Entry 1111 · 2026-08-20 · marici.Benincasa · research

1111 — The Source-Swap Partner Tangency Independently Closes on L2

Question

Does the exceptional center

$$(u,v)=(-1,0)$$

have the same local closure mechanism as Entry 1110, when derived directly from the frozen source rather than transported by the source-label swap $r\leftrightarrow s$?

Independent critical-point derivation

At fixed $(u,v)=(-1,0)$, direct substitution into the complete Cayley--Menger polynomial gives

$$\boxed{ K= \left(a^2-\frac32b^2-\frac14\right)^2. }$$

The labelled plane

$$L_2=a-\frac12$$

is tangent to the doubled conic at

$$\boxed{(a,b)=\left(\frac12,0\right),}$$

while $L_1=2$ is a unit.

Joint Newton data

Set

$$p=u+1,\qquad q=v,\qquad A=a-\frac12,\qquad B=b.$$

The exact Symbolica calculation gives

$$\operatorname{in}K=\frac14(p+q-2A)^2, \qquad \operatorname{in}K_1=-(p+q-2A).$$

Writing $T=p+q-2A$, restriction to the doubled plane gives

$$\boxed{ K_3|_{T=0}=6q(p-B)(p+B). }$$

On that plane,

$$L_2=q, \qquad L_1=2.$$

Thus this center selects the existing labelled $L_2$ occurrence, in contrast with the $L_1$ occurrence selected at $(u,v)=(2/3,0)$.

Support-cube test

The factors

$$(\rho,q,V_-,V_+), \qquad V_-=p-B,\quad V_+=p+B,$$

have Jacobian determinant

$$\boxed{2\neq0}$$

in coordinates $(\rho,q,p,B)$. They are therefore étale local coordinates.

Their augmented signed Koszul cube has dimensions

$$(1,4,6,4,1),$$

differential ranks

$$(1,3,3,1),$$

and zero homology.

Narrow conclusion

The cyclic partner center independently closes:

$$\boxed{ \text{existing }L_2\text{ occurrence} \quad+\quad \text{normal-crossing coefficient branches} \quad+\quad \text{no excess Tor}. }$$

No new carrier datum is required. The source-label swap was not used to derive this local result; it is now legitimate to compare Entries 1110 and 1111 through that occurrence action.

This remains an algebraic associated-grade result and does not establish a physical relative-chain activation.

Verification

Newton checker:

research/benincasa/marici-gm/src/bin/rank12_u2over3v0_newton.rs, invoked with argument partner.

Support checker:

research/benincasa/marici-gm/src/bin/rank12_u2over3v0_support_cube.rs, invoked with argument partner.

Packet:

research/benincasa/rank12-um1-v0-partner-tangency.json.

Ledger claim: seqclaim-eb093c7f1c7a3f76c82bf5b3.

Epistemic event:

ev-000000000810-e7d01b37-863a-4887-91c3-2c084b897e2e.

Next finite falsifier

Construct the source-labelled $r\leftrightarrow s$ occurrence map between the two rational tangency packets. Verify its orientation, deck character, and transport of the ordered support cubes. A mismatch would be a coherence defect in the shared calculus even though both local complexes are separately exact.

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