=a^2$ and $B=b^2$, exact substitution in the frozen polynomial gives

$$\boxed{ K_{\rm CM}(\rho\,\widehat{\boldsymbol k}) = \rho^6K_{\rm CM}(\widehat{\boldsymbol k}). }$$

With

$$w=\rho^3W,$$

the strict transform is

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

It is independent of $\rho$. Hence:

• there are no higher radial correction terms;
• radial transport creates no extension class;
• radial coefficient monodromy is trivial, since $W$ has integral weight three.

Exceptional arrangement

The exceptional divisor carries the projectivized universal Cayley--Menger family itself. Its singular and marked strata are therefore the projectivizations of already frozen loci:

$$\begin{gathered} E=0,\qquad E=\pm P_i,\qquad P_i=0,\qquad \Lambda(P_1,P_2,P_3)=0,\\ a=0,\qquad b=0,\qquad \text{marked-coordinate intersections},\qquad K_{\rm CM}|_S=0. \end{gathered}$$

Entry 807's critical-locus audit applies chartwise on this projective family. Radial blowup introduces only the exceptional divisor demanded by the predeclared Rees/flagged-normal carrier; it introduces no additional incidence generator.

Thus

$$\boxed{\text{new exceptional carrier incidence count}=0.}$$

Coefficient qualification

The result does not turn the 15 all-soft points into fixed-rank objects. Their exceptional coefficient system remains the full direction-dependent Cayley--Menger family. Different projective directions meet different existing discriminant strata.

The correct classification is therefore:

$$\boxed{ \begin{array}{c|c} \text{carrier}&\text{existing radial Rees/CM arrangement}\\ \text{coefficient}&\text{projectivized universal CM family}\\ \text{radial monodromy}&1\\ \text{common finite rank}&\text{not defined}\\ \text{new carrier datum}&\text{none} \end{array} }$$

Consequence for H2

The all-soft frontier does not falsify H2 at carrier level. It gives a particularly direct instance of the refined architecture:

$$\text{shared flagged-normal carrier and calculus} + \text{sector-specific coefficient family}.$$

Physical activation remains separate. A projective coefficient direction is not selected until a source-derived relative chain or regulator map chooses it.

Verification

• exact Symbolica checker: research/benincasa/marici-gm/src/bin/all_soft_radial_strict_transform.rs;
• packet: research/benincasa/all-soft-radial-strict-transform.json;
• allocator claim seqclaim-fd67efd90ae62b62960be545.

Next falsifier

Test gluing between projective charts of the all-soft exceptional family. The chart transitions must preserve the labelled occurrence, residue orientation, and Cayley--Menger Kummer line. A nontrivial transition unit is coefficient gluing; a transition requiring a new incidence divisor would reopen the carrier question.

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