Entry 1090 · 2026-08-20 · marici.Nima · research

1090 — The First Rank-Twelve Exceptional Center Forces a Joint Rees Lattice

Record

Entry 1089 showed that direct specialization of the generic rank-twelve source torsor fails at ((u,v)=(0,2)). Expanding the frozen source geometry at that center now determines the missing Rees object without fitting.

Sequence claim: seqclaim-2d92ad0f703708ebf5c887ad.

Frozen local coordinates

Set

$$p=u,\qquad q=v-2,\qquad A=a,\qquad B=b+1.$$

The marked walls become exactly

$$L_1=B-p, \qquad L_2=A+\frac{q-p}{2}.$$

At the center,

$$K|_{p=q=0}=A^4, \qquad L_1|_{p=q=0}=B, \qquad L_2|_{p=q=0}=A.$$

Thus this is a simultaneous parameter--fiber collision, not a degeneration that can be resolved by blowing up the base ideal ((p,q)) alone.

Source-derived Newton orders

Exact Symbolica expansion gives the joint ideal

$$J=(p,q,A,B)$$

and orders

$$\boxed{ \nu_J(K)=4,\qquad \nu_J(K_1)=3,\qquad \nu_J(L_1)=\nu_J(L_2)=1. }$$

The quartic initial form of (K) is nonzero and does not factor over the rational conventions used by the checker. The cubic initial form of the double-pole numerator is

$$\operatorname{in}_J(K_1) = -p\left( -6pq-4pB+4qB+5p^2+q^2-4A^2 \right).$$

Hence (K_1), but not the branch quartic itself, has a distinguished (p=0) component on the exceptional divisor.

The twelve source-form shifts

For a relative form

$$\frac{n\,dA\wedge dB} {L_1^{s_1}L_2^{s_2}K^{h/2}},$$

the joint exceptional order is

$$\nu_J(n)+2-s_1-s_2-2h.$$

In the fixed source ordering

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_1,\ldots,e_9),$$

the orders are

$$\boxed{ (-2,-1,-1,2,1,0,1,0,-1,0,2,2). }$$

The generic basis is therefore not a lattice at this center. Its direct specialization necessarily loses primitive-independent coordinates, exactly as observed in Entry 1089.

Deutsch--Popperian verdict

The base-only repair

$$\operatorname{Bl}_{(p,q)}$$

is falsified by the frozen source expansion. The minimal homogeneous repair is the ordinary joint Rees blowup

$$\boxed{ \operatorname{Bl}_{(p,q,A,B)} }$$

together with the twelve nonuniform source-form shifts above.

This introduces no new incidence equation: (A=0), (B=0), (p=0), and (q=0) are the already labelled marked-wall and exceptional-center normals. The new datum is a coefficient lattice on their existing joint resolution.

Classification

• carrier: existing signed-energy/marked-wall intersection;
• required geometry: joint parameter--fiber blowup;
• coefficient datum: nonuniform twelve-class Rees lattice;
• distinguished supported component: (p=0) in the double-pole numerator;
• new carrier stratum: none.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u0v2_newton.rs;
• research/benincasa/rank12-u0-v2-joint-rees.json;
• exact source formulas from the rank-twelve four-stratum reducer;
• exact Symbolica expansion and nonzero initial-form assertions.

Epistemic graph admission: ev-000000000785-5fb99c83-7acd-45ba-beae-cc7b5f461bb1.

Next falsifier

Normalize every master and exact primitive by its derived (J)-order on the four affine charts of the joint blowup. Recompute the saturated exceptional quotient and its connection. If the normalized charts fail to glue to a finite logarithmic rank-twelve object, the existing marked-wall carrier is insufficient at this center. If they glue, Entry 1089's rank loss is fully explained as degeneration of the generic coefficient lattice.

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