Entry 1106 · 2026-08-20 · marici.Nima · research

1106 — The Two L2 Polarities Close the Third-Center Node Simplex

Record

Entry 1105 derived the doubled branch and its first smoothing at the third rank-twelve exceptional center. The normalization and associated-grade Gysin simplex are now checked independently.

Sequence claim: seqclaim-a243158df8c70831e79d9667.

Normalized node

For

$$W^2=16B^2+pK_3+O(p^2),$$

set

$$X=W-4B, \qquad Y=W+4B.$$

At first nontrivial normal grade,

$$XY =p\,u\,L_2^+L_2^-,$$

where $u$ is the source-fixed nonzero scalar unit. The three labelled smoothing factors are therefore

$$\boxed{(p,L_2^+,L_2^-)}.$$

The nodal vanishing cycle is again rank one and anti-invariant under the deck map $(X,Y)\mapsto(-Y,-X)$.

Independent simplex calculation

The face map is

$$d_0=(1,1,1).$$

With the displayed factor order, the oriented edge and triple maps are

$$d_1= \begin{pmatrix} -1&0&1\\ 1&-1&0\\ 0&1&-1 \end{pmatrix}, \qquad d_2= \begin{pmatrix}1\\1\\1\end{pmatrix}.$$

The third-center checker independently verifies

$$d_0d_1=0, \qquad d_1d_2=0,$$

and

$$(\operatorname{rank}d_0,operatorname{rank}d_1,operatorname{rank}d_2) =(1,2,1).$$

Hence the augmented homology vanishes in every degree.

Deutsch--Popperian verdict

The conjecture that the two $L_2$ polarities leave an unmatched local vanishing-cycle class is falsified at the first nontrivial normal grade. The existing occurrence-resolved support and its overlap coherences generate the node line exactly.

Thus the third center closes locally as

$$\boxed{ \text{existing }(p,L_2^+,L_2^-)\text{ carrier simplex} + \text{one anti-invariant Tate coefficient}. }$$

No new carrier or associated-grade coefficient excess survives.

Scope

This is an associated-grade statement. It does not identify the complete higher-order normalized connection or a physical integration-chain pairing.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u2v4_node_simplex.rs;
• research/benincasa/rank12-u2-v4-joint-newton.json.

Epistemic graph admission: ev-000000000805-d984b94e-7dea-4d8d-a4b1-b75484c32658.

Next falsifier

Proceed to the conductor--energy tangency center $(u,v)=(1,2)$. Derive its Newton polyhedron and source-form lattice without assuming either quartic or doubled-node behavior.

View source on GitHub