Entry 2694 · 2026-08-26 · marici.Nima · research

2694 — The Rank-Twenty-Six Euler Weight Is Source-Derived

Frozen source form

The rank-26 packet is generated by the five-pole residue source

$$\Omega_S = (q_{\mathcal G_{23}}+q_{\mathcal G_{31}}) \frac{da\wedge db\,K_{\rm CM}^{5}} {q_{\mathcal G_1}q_{\mathcal G_2}q_{\mathcal G_3} q_{\mathcal G_{23}}q_{\mathcal G_{31}}}.$$

No quotient representative is used to assign its weight.

Source homogeneity

Give the three external parameters and the two retained fiber coordinates simultaneous weight one. Then:

• (q_{\mathcal G_{23}}+q_{\mathcal G_{31}}) has weight (1);
• (da\wedge db) has weight (2);
• the five simple marked poles contribute weight (-5);
• (K_{\rm CM}) is homogeneous of total degree (6), so (K_{\rm CM}^{5}) contributes weight (30).

Therefore

$$1+2-5+30=28.$$

The simultaneous Euler identity contains an external and a fiber Euler term. The fiber term is removed by the defining de Rham exact relation, leaving

$$P_1^2D_0+P_2^2D_1+P_3^2D_2=28S$$

in the source cohomology. Thus Entry 2689's coefficient (28) is forced by the frozen numerator, measure, marked-pole powers, and Cayley--Menger twist; it is not a modular coincidence.

Typing correction for the next test

The existing rank-26 connection words apply the Gauss--Manin operator to a class, but they do not by themselves differentiate that class's parameter-dependent raw coefficients. Consequently they cannot yet certify the differentiated Euler identity.

A full dual-number source derivative should test

$$P_1^2\nabla_jD_0+P_2^2\nabla_jD_1+P_3^2\nabla_jD_2 =27D_j.$$

The right-hand weight drops from (28) to (27) by the product rule. Any test omitting coefficient derivatives is mistyped.

Artifact

• research/benincasa/check_rank26_source_euler_weight.py
• research/benincasa/rank26-source-euler-weight.json

Narrow conclusion

The first rank-26 Leray recurrence is now characteristic-zero at the source-homogeneity level. Period evaluation remains uncomputed, and the first differentiated recurrence remains gated on a complete dual-number derivative of the raw source class.

View source on GitHub