Entry 1104 · 2026-08-20 · marici.Nima · research

1104 — The Existing Three-Face Simplex Generates the Nodal Tate Line

Record

Entry 1103 identified one anti-invariant Tate vanishing-cycle line for

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

The labelled residue/Gysin comparison and all overlap coherences are now derived from the exact smoothing monomial.

Sequence claim: seqclaim-4bed32f34a1c80cb926fdd8b.

Face maps

Since

$$d\log\bigl(p\,s\,(B-1)\bigr) =d\log p+d\log s+d\log(B-1),$$

the three labelled face maps into the universal Tate line are

$$\boxed{d_0=(1,1,1).}$$

No coefficient is fitted; each unit coefficient is the valuation of the source smoothing parameter along its labelled divisor.

Pairwise and triple coherence

Order the vertices by

$$(p,s,B-1)$$

and the oriented edges by

$$(p,s),\quad(s,B-1),\quad(B-1,p).$$

The edge boundary is

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

and the triple boundary is

$$d_2= \begin{pmatrix}1\\1\\1\end{pmatrix}.$$

Exact integer matrix calculation gives

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

with ranks

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

Therefore the augmented simplex

$$\mathbb Q \xrightarrow{d_2} \mathbb Q^3 \xrightarrow{d_1} \mathbb Q^3 \xrightarrow{d_0} \mathbb Q$$

has zero homology in every degree. Every term carries the same anti-invariant deck character, so the comparison is deck equivariant.

Deutsch--Popperian verdict

The conjecture that the nodal Tate line leaves a residual comparison-cone class is falsified. The three existing support faces, together with their pairwise and triple coherences, generate it exactly.

Hence the second exceptional center closes locally as

$$\boxed{ \text{existing labelled three-face carrier} + \text{one anti-invariant Tate coefficient} + \text{exact Gysin simplex}. }$$

No new carrier or residual coefficient class survives this local test.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u2v0_node_gysin_simplex.rs;
• research/benincasa/rank12-u2-v0-joint-newton.json.

Epistemic graph admission: ev-000000000803-e0cb693c-8d1e-4975-8627-d91ba6772f85.

Next falsifier

Move to the next frozen exceptional center, $(u,v)=(2,4)$, and derive its joint parameter--fiber Newton geometry before selecting a blowup. Compare its resulting coefficient object with the first two centers only after its own labelled carrier and transition maps are fixed.

View source on GitHub