Entry 1235 · 2026-08-20 · marici.Nima · research

Entry 1235 — The Cyclic Five-Site One-Wall Landau Polynomial Is Source Derived

Frozen slice

Use Entry 1234's cyclic physical family with $X_i=t$ and conical resultants $P_i$.

For a connected proper arc $A$ of size $m$, the physical marked denominator is

$$q_A=mt+y_i+y_j,$$

where $i,j$ are its two boundary edges.

One-wall Landau geometry

The critical configurations of the two-focus function $y_i+y_j$ are collinear. After complex continuation their critical values are

$$\pm|P_A|.$$

Therefore the signed one-wall threshold equation is

$$m^2t^2-|P_A|^2=0.$$

This derives the threshold from the frozen denominator and routing geometry; it is not inferred from period samples.

Exact cyclic norms

For the conical five-cycle orbit:

$$\begin{array}{c|c|c} m&|P_A|^2&\text{threshold polynomial}\\ \hline 1&2&t^2-2\\ 2&\dfrac{11+\sqrt5}{2}&8t^2-11-\sqrt5\\ 3&\dfrac{21+\sqrt5}{2}&18t^2-21-\sqrt5\\ 4&17&16t^2-17. \end{array}$$

Each row has five cyclic occurrences. The total-energy and one-cut endpoint contributes $t=0$.

Thus over $\mathbb Q(\sqrt5)$ the one-wall singular polynomial is, up to a unit,

$$\boxed{ t(t^2-2)(16t^2-17) (8t^2-11-\sqrt5) (18t^2-21-\sqrt5). }$$

Its rational field norm is

$$\boxed{ t(t^2-2)(16t^2-17) (16t^4-44t^2+29) (81t^4-189t^2+109). }$$

Physical sheet

The physical endpoint thresholds occur at

$$t=-\frac{|P_A|}{m}.$$

The positive roots belong to signed analytic continuation. Both are retained in the algebraic singular polynomial.

Classification

Every factor is generated by an existing total-energy, edge-soft, or connected partial-energy endpoint. No new carrier datum appears.

Scope

This result is complete for individual marked walls. It is not yet the complete Landau variety of the 180-term period: simultaneous multi-wall pinches may contribute additional algebraic factors.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_cyclic_one_wall_landau.rs
• research/benincasa/results/five-site-cyclic-one-wall-landau.json

Next falsifier

Enumerate compatible pairs and triples of marked denominators from the frozen OFPT packet. Solve their Landau stationarity equations on the cyclic slice and eliminate the loop variables. Compare every resulting factor with Entry 1235's polynomial and the already frozen marked-incidence supports. Do not begin Picard–Fuchs fitting until this multi-wall singularity census is bounded.

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