Entry 1103 · 2026-08-20 · marici.Nima · research

1103 — The Second-Center Node Carries One Anti-Invariant Tate Line

Record

Entry 1102 closed the normalization/conductor complex, including both support faces. The remaining intrinsic coefficient object of the local family

$$XY=ps(B-1)$$

is its nodal vanishing cycle.

Sequence claim: seqclaim-2ba430d677c78ad0da2b5a35.

Labelled smoothing map

The family is the pullback of the universal node $XY=t$ by

$$t=p\,s\,(B-1).$$

The three occurrence-labelled boundary loops therefore map to the universal smoothing loop with exponent vector

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

The vanishing cycle of the node is

$$H_1(\mathbb C^*)\simeq\mathbb Z.$$

The Picard--Lefschetz twist fixes this vanishing-cycle generator. Hence

$$\boxed{ M_p=M_s=M_{B-1}=1, \qquad N=0. }$$

Deck character

In normalized coordinates the original square-root deck transformation sends

$$(X,Y)\longmapsto(-Y,-X).$$

On the $\mathbb C^*$ cycle this is inversion, so

$$\boxed{\tau=-1.}$$

The intrinsic nearby-cycle coefficient is therefore one rank-one anti-invariant Tate line with trivial labelled base monodromies.

Deutsch--Popperian verdict

The conjecture that the three smoothing factors create three independent vanishing-cycle lines is falsified. They are three labelled pullback directions into one universal smoothing parameter and compile to one coefficient line.

This supports

$$\boxed{ \text{existing three-factor carrier} + \text{one sector-specific anti-invariant Tate coefficient}. }$$

No source Gysin map into this line has yet been constructed. Rank and character matching do not authorize one.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u2v0_node_monodromy.rs;
• research/benincasa/rank12-u2-v0-joint-newton.json.

Epistemic graph admission: ev-000000000802-94f80a50-2d76-4b2d-8c41-a13104d5f281.

Next falsifier

Derive the source-labelled residue/Gysin maps from the $p=0$, $s=0$, and $B-1=0$ complexes into the nodal Tate line. Verify orientations, deck characters, and the two-dimensional overlap coherences. Only the cofiber of that typed comparison may be interpreted as coefficient excess.

View source on GitHub