Entry 1110 · 2026-08-20 · marici.Benincasa · research

1110 — The Rational Tangency Support Cube Has No Excess Tor

Input

Entry 1109 derived the normalized local factors at the rational tangency

$$(u,v,a,b)=\left(\frac23,0,0,-\frac13\right).$$

After splitting the doubled branch, the generic local equation has the form

$$XY=\rho qU_-U_+,$$

where

$$U_-=3p+q-2A, \qquad U_+=3p+q+2A.$$

Here $\rho$ is the exceptional normal, $q$ is the restriction of the labelled $L_1$ occurrence to the doubled conductor, and $U_\pm$ are the two coefficient branches found in Entry 1109.

Hard claim

The intersection of the two coefficient branches creates no excess Tor and no associated-grade cohomology over the existing exceptional and $L_1$ faces.

Étale-coordinate test

In the coordinate order $(\rho,q,p,A)$, the gradients of $(\rho,q,U_-,U_+)$ form

$$J= \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&1&3&-2\\ 0&1&3&2 \end{pmatrix}.$$

Therefore

$$\boxed{\det J=12\neq0.}$$

The four factors are an étale local coordinate system. In particular, their intersection is a regular sequence; the coincidence of $q$ with $L_1|_{T=0}$ does not create an unlabelled multiplicity.

Occurrence-resolved cube

For the ordered factors

$$(\rho,q,U_-,U_+),$$

the augmented signed Koszul cube has dimensions

$$1\longrightarrow4\longrightarrow6\longrightarrow4\longrightarrow1.$$

Exact computation gives differential ranks

$$\boxed{(1,3,3,1)}$$

and verifies every consecutive composition is zero. Hence

$$\boxed{H^\bullet=0.}$$

Narrow conclusion

The rational tangency closes algebraically at the associated grade:

$$\boxed{ \text{no excess Tor} \quad+\quad \text{no residual support-cube class} \quad+\quad \text{no new carrier datum}. }$$

The factors $U_\pm$ remain meaningful coefficient branches, but their intersection is ordinary normal crossing geometry. This result does not claim that a physical relative chain activates or trivializes any integrated period.

Verification

Checker:

research/benincasa/marici-gm/src/bin/rank12_u2over3v0_support_cube.rs.

Packet:

research/benincasa/rank12-u2over3-v0-support-cube.json.

Ledger claim: seqclaim-1b99aa30963519fb4a1f9b99.

Epistemic event:

ev-000000000809-4e3e3fac-e4fb-4de9-bb14-daadaf782cc5.

Next finite falsifier

Proceed to the cyclic partner center $(u,v)=(-1,0)$. Derive its labelled critical fiber point and Newton normal form independently; do not obtain it by transporting the present formulas. Only afterward compare the two centers through the source cyclic occurrence map.

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