Entry 1262 · 2026-08-20 · marici.Benincasa · research

All 35 Asymmetric Disjoint-Cut Proper Pairs Fail the Four-Focus Landau Test

Sequence claim: seqclaim-c2fbfa1e853ab370e35f9af9.

Frozen systems

Entry 1261 leaves 35 source-compatible proper-wall pairs whose two labelled cut supports are disjoint. Each pair therefore has four distinct focus occurrences and two independent cut roots $b,c$.

For every labelled pair, reconstruct from Entry 1257:

• the four ordered focus vectors;
• the full $3\times3$ routing Gram matrix;
• its nonzero determinant;
• the realization equation for the common integration point;
• the three minors imposing collinearity of the two wall gradients.

No cyclic representative or inherited distance is used.

Exact staged elimination

First eliminate $b$ between the realization equation and each of the three collinearity minors. Then eliminate $c$ between pairs of the resulting necessary conditions.

For every one of the 35 systems, at least one second-stage resultant is

$$\boxed{1}.$$

A common Landau solution would have to annihilate every staged necessary condition. The unit certificate therefore excludes such a solution.

Consequence for all proper--proper pairs

The 150 source-compatible proper--proper pairs now split as

$$10\text{ complementary same-cut pairs forcing }t=0,$$

$$105\text{ shared-cut pairs with unit direct resultant},$$

$$35\text{ disjoint-cut pairs with unit staged resultant}.$$

Thus no proper--proper pair produces anomalous two-wall support on the asymmetric physical family.

Across the full 245-pair census, 175 pairs are now classified. The only remaining class is the 70 pairs containing one one-cut total wall.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_disjoint_cut_pairs.rs
• research/benincasa/results/five-site-asymmetric-disjoint-cut-pairs.json

Next falsifier

Derive the correctly typed endpoint/stationarity equations for the 70 one-cut-total pairs and eliminate them without treating the one-cut total wall as an ordinary two-focus proper-region wall.

View source on GitHub