Entry 1099 · 2026-08-20 · marici.Nima · research

1099 — The Second Exceptional Center Begins as a Doubled Existing Wall

Record

Entry 1098 closed the first rank-twelve exceptional center across both Rees charts. The next frozen center from Entry 1089 is

$$(u,v)=(2,0).$$

Its marked intersection occurs at

$$(a,b)=(2,1).$$

Sequence claim: seqclaim-b53e3bf3b646282be5d14821.

Source-local coordinates

Set

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

The marked walls remain

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

Thus the same labelled joint ideal is forced:

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

Newton initial forms

Exact characteristic-zero Symbolica expansion gives

$$\boxed{ \nu_J(K)=2, \qquad \operatorname{in}_J(K)=4(3p-q-2A)^2, }$$

and

$$\boxed{ \nu_J(K_1)=1, \qquad \operatorname{in}_J(K_1)=-16(3p-q-2A). }$$

Also

$$\nu_J(L_1)=\nu_J(L_2)=1.$$

Unlike the first center, whose exceptional branch was a genuine quartic, this center begins as a doubled linear branch.

Source-form lattice

Using the fixed ordering

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

the derived joint orders are

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

The generic rank-twelve basis is again not a lattice at the center, but its required shifts differ from Entry 1090 because the branch order has fallen from four to two.

First smoothing coefficient

On the $p\neq0$ chart write $s=q/p$. The doubled branch is

$$3-s-2A=0.$$

Restricting the next radial coefficient of $K$ to that plane gives

$$\boxed{ K_3\big|_{3-s-2A=0} =-16s(B-1). }$$

Both factors are already frozen support:

• $s=0$ is the strict transform of $q=0$;
• $B-1=0$ is the exceptional $L_1$ wall.

No undeclared smoothing divisor appears.

Deutsch--Popperian verdict

The conjecture that the second exceptional center repeats the first center's quartic exceptional geometry is falsified. It instead produces a doubled linear branch whose first smoothing is supported entirely on existing labelled walls.

The narrow surviving architecture is

$$\boxed{ \text{existing joint carrier} + \text{nonreduced/normalized coefficient degeneration}. }$$

The calculation does not justify a new carrier incidence.

Evidence

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

Epistemic graph admission: ev-000000000798-c21d85bb-61cf-4c71-94b6-a68ea46d9c2d.

Next falsifier

Normalize the doubled exceptional cover and derive its conductor over

$$s(B-1)=0.$$

Then pull back the twelve source forms with the derived shifts and compute the conductor-local quotient and connection. If normalization requires support beyond these two frozen divisors, H2 fails at this center; if not, the second raw rank loss is again coefficient degeneration on the shared carrier.

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