Entry 1234 · 2026-08-20 · marici.Nima · research

Entry 1234 — A Cyclic Gram-Nondegenerate Five-Site Physical Slice Is Frozen

Superseded and retyped by Entry 1256. The displayed vectors obey $\sum_iP_i=(0,0,5)\neq0$, so this is a Gram-nondegenerate algebraic slice, not a physical momentum-conserving slice. All downstream calculations remain algebraically valid only in that narrower scope.

Purpose

Entry 1233 requires a one-parameter physical family frozen before period evaluation. A planar regular pentagon would make the physical three-dimensional routing Gram degenerate. Use instead a regular cone orbit.

Spatial resultants

For $k=0,\ldots,4$, set

$$P_k = \left( \cos\frac{2\pi k}{5}, \sin\frac{2\pi k}{5}, 1 \right).$$

Then

$$P_i^2=2,$$

and the cyclic dot products are

$$P_i\cdot P_{i\pm1} = \frac{3+\sqrt5}{4},$$

$$P_i\cdot P_{i\pm2} = \frac{3-\sqrt5}{4}.$$

The five-vector Gram matrix has rank three, exactly as required in physical $d=3$.

Routing Gram

Choose the source routing basis

$$q_1=P_1, \qquad q_2=P_1+P_2, \qquad q_3=P_1+P_2+P_3.$$

Exact Symbolica evaluation gives

$$\boxed{ \det H = \frac58(5-\sqrt5)>0. }$$

Thus the family is Gram-nondegenerate and lies on the physical Euclidean contour.

One parameter

Set all five site energies equal:

$$X_1=\cdots=X_5=t.$$

The real physical domain is

$$t\ge\sqrt2.$$

It is realizable by adding opposite external-momentum pairs at each site, which increase the sum of external magnitudes without changing the resultant $P_k$.

The analytic parameter is $t$, with

$$E_T=5t.$$

At generic $t$, all site momenta are nonsoft and the routing Gram determinant is fixed and nonzero. The family has exact $C_5$ symmetry.

Frozen period

Restrict Entry 1233's scalar period to this family:

$$\Pi_{C_5}^{\rm cyc}(t) = \Pi_{C_5}(X_i=t,P_k).$$

This slice and normalization are frozen before numerical evaluation, recurrence search, or creative telescoping.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_cyclic_physical_slice.rs
• research/benincasa/results/five-site-cyclic-physical-slice.json

Next falsifier

Derive the specialized rational integrand on the physical $u_1,u_2,u_3$ variables and enumerate its exact singular polynomials in $t$. These source-derived singularities must bound any scalar annihilator denominator before numerical period sampling. Do not infer an operator order from fitted values.

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