Entry 1238 · 2026-08-20 · marici.Nima · research

Entry 1238 — Shared-Cut Mixed Five-Site Pairs Add No Landau Factor

Source-present class

Among Entry 1236's compatible pairs, consider a one-cut wall and a connected-region wall that share the cut occurrence $i$:

$$q_e=5t+2y_i, \qquad q_A=mt+y_i+y_j, \qquad m=|A|.$$

The frozen census contains eight free $C_5$-orbits of this type, or 40 labelled pairs.

Wall equations

On their common zero locus,

$$y_i=-\frac52t, \qquad y_j=\frac{5-2m}{2}t.$$

Let $n_i$ and $n_j$ denote the signed unit gradients of the two roots. Pair stationarity has the form

$$(2\alpha+\beta)n_i+\beta n_j=0.$$

For a genuine two-wall pinch, this forces

$$n_j=\sigma n_i, \qquad \sigma\in\{+1,-1\}.$$

Routing geometry

The difference between the two labelled routing foci is the region resultant $P_A$. Hence

$$P_A^2=(y_i-\sigma y_j)^2.$$

For $\sigma=-1$, this gives

$$P_A^2=m^2t^2.$$

For $\sigma=+1$, it gives

$$P_A^2=(5-m)^2t^2.$$

These are exactly the one-wall threshold equations for $A$ and its connected complement $A^c$, respectively.

Result

Therefore

$$\boxed{ \text{the eight shared-cut }M1+A_m\text{ pair orbits add no new nonzero-}t\text{ factor.} }$$

Together with Entry 1237, 15 of the 49 compatible pair orbits are now classified against new nonzero-$t$ support. This does not remove the shared-cut pairs from the Landau complex; it identifies their projected support with already frozen one-wall divisors.

Scope

The result does not apply to the six disjoint-cut $M1+A_m$ orbits. Their stationarity involves three distinct routing foci and may produce a genuine anomalous threshold.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_mixed_pair_landau_overlap.rs
• research/benincasa/results/five-site-mixed-pair-landau-overlap.json

Next falsifier

Derive the three-focus elimination for the six disjoint-cut $M1+A_m$ representatives. Preserve the exact labelled routing Gram data and signed roots. Test whether each elimination factor belongs to the existing one-wall/Gram carrier or defines new coefficient support.

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