Entry 1261 · 2026-08-20 · marici.Benincasa · research

All 105 Asymmetric Shared-Cut Proper Pairs Have Unit Landau Resultant

Sequence claim: seqclaim-f56fde48e742f3fa7f86e35f.

Frozen source-compatible pair census

The 180 source OFPT terms admit 245 distinct labelled wall pairs:

$$\begin{array}{c|r} \text{first-gate class}&\text{count}\\ \hline G\text{-containing}&25\\ \text{same-cut proper}&10\\ \text{one-cut-total containing}&70\\ \text{disjoint-cut proper}&35\\ \text{shared-cut proper}&105 \end{array}$$

No cyclic quotient is used.

The 25 $G$-containing pairs lie on $t=0$. For each of the ten same-cut proper pairs, the complementary walls have equations

$$mt+s=0, \qquad (5-m)t+s=0,$$

so they also force $t=0$.

Exact shared-cut elimination

For every shared-cut proper pair, retain the shared cut root $b$ and substitute

$$y_{\rm shared}=b, \qquad y_{\rm left}=-mt-b, \qquad y_{\rm right}=-nt-b.$$

All three squared focus distances are recomputed from Entry 1257's conserved integer resultants. Eliminating $b$ between the exact coplanarity and stationary-collinearity equations gives

$$\boxed{\operatorname{Res}_b(C,L)=1}$$

for every one of the 105 labelled pairs.

Therefore none of these shared-cut systems has a stationary solution over the complexified asymmetric family.

Narrow conclusion

Of the 245 source-compatible pairs, 140 are now classified:

$$25+10\text{ force existing }t=0\text{ support}, \qquad 105\text{ have empty stationary locus}.$$

The unresolved pair census is exactly

$$70\text{ one-cut-total pairs} \quad+\quad 35\text{ disjoint-cut proper pairs}.$$

No anomalous two-wall support and no new carrier datum have appeared in the classified portion.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_compatible_pairs.rs
• research/benincasa/results/five-site-asymmetric-compatible-pairs.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_shared_cut_pairs.rs
• research/benincasa/results/five-site-asymmetric-shared-cut-pairs.json

Next falsifier

Construct and eliminate the 35 labelled disjoint-cut four-focus systems using the same asymmetric routing. Keep the 70 one-cut-total systems separate because their incidence and endpoint variance differ.

View source on GitHub