Entry 1660 · 2026-08-21 · marici.Nima · research

1660 — The Complete First Quantum Cut Defect Needs No Ternary Cell

Decisive coherence test

Entry 1659 computes the complete eight-term binary defect $\Theta_D(p^3)$, including the central cross term $4\hbar_L\hbar_R$. Test its full three-occurrence co-Hochschild identity.

Exact three-factor Weyl calculation

Use occurrence-resolved generators

$$(q_i,p_i,\hbar_i), \qquad i=1,2,3,$$

with

$$[q_i,p_i]=i\hbar_i,$$

vanishing cross-occurrence commutators, and primitive coproducts.

The checker normal-orders both routes:

$$L = (\Delta\otimes1)\Theta_D +(\Theta_D\otimes1)\Delta,$$

$$R = (1\otimes\Delta)\Theta_D +(1\otimes\Theta_D)\Delta.$$

For $p^3$, each route contains 39 nonzero normal-ordered terms, of which three are pure-central occurrence terms.

The result is

$$\boxed{L=R}$$

coefficientwise, with no scalar residual.

Narrow result

$$\boxed{ \text{The complete first quantum co-Leibniz defect is already a coherent cocycle; no independent ternary quantum cell is required.} }$$

Primitive occurrence-labelled $\hbar_i$ absorbs both ordering and sewing corrections. The quantum coefficient filtration enriches the binary defect but preserves its inherited higher coherence.

This closes the first quantum-ordering falsifier for the cubic moment $p^3$. It does not prove all-arity closure for arbitrary interaction polynomials or renormalized field-theory products.

Architectural consequence

At this tested grade, the cosmological process object is a coherent lax differential coalgebra in the homogenized CCR category:

• interaction determines the binary defect;
• normal ordering contributes central occurrence terms;
• coassociativity determines the ternary identity;
• fixed-$\hbar$ specialization must occur only afterward.

No new carrier primitive is supported.

Durable artifacts

• research/benincasa/checkers/weyl_p3_cohochschild_cocycle.rs
• research/benincasa/results/weyl-p3-cohochschild-cocycle.json
• research/benincasa/weyl-p3-cohochschild-cocycle.md

Next falsifier

Generalize from the cubic potential to $q^d/d$ and compute the first normal-ordering grades and co-Hochschild closure uniformly in $d$. Determine whether the partition/binomial rule remains arity-uniform or whether some interaction degree introduces an independent quantum coherence.

View source on GitHub