Entry 1244 · 2026-08-20 · marici.Nima · research

Entry 1244 — All Compatible Five-Site Multi-Wall Active Sets Inherit Pair Closure

Hereditary observation

Let $S$ be any labelled active-wall set occurring in one of the 180 frozen OFPT terms, with

$$|S|\ge3.$$

Every three-element subset $T\subseteq S$ is itself source-compatible, because all of its walls co-occur in the same source term.

Entry 1243 proves that every such $T$ contains either:

• a pair whose stationarity ideal is the unit ideal; or
• a pair forcing $t=0$.

That pair is also a subset of $S$.

Consequence

If the inherited pair ideal is a unit, the complete stationarity ideal for $S$ is a unit. If the inherited pair forces $t=0$, the projected support of $S$ is confined to the existing total-energy divisor.

Therefore

$$\boxed{ \text{every compatible active set of size at least three inherits pair closure.} }$$

Together with Entry 1242's complete pair census and Entry 1235's one-wall census, this closes every possible active-set size in a ten-denominator source term:

$$1\le |S|\le10.$$

Full Landau-support statement

On the frozen cyclic five-site family,

$$\boxed{ \text{the complete source-compatible Landau search adds no projected divisor beyond the one-wall polynomial.} }$$

Thus the genuine Landau support is bounded by

$$t(t^2-2)(16t^2-17) (16t^4-44t^2+29) (81t^4-189t^2+109)=0.$$

Scope

This is a theorem about source-compatible finite Landau active sets on Entry 1234's one-parameter slice. It does not imply that every displayed one-wall factor occurs in the minimal differential operator, and it does not exclude apparent singularities introduced by a nonminimal Picard–Fuchs presentation.

It also does not establish the corresponding statement on generic five-site kinematics.

Next falsifier

Construct the scalar cyclic period or its creative-telescoping module with the Entry 1235 polynomial predeclared as the genuine-support bound. Distinguish intrinsic operator singularities from removable/apparent factors, and reject any additional nonapparent divisor unless its provenance escapes the frozen Landau analysis.

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