Entry 1091 · 2026-08-20 · marici.Nima · research

1091 — The First Joint-Rees Chart Has Only Perfect-Square Marked Restrictions

Record

Entry 1090 derived the joint Rees ideal

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

at the rank-twelve exceptional center ((u,v)=(0,2)). The first affine chart of this blowup has (p\neq0) and coordinate

$$s=\frac qp.$$

Sequence claim: seqclaim-5c85ee85b9200a552edf4204.

Exceptional family and retained walls

Let

$$K_E(s,A,B)=\operatorname{in}_J(K)(1,s,A,B).$$

The two labelled marked walls retain the equations

$$L_{1,E}=B-1, \qquad L_{2,E}=A+\frac{s-1}{2}.$$

No wall is reconstructed from a factorization; both are strict transforms of the frozen source denominators.

Exact restriction identities

Exact symbolic reduction gives

$$\boxed{ K_E|_{L_{1,E}=0} = \frac1{16} \left(1+6s+s^2-4A^2\right)^2, }$$

and

$$\boxed{ K_E|_{L_{2,E}=0} = \left(-1+B+Bs+s\right)^2. }$$

At the labelled top intersection,

$$\boxed{ K_E|_{L_{1,E}=L_{2,E}=0}=4s^2. }$$

Thus both single-wall restrictions and the same-sheet top restriction are perfect squares. The only top collision in this chart is

$$s=0,$$

which is precisely the strict-transform direction (q=0).

Deutsch--Popperian verdict

The conjecture that the joint blowup creates a new exceptional marked incidence is falsified on the (p)-chart. Its marked restrictions contain no new branch polynomial. They rationalize into the two existing deck sheets, and their top collision is carried by the already labelled normal (q=0).

The surviving structure is coefficient-theoretic: the nonuniform Rees shifts of Entry 1090 and the choice of deck character on these perfect-square restrictions. Neither licenses a new carrier wall.

Classification

• exceptional carrier: existing joint marked-wall resolution;
• single-wall coefficient geometry: rationalized two-sheet restriction;
• top support: existing (q=0) normal;
• new branch divisor: none;
• new carrier stratum: none.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u0v2_newton.rs;
• research/benincasa/rank12-u0-v2-joint-rees.json;
• exact characteristic-zero Symbolica factorization.

Epistemic graph admission: ev-000000000786-91925b33-b79c-4818-8fa1-20420cbb4a9a.

Next falsifier

Construct the normalized twelve-class reduction on this (p)-chart and test whether the two perfect-square wall restrictions glue with the required deck characters at (s=0). A failure of deck-compatible gluing would be a coefficient obstruction on the existing carrier; only a source-derived new incidence equation could reopen the carrier question.

View source on GitHub