Entry 1109 · 2026-08-20 · marici.Benincasa · research

1109 — The Rational Tangency Is a Doubled Conic with One Labelled Marked Face

Question

At the exceptional parameter center

$$(u,v)=\left(\frac23,0\right),$$

what is the source-labelled Cayley--Menger degeneration, and does its first smoothing require a new carrier wall?

Frozen source

Use the same homogeneous rank-twelve Cayley--Menger polynomial $K$, residue coefficient $K_1$, and marked planes

$$L_1=b+1-u, \qquad L_2=a+\frac{v-2-u}{2}$$

as in Entries 1098--1108. No wall, quotient, or support summand is added after the calculation.

The naive marked intersection is not the critical point

Simultaneously imposing $L_1=L_2=0$ gives $(a,b)=(4/3,-1/3)$, but direct substitution yields

$$K=\frac{256}{81}, \qquad K_1=\frac{128}{81}.$$

That point is therefore rejected.

At fixed $(u,v)=(2/3,0)$, the complete fiber polynomial instead factors as

$$\boxed{ K = \frac1{729} \left(27a^2-18b^2+2\right)^2. }$$

The labelled plane $L_1=b+1/3$ is tangent to this doubled conic at

$$\boxed{(a,b)=\left(0,-\frac13\right),}$$

while $L_2=-4/3$ is a unit there.

Joint Newton audit

Set

$$p=u-\frac23, \qquad q=v, \qquad A=a, \qquad B=b+\frac13.$$

The exact Symbolica expansion gives

$$\nu K=2, \qquad \operatorname{in}K = \frac{16}{81}(p+q-B)^2,$$

and

$$\nu K_1=1, \qquad \operatorname{in}K_1 = -\frac{32}{81}(p+q-B).$$

Writing $T=p+q-B$, the next coefficient on the doubled plane is

$$\boxed{ K_3|_{T=0} = -\frac49 q(3p+q-2A)(3p+q+2A). }$$

Moreover,

$$L_1|_{T=0}=q, \qquad L_2|_{T=0}=-\frac43.$$

Narrow conclusion

The first smoothing contains the already-labelled $L_1$ occurrence. The two additional linear factors

$$3p+q-2A, \qquad 3p+q+2A$$

are branches of the coefficient smoothing polynomial. This calculation does not type them as new carrier walls.

Thus the surviving statement is

$$\boxed{ \text{doubled conic} + \text{exceptional normal} + \text{one existing labelled }L_1\text{ face}, }$$

with no new carrier datum found.

This entry does not yet claim that the resulting support complex is exact.

Verification

The durable checker is

research/benincasa/marici-gm/src/bin/rank12_u2over3v0_newton.rs.

Its machine-readable packet is

research/benincasa/rank12-u2over3-v0-rational-tangency.json.

Ledger claim: seqclaim-d727929531ddbd13dfba88a8.

Epistemic event:

ev-000000000808-40cf5c73-4a73-48c8-b70b-565d588bcb45.

Next finite falsifier

Normalize the doubled conic and construct the occurrence-resolved support complex for the exceptional normal and $L_1$. Retain the two coefficient branches separately, and test whether their intersection creates Tor or a nontrivial nearby-cycle extension. Only such surviving cohomology could justify new coefficient complexity; it would still not by itself define a new carrier wall.

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