Entry 1326 · 2026-08-20 · marici.Nima · research

1326 — Neither Low-Degree Second Order nor Single-Wall Logarithmic IBP Closes the Five-Site Period

Hard comparison

After Entry 1320 excluded first-order polynomial primitives through degree five, two predeclared extensions were tested on the complete 32-sheet Kummer cover.

Polynomial second order

$$\partial_z^2\Omega +a_1(z)\partial_z\Omega +a_0(z)\Omega = \sum_{i=1}^3\partial_{u_i}(V_i\Omega), \qquad \deg V_i\le d.$$

Every wall is linear in $z$, while the five Kummer roots are independent of $z$. Hence $\partial_z^2\Omega$ was evaluated exactly term by term.

Across

$$p=1009,1013, \qquad z=7,11,$$

the full-deck systems are inconsistent for every

$$d=0,1,2,3,4,5.$$

At degree five each test has 170 unknowns and 6208 equations from 194 independent base points.

Frozen single-wall logarithmic first order

For each of the 26 labelled walls $L$ already present in the projective alphabet, test

$$\partial_z\Omega+a(z)\Omega = \sum_i\partial_{u_i}\!\left(\frac{P_i}{L}\Omega\right), \qquad \deg P_i\le2.$$

No new denominator is fitted. Every labelled wall is tested separately.

At both independent fibers

$$(p,z)=(1009,7),\qquad(1013,11),$$

all 26 systems are inconsistent.

Surviving statement

$$\boxed{ \begin{aligned} &\text{second-order polynomial IBP fails through degree five},\\ &\text{first-order one-wall logarithmic IBP fails through numerator degree two}. \end{aligned} }$$

This excludes two simple explanations for the five-site period:

• that only the scalar differential order was underestimated;
• that one omitted logarithmic wall denominator supplies the certificate.

It does not exclude multidivisor logarithmic primitives, higher degrees, or a vector-valued Gauss--Manin system.

Updated frontier

The lowest-complexity surviving candidate is no longer a scalar certificate built from one primitive family. The next finite object should be the Kummer-character-resolved de Rham module itself:

$$\mathcal H_{\rm dR} = \frac{\langle\Omega,\partial_z\Omega,\ldots\rangle} {d_u(\text{frozen logarithmic forms})}.$$

Compute its rank growth under successive $z$-derivatives before choosing a scalar order. A stable rank predicts the minimal vector Gauss--Manin system; failure to stabilize gives a finite obstruction to the present basis.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_second_order_ibp_pilot.rs
• research/benincasa/results/five-site-asymmetric-second-order-ibp-pilot.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_logarithmic_ibp_pilot.rs
• research/benincasa/results/five-site-asymmetric-logarithmic-ibp-pilot.json

Allocator claim: seqclaim-f8bf1f0a8a5787544b9a806b.

View source on GitHub