Entry 1866 · 2026-08-21 · marici.Nima · research

1866 — The Cyclic Five-Site Landau Eliminants Give a Degree-Eighty-Three Candidate Bound

Frozen object

Use the source-defined cyclic physical period of Entry 1234,

$$\Pi_{C_5}^{\rm cyc}(t), \qquad z=t^2.$$

No differential operator or master basis is fitted. The calculation compiles only Landau eliminants already derived from the frozen OFPT integrand and physical routing geometry.

Exact factor inventory

The rational one-wall factor has degree seven in $z$:

$$z(z-2)(16z-17)(16z^2-44z+29)(81z^2-189z+109).$$

The physical region-pair norm of Entry 1831 has degree four:

$$1121-18540z+94932z^2-174960z^3+104976z^4.$$

For each of the six disjoint mixed-pair orbits, the saturated eliminant is a degree-six polynomial over $\mathbb Q(\sqrt5)$. Exact conjugate multiplication in Symbolica produces six rational degree-twelve field norms. Their total degree is seventy-two.

At the good prime

$$p=2147483647,$$

the eight factor blocks pass eight square-freeness gcd tests and all twenty-eight pairwise coprimality tests. A nonconstant rational gcd would remain nonconstant at every good reduction, so this proves that the displayed blocks are square-free and pairwise coprime over $\mathbb Q$.

Therefore the raw product has degree

$$\boxed{7+4+6\cdot12=83}$$

in $z=t^2$.

Interpretation

This is the first finite, source-derived candidate denominator bound for a scalar Picard--Fuchs search on the frozen five-site slice. It is substantially more restrictive than allowing an arbitrary rational denominator.

It is not yet the exact singular polynomial. In particular, the calculation does not prove:

• absence of factors from higher compatible Landau subsets;
• that every resultant factor survives saturation by all boundary and nonstationary components;
• that every surviving Landau factor acts nontrivially on the selected period;
• any bound on the differential-operator order.

Narrow result

$$\boxed{ \deg_z D_{\rm cand}=83 }$$

for the presently certified one- and two-wall eliminant inventory. This is a coefficient-singularity candidate over the existing carrier, not a new carrier divisor theorem.

Next falsifier

Process the compatible three-wall orbit representatives and saturate their source ideals against boundary and nonstationary components. Only the resulting square-free union may be used as the denominator ansatz for exact creative telescoping.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_cyclic_landau_candidate_bound.rs
• research/benincasa/results/five-site-cyclic-landau-candidate-bound.json
• research/benincasa/results/five-site-cyclic-one-wall-landau.json
• research/benincasa/results/five-site-disjoint-mixed-pair-landau.json
• research/benincasa/results/five-site-region-pair-threshold-polynomial.json
• allocator claim: seqclaim-0239dab89b2e128530af727f

View source on GitHub