Entry 1100 · 2026-08-20 · marici.Nima · research

1100 — The Second Exceptional Cover Normalizes to an Existing-Support Node

Record

Entry 1099 found that the second rank-twelve exceptional center begins with the doubled branch

$$W^2=4T^2, \qquad T=3-s-2A$$

on the $p\neq0$ chart. The normalization/conductor model is now derived from the next exact radial coefficient.

Sequence claim: seqclaim-9371d13bb9be8b0bac70b34a.

Normalized node coordinates

Set

$$X=W-2T, \qquad Y=W+2T.$$

Since the strict-transform branch equation has the form

$$W^2=4T^2+pK_3+p^2K_4+\cdots,$$

the complete local equation becomes

$$\boxed{ XY=pU, }$$

where

$$U|_{p=0,T=0}=-16s(B-1).$$

Away from $s(B-1)=0$, this is the standard semistable node. Its special fiber is the union $X=0$ and $Y=0$, and the conductor is their labelled intersection.

Marked-wall restriction

On the conductor plane $T=0$, exact substitution gives

$$\boxed{ L_1|_{T=0}=B-1, \qquad L_2|_{T=0}=1. }$$

Thus the conductor meets the first marked wall, while the second marked wall is a unit. The only failure of the generic node model occurs on

$$s=0 \qquad\text{or}\qquad B-1=0,$$

both already declared carrier supports.

Deutsch--Popperian verdict

The conjecture that normalization of the doubled branch exposes an additional marked or branch divisor is falsified. The normalization is semistable away from the existing $q=0$ and $L_1$ supports, and $L_2$ disappears from the conductor geometry as a unit.

Therefore the second center currently has the typed local form

$$\boxed{ \text{two normalized coefficient sheets} \xleftarrow{\text{conductor}} \text{existing }(q=0)\cup(L_1=0)\text{ support}. }$$

No new carrier incidence is required.

Epistemic status

• source branch and radial coefficients: exact characteristic-zero algebra;
• node factorization and wall restrictions: exact Symbolica assertions;
• conductor-local rank-twelve quotient and connection: not yet computed;
• new carrier datum: none.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u2v0_newton.rs;
• research/benincasa/rank12-u2-v0-joint-newton.json.

Epistemic graph admission: ev-000000000799-47608461-9926-4115-a9ab-24834216680c.

Next falsifier

Construct the conductor-local source complex with its two normalized sheet restrictions and their difference map. Apply the twelve derived Rees shifts from Entry 1099, then compute the kernel/cokernel and induced connection away from and on $s(B-1)=0$. Any surviving class must be classified as sheet anti-invariance, existing-support coefficient data, or genuinely new carrier structure before physical interpretation.

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