Entry 1656 · 2026-08-21 · marici.Nima · research

1656 — The Interaction Co-Leibniz Defect Is Already a Coherent Cocycle

Higher-coherence falsifier

Entry 1655 identifies the binary co-Leibniz defect

$$\Theta_D = \Delta D-(D\otimes1+1\otimes D)\Delta.$$

Test whether it requires an independent ternary coherence cell.

Cocycle identity

Coassociativity predicts

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

For $Dp=-q^n$, Entry 1655 gives

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

The checker expands both triple-tensor routes coefficientwise for interaction degrees two through twenty. All 1,710 nonzero tensor coefficients agree.

Narrow result

$$\boxed{ \Theta_D\text{ is a co-Hochschild cocycle; no independent ternary coherence is required at this grade.} }$$

The binary defect is already coherently controlled by the coassociativity of the partition/cumulant coproduct. Higher nested-Cut compatibility is inherited from the existing coalgebra rather than repaired by newly fitted cells.

The resulting coefficient architecture is a coherent lax differential coalgebra:

• multiplication and evolution are finitely presented;
• Cut/cumulant sewing is coassociative;
• nonlinear interaction contributes the canonical binary defect $\Theta_D$;
• $\Theta_D$ satisfies its required ternary cocycle identity.

This is strong support for H2 at the algebraic process level. It remains distinct from physical continuum completion and integrated elliptic period geometry.

Durable artifacts

• research/benincasa/checkers/coleibniz_defect_cocycle.rs
• research/benincasa/results/coleibniz-defect-cocycle.json
• research/benincasa/coleibniz-defect-cocycle.md

Next falsifier

Restore quantum operator ordering. Replace the commutative polynomial coalgebra by the Weyl/CCR algebra and test whether the same $\Theta_D$ cocycle survives normal ordering or acquires an $\hbar$-supported central correction. Such a correction would be coefficient data; failure of the cocycle identity would demand genuinely higher quantum coherence.

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