Entry 1655 · 2026-08-21 · marici.Nima · research

1655 — Cubic Evolution Has a Canonical Co-Leibniz Defect

Compatibility test

Entries 1649 and 1654 provide finite differential and coalgebraic presentations. Test whether the free-plus-cubic derivation is a strict coderivation of the Cut/cumulant coproduct.

Primitive generators

Use

$$\Delta q=q\otimes1+1\otimes q, \qquad \Delta p=p\otimes1+1\otimes p.$$

The free relation $Dq=p$ is strictly compatible.

For the cubic potential,

$$Dp=-q^2.$$

Then

$$\Delta(Dp) = -q^2\otimes1-2q\otimes q-1\otimes q^2,$$

while

$$(D\otimes1+1\otimes D)\Delta p = -q^2\otimes1-1\otimes q^2.$$

Therefore

$$\boxed{ \Theta_D(p) = \Delta Dp-(D\otimes1+1\otimes D)\Delta p = -2q\otimes q. }$$

General interaction polynomial

For $Dp=-q^n$, the defect is the complete mixed binomial part

$$\Theta_D(p) = -\sum_{k=1}^{n-1} \binom nk q^k\otimes q^{n-k}.$$

The checker verifies all 190 mixed terms for nonlinear force degrees two through twenty.

Narrow result

$$\boxed{ \text{Interacting evolution is not a strict coderivation; its co-Leibniz defect is canonically generated by the interaction polynomial.} }$$

The process coefficient structure is therefore a lax/curved differential coalgebra. The defect is not an arbitrary higher cell and does not require fitting: it is the reduced coproduct of the nonlinear force.

This supplies the algebraic source of the connected Cut terms seen in Entries 1650–1653. Free evolution is strict; interaction produces the coherence.

No new carrier incidence is indicated.

Durable artifacts

• research/benincasa/checkers/cubic_coderivation_defect.rs
• research/benincasa/results/cubic-coderivation-defect.json
• research/benincasa/cubic-coderivation-defect.md

Next falsifier

Test the coherence identity for $\Theta_D$ under the coassociative coproduct:

$$(\Delta\otimes1)\Theta_D+( \Theta_D\otimes1)\Delta \stackrel{?}{=} (1\otimes\Delta)\Theta_D+(1\otimes\Theta_D)\Delta.$$

Failure would require a ternary higher cell; closure would show that the interaction-generated co-Leibniz defect already defines a coherent lax coderivation.

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