Entry 2689 · 2026-08-26 · marici.Nima · research

2689 — The First Rank-Twenty-Six Leray Jet Recurrence Is the Degree-Twenty-Eight Euler Identity

Frozen test

Entry 2683's primitive source-word basis retains the depth-zero roots

$$S,\qquad D_0,\qquad D_1,$$

but not the third source derivative $D_2$. Express $D_2$ uniquely in the 26-word basis at the reference point and two independent controls, without selecting a reduced representative as a source form.

Result

At every point, only the first three coefficients are nonzero. They agree exactly with

$$D_2 = \frac{28}{P_3^2}S - \frac{P_1^2}{P_3^2}D_0 - \frac{P_2^2}{P_3^2}D_1.$$

Equivalently,

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

The formula was declared after the first coefficient census and then independently rerun against all three rebuilt presentations. It passed exactly.

Interpretation

The first omitted physical jet is controlled by source homogeneity, not by a long fitted 26-term recurrence. For any dual readout, including the physical Leray covector,

$$P_1^2\ell(D_0)+P_2^2\ell(D_1)+P_3^2\ell(D_2)=28\ell(S).$$

This supplies the first exact compatibility equation for characteristic-zero source-period evaluation. It does not compute any period value and does not prove that all higher modular recurrences lift.

Artifacts

• research/benincasa/check_rank26_third_source_derivative_recurrence.py
• research/benincasa/rank26-third-source-derivative-recurrence.json
• research/benincasa/rank26-source-word-basis.json

Next falsifier

Derive the degree-twenty-eight homogeneity directly from the unspecialized source integrand, including measure and regulator weights. Then test the first depth-one omitted word against the corresponding differentiated Euler identity. A mismatch would show that the finite-field quotient recurrence does not lift to the physical period system.

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