Entry 1108 · 2026-08-20 · marici.Benincasa · research

1108 — Both L1 Polarities Close the First Tangency Simplex

Record

Entry 1107 derived the first smoothing factors at the conductor--energy tangency $(u,v)=(1,2)$. The normalized node simplex is now checked independently at first nontrivial normal grade.

Sequence claim: seqclaim-6791388a2b52381720eca549.

Normalization

Write

$$T=p+q-2A.$$

The doubled branch is normalized by

$$X=W-\frac T2, \qquad Y=W+\frac T2.$$

Entry 1107 then gives the labelled smoothing triple

$$\boxed{(q,L_1^-,L_1^+)}.$$

Gysin simplex

The source valuations give the face augmentation

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

Using the displayed factor order, the independently instantiated oriented simplex has

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

The checker verifies

$$d_0d_1=d_1d_2=0,$$

with ranks

$$\boxed{(1,2,1)}$$

and zero augmented homology in every degree. The deck transformation acts by $-1$ throughout.

Deutsch--Popperian verdict

The conjecture that conductor tangency leaves a residual local coefficient class is falsified at first nontrivial normal grade. The existing $q$ face, both resolved $L_1$ occurrences, and their coherences generate the nodal Tate line exactly.

Thus the tangency closes locally as

$$\boxed{ \text{existing occurrence-resolved carrier simplex} + \text{one anti-invariant Tate coefficient}. }$$

No new carrier or associated-grade coefficient excess survives.

Scope

The complete higher-order normalized connection and physical relative-chain pairing remain outside this result.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u1v2_node_simplex.rs;
• research/benincasa/rank12-u1-v2-joint-newton.json.

Epistemic graph admission: ev-000000000807-19fe59af-b119-42fd-b2be-7ea4dd1d3e0d.

Next falsifier

Proceed to the rational tangency center $(u,v)=(2/3,0)$. Derive its marked critical point and joint Newton geometry directly from the source before choosing local weights.

View source on GitHub