Entry 1243 · 2026-08-20 · marici.Nima · research

Entry 1243 — Every Compatible Five-Site Triple Inherits a Closed Pair

Frozen census

Entry 1236 supplies

$$1210$$

compatible labelled triples, forming

$$242$$

free $C_5$-orbits.

Each triple contains three source-compatible labelled pairs. Classify those subpairs using the exact outcomes of Entries 1237–1242:

• unit: the pair stationarity ideal is the unit ideal;
• t_zero: the pair equations force $t=0$;
• old_threshold: the pair projects only to an existing one-wall threshold.

Exact inheritance census

The 242 triple orbits divide as

$$\begin{array}{c|c} \text{inherited pair content}&\text{triple orbits}\\ \hline \text{unit subpair only}&172\\ \text{unit and }t=0\text{ subpairs}&56\\ t=0\text{ subpair only}&14 \end{array}$$

There is no unresolved orbit.

Equivalently, 228 triple orbits contain a unit pair ideal. Their three-wall stationarity ideals are therefore units without further elimination. The remaining 14 orbits inherit total-energy support from a $t=0$-forcing pair.

Result

$$\boxed{ \text{no compatible triple requires a new three-wall stationarity calculation.} }$$

Since every orbit is free under $C_5$, the corresponding labelled counts are

$$860, \qquad 280, \qquad 70,$$

respectively, summing to 1210.

Thus the complete source-compatible pair-and-triple Landau census on the frozen cyclic family adds no divisor beyond total-energy and the existing one-wall supports.

Scope

This is an inheritance theorem for the pair/triple subsets occurring in the frozen 180-term OFPT packet. It does not classify active sets of four or more walls, nor does it prove that the scalar period's minimal Picard–Fuchs operator has no apparent singularities.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_triple_landau_inheritance.rs
• research/benincasa/results/five-site-triple-landau-inheritance.json

Next falsifier

No additional hypergraph census is required. Every compatible active set of size at least four contains a compatible triple, and every compatible triple contains a closed pair by this entry. Record the hereditary closure for all active-set sizes, then return to the scalar period problem: determine whether the resulting one-wall support bound controls a derived differential equation or only its genuine Landau divisor.

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