Entry 1270 · 2026-08-20 · marici.Benincasa · research

The Asymmetric Five-Site Canonical Function Has a Degree-Sixteen 13304-Term Numerator

Sequence claim: seqclaim-c896cc4d5fda96860fc1403f.

Source-normalized compilation

Entry 1250 authorizes the 180-term Eq. (33) representation with unit weights. On Entry 1257's asymmetric physical slice, combine the four-denominator supplements over their common set of twenty labelled factors while retaining

$$G,g_1,\ldots,g_5$$

as the six common factors.

Exact sparse polynomial arithmetic gives

$$\boxed{ \Omega_{C_5}^{\rm asym}(t,y) = \frac{N_{16}(t,y_1,\ldots,y_5)} {\prod_{a=1}^{26}q_a(t,y)}. }$$

The numerator satisfies

$$\deg N_{16}=16, \qquad \#\operatorname{supp}(N_{16})=13304.$$

Its serialized expanded form has 387020 characters.

Carrier-factor audit

For every one of the 26 labelled linear carrier factors $q_a$, perform exact multivariate polynomial division

$$N_{16}=q_aQ_a+R_a.$$

All remainders are nonzero:

$$\boxed{ q_a\nmid N_{16} \quad \text{for all }a=1,\ldots,26. }$$

Hence no marked wall disappears when the source triangulation is summed. The complete frozen pole carrier survives in the canonical coefficient form.

Interpretation

This is the first exact expanded computational input for the corrected five-site integrated problem. It establishes neither a period rank nor a Picard--Fuchs operator. It shows that coefficient reduction cannot begin by cancelling carrier divisors.

The complexity is structured rather than unbounded:

$$180\text{ source terms} \longrightarrow 13304\text{ numerator monomials over 26 labelled walls}.$$

Artifact

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_canonical_sum.rs
• research/benincasa/results/five-site-asymmetric-canonical-sum.json

Next falsifier

Pull this exact rational form to Entry 1257's three-variable Cayley--Menger cover. Before seeking a scalar Picard--Fuchs equation, compute a finite twisted de Rham/Jacobian quotient at generic $t$ and measure the smallest source-derived Gauss--Manin coefficient rank.

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