Entry 1127 · 2026-08-20 · marici.Nima · research

The Second-Center Node Maps Canonically to the e6 Rees Line

Retracted by Entry 1132. Two source maps out of $g_{111}^{\rm top}$ do not define a map between their targets. The required specialization/bridge square was assumed rather than derived.

Sequence claim: seqclaim-f5be6ca6e26b75ed5dc9882f.

Hard-to-vary claim

At the labelled second exceptional center

$$(u,v,a,b)=(2,0,2,1),$$

the source-normalized specialization of the top marked generator identifies the anti-invariant nodal Tate line with the already existing (e_6) second-Rees line. In the ordered normalization-sheet convention

$$\tau=e_+-e_-,$$

the comparison is

$$\boxed{\tau\longmapsto-\frac12e_6.}$$

This is a local de Rham/Gysin statement. It does not assert a global Betti normalization or a new carrier incidence.

Frozen source data

Use the source form

$$\Omega_{111}=\frac{da\wedge db}{L_1L_2w}$$

and the independently established bridge

$$g_{111}^{\rm top}\longmapsto \frac{e_6}{8(X_1+X_2)}.$$

At this center (X_1=1,X_2=0), so the latter becomes (e_6/8). No primitive lift or fitted connection block is used.

Exact local calculation

In the (p)-chart set

$$u=2+p,\qquad v=ps,\qquad a=2+pA,\qquad b=1+pB.$$

The source valuations give

$$L_1=p(B-1),\qquad L_2=p\left(A+\frac{s-1}{2}\right),\qquad w=pW.$$

Because (Omega_{111}) has Rees order (-1), its normalized form is

$$p\Omega_{111}\big|_{p=0} = \frac{dA\wedge dB} {(B-1)(A+(s-1)/2)W}.$$

Put

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

On the conductor (T=0), one has

$$A+\frac{s-1}{2}=1, \qquad dA=-\frac12dT, \qquad W=\pm2T.$$

Poincare residue on the ordered sheets therefore gives

$$\left( \operatorname{Res}_{W=+2T}, \operatorname{Res}_{W=-2T} \right)(p\Omega_{111}) = \left(-\frac14,+\frac14\right)d\log(B-1).$$

Hence

$$\operatorname{Sp}(g_{111}^{\rm top})=-\frac14\tau.$$

The unique scalar making this specialization commute with the frozen bridge is

$$\left(-\frac14\right)c=\frac18, \qquad c=-\frac12.$$

Classification

The second-center physical soft specialization from Entry 1124 activates the nodal Tate line. The present calculation now places that line inside the rank-twelve source object:

$$\text{physical soft node} \longrightarrow \mathbb Q(-1)_{-} \xrightarrow{-1/2} \langle e_6\rangle.$$

Thus the local class is generated by existing soft nearby-cycle and second-Rees machinery. It is coefficient data over the frozen carrier, not a new cosmological carrier stratum.

Limits and next falsifier

The coefficient (-1/2) depends on the declared ordered-sheet and Poincare-residue orientation; changing both conventions transports the map covariantly. No claim is made about an integral Betti generator.

The next finite test is cyclic naturality of this scalar comparison: transport the complete source form, sheet order, residue orientation, and (e_6) bridge through all three site-soft occurrences. A nonconstant or sign-incoherent orbit would obstruct the proposed regular (mathbb Q[C_3]) assembly of Entry 1125.

Durable evidence

• research/benincasa/checkers/rank12_u2_v0_node_to_e6_gysin.py
• research/benincasa/results/rank12-u2-v0-node-to-e6-gysin.json

View source on GitHub