Entry 1237 · 2026-08-20 · marici.Nima · research

Entry 1237 — Seven Five-Site Pair Orbits Are Confined to Total-Energy Support

Frozen input

Use the 49 compatible labelled pair orbits admitted by Entry 1236. On the cyclic physical slice, write the total-energy wall as

$$q_G=5t$$

and a connected-region wall as

$$q_A=|A|t+y_i+y_j,$$

where $i,j$ are the two labelled cut occurrences of the connected region $A$.

First algebraic gate

Two source-derived pair classes force $t=0$ before any Landau stationarity equation is imposed.

First, every pair containing $G$ obeys

$$q_G=0\quad\Longrightarrow\quad t=0.$$

There are five such free $C_5$-orbits.

Second, suppose two distinct connected regions have the same two cut occurrences. For the five-cycle they are complementary, so their cardinalities are unequal. Their two wall equations have the form

$$|A|t+y_i+y_j=0, \qquad (5-|A|)t+y_i+y_j=0.$$

Subtracting gives

$$(2|A|-5)t=0.$$

Since $2|A|-5\neq0$, these pairs also force $t=0$. There are two such free $C_5$-orbits.

Exact census consequence

Therefore

$$\boxed{ 7\text{ of }49\text{ compatible pair orbits} }$$

or equivalently

$$\boxed{35\text{ of }245\text{ labelled compatible pairs}}$$

cannot generate a nonzero-$t$ Landau factor.

The remaining search has

$$\boxed{42\text{ pair orbits}}$$

before stationarity and boundary-admissibility tests.

Narrow interpretation

This is a necessary-support elimination only. It does not assert that the seven classes possess a Landau solution at $t=0$, and it does not assert that any of the remaining 42 classes possesses one.

The result excludes only a new nonzero-$t$ factor from these seven source classes. Their possible support is confined to the already frozen total-energy locus and its marked intersections.

Artifact update

research/benincasa/results/five-site-compatible-landau-subsets.json now preserves, for every representative, its exact walls, cut supports, source multiplicity, and first-gate disposition. The checker no longer emits the entire packet to stdout.

Next falsifier

The frozen census contains no pair of distinct one-cut walls, so such a pair is not an admissible next test. Solve instead the source-present mixed pairs

$$q_e=5t+2y_e, \qquad q_A=|A|t+y_i+y_j,$$

separately for $e\in\{i,j\}$ and $e\notin\{i,j\}$. Retain signed roots and exact labelled routing data, and compare every elimination factor with the one-wall polynomial and existing incidence supports.

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