Entry 1240 · 2026-08-20 · marici.Nima · research

Entry 1240 — Shared-Cut Connected Five-Site Pairs Have Unit Stationarity Resultant

Frozen class

After Entries 1237–1239, consider the 21 compatible connected-region pair orbits whose two cut supports share exactly one labelled occurrence.

Write their walls as

$$q_A=mt+y_0+y_1, \qquad q_B=nt+y_0+y_2.$$

On the common wall intersection, retain the shared signed root $b$:

$$y_0=b, \qquad y_1=-mt-b, \qquad y_2=-nt-b.$$

Typed stationarity system

The pair-gradient equation is

$$\alpha(n_0+n_1)+\beta(n_0+n_2)=0.$$

Thus the two vectors

$$u=n_0+n_1, \qquad v=n_0+n_2$$

must be collinear.

The exact elimination retains two conditions:

1. Cayley–Menger coplanarity of the loop point and the three labelled routing foci;
2. the cleared Gram determinant

$$(u\cdot v)^2-u^2v^2=0.$$

All focus distances are source-labelled distances on Entry 1234's conical five-cycle. No coarse profile is used in place of them.

Exact result

For every one of the 21 representatives, Symbolica gives

$$\boxed{ \operatorname{Res}_b(C_{\rm pl},C_{\parallel})=1 }$$

over $\mathbb Q(z,t)$, before imposing $z^2=5$.

Therefore none of these shared-cut connected-region pairs has a stationary Landau solution on the frozen cyclic family.

Pair-census update

Entries 1237–1240 now classify

$$\boxed{42\text{ of }49\text{ compatible pair orbits}.}$$

The only unresolved pair class consists of seven connected-region pairs with disjoint cut supports.

No new carrier or coefficient divisor has appeared.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_connected_pair_landau_shared_cut.rs
• research/benincasa/results/five-site-connected-pair-landau-shared-cut.json

Next falsifier

Compute the exact four-focus stationarity elimination for the seven disjoint-cut connected-region pair representatives. Preserve the four labelled routing foci and signed roots. A projected factor is admissible only if it survives the full vector stationarity system and is not generated by the existing one-wall, Gram, or incidence supports.

View source on GitHub