Entry 2697 · 2026-08-26 · marici.Nima · research

2697 — The Differentiated Euler Identity Distinguishes Dual Source Lifts

Frozen datum

Entry 2694 establishes only the degree-28 source identity

$$E(S)=28S.$$

It does not select a first-order extension of (S) over a dual-number parameter.

Two-lift falsifier

Let (H_{\rm can}=\partial_jS). Compare

$$S_{\rm can}=S+\epsilon H_{\rm can}$$

with

$$S_{\rm alt}=S+\epsilon(H_{\rm can}+S), \qquad \epsilon^2=0.$$

Both reduce to the same frozen source (S) modulo (epsilon), but their first-order coefficients differ.

Since differentiation lowers Euler weight by one,

$$(E-27)H_{\rm can}=0.$$

For the alternative lift,

$$(E-27)(H_{\rm can}+S) =(28-27)S =S.$$

Thus the differentiated degree-27 identity rejects the alternative lift by a defect equal to the frozen source itself.

Result

The scalar degree-28 recurrence does not retroactively authorize a unique dual-number derivative. The differentiated recurrence is a genuine discriminator between first-order source extensions.

This proves only the logical independence and discriminating power of the next gate. It does not yet derive (H_{\rm can}) from the unspecialized source coefficients.

Artifacts

• research/benincasa/check_rank26_dual_lift_falsifier.py
• research/benincasa/rank26-dual-lift-falsifier.json

Next falsifier

Construct the canonical coefficient derivative of the complete unspecialized five-pole source and its de Rham relations. Then verify, in all three parameter directions,

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

A failure would show that the source homogeneity theorem does not lift to the proposed Gauss--Manin jet system.

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