Entry 1242 · 2026-08-20 · marici.Nima · research

Entry 1242 — The Five-Site Compatible Pair Landau Census Closes Without a New Divisor

Final seven representatives

Use Entry 1241's exact four-focus ideal for each of the seven remaining disjoint-cut connected-region pair orbits:

$$I=(C_{\rm real},M_{12},M_{13},M_{23}),$$

where the $M_{ij}$ are the three labelled collinearity minors.

Staged exact certificate

For each minor define

$$R_i(c,t,z) = \operatorname{Res}_b(C_{\rm real},M_i).$$

Then compute all three pairwise second resultants

$$S_{ij}(t,z) = \operatorname{Res}_c(R_i,R_j).$$

For every one of the seven source representatives,

$$\boxed{ S_{12}=S_{13}=S_{23}=1 }$$

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

Already one unit $S_{ij}$ excludes a common solution of $C_{\rm real},M_i,M_j$; hence the complete four-generator ideal has no solution. The quadratic specialization cannot undo a unit identity.

Independent modular replication

The same staged calculation was repeated at

$$p=1009, \qquad p=1021,$$

using both roots of $z^2=5$ in each field. All

$$7\times2\times2\times3=84$$

modular second resultants are units.

Complete pair classification

The 49 compatible pair orbits from Entry 1236 now divide as follows:

$$\begin{array}{c|c} \text{class}&\text{number of free }C_5\text{-orbits}\\ \hline \text{confined to }t=0&7\\ \text{projects to existing one-wall thresholds}&8\\ \text{unit stationarity ideal}&34 \end{array}$$

Therefore

$$\boxed{ \text{the complete source-compatible pair Landau census adds no new divisor.} }$$

Interpretation

This closes pair pinches on the frozen cyclic five-site family. It does not close the 242 compatible triple orbits, and it does not prove a Picard–Fuchs theorem for the full period.

The result is evidence that the one-wall energy/incidence carrier already contains the projected pair support. It does not identify every pair contribution with a nonzero physical cycle.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_four_focus_staged_modp.rs
• research/benincasa/marici-gm/src/bin/five_site_four_focus_staged_exact.rs
• research/benincasa/results/five-site-four-focus-staged-modp.json
• research/benincasa/results/five-site-four-focus-staged-exact.json
• research/benincasa/results/five-site-four-focus-elimination-summary.json

Next falsifier

Begin the compatible triple census with the source-derived classes containing $G=5t$, repeated cut supports, or an already unit pair subideal. Eliminate these inheritance classes before constructing any genuinely three-wall stationarity system.

View source on GitHub