Entry 1662 · 2026-08-21 · marici.Nima · research

1662 — Cut Coherence Cannot Select the Quantum Ordering of a Derivative Interaction

Hostile test

Entry 1661 proves arity-uniform coherence for position-polynomial interactions, whose primitive force has no ordering ambiguity. Test a derivative interaction involving both $q$ and $p$.

Classically equivalent quantum family

The Weyl quantization of the classical symbol $q^2p$ is

$$H_W = \frac13(q^2p+qpq+pq^2) = qpq.$$

The Hermitian family

$$H_\lambda = qpq+\lambda\hbar q, \qquad \lambda\in\mathbb R,$$

has the same classical symbol.

Its primitive equations are

$$Dq=q^2,$$

$$\boxed{ Dp=-(pq+qp)-\lambda\hbar. }$$

Thus $\lambda$ changes the primitive quantum force while disappearing in the classical limit.

Cut-coherence blindness

In the homogenized CCR coalgebra, $\hbar$ is primitive. Therefore

$$\Delta(-\lambda\hbar) -\left(-\lambda\hbar_L-\lambda\hbar_R\right) =0.$$

The ordering shift contributes nothing to the co-Leibniz defect. The checker verifies this for 257 integer values of $\lambda$; all 256 nonzero values change the primitive force, while all 257 leave the Cut defect unchanged.

Narrow result

$$\boxed{ \text{Cut sewing and higher coherence cannot determine the quantum ordering of a derivative interaction.} }$$

Once the ordering is frozen, the homogenized Weyl calculus remains coherent. But the choice itself must come from independent source authority:

• the primary Hamiltonian;
• a regulator;
• a symmetry or self-adjointness prescription;
• renormalization conditions.

This is a sharp boundary of the shared-carrier program. Carrier geometry and Cut calculus constrain transport of coefficient data; they do not manufacture missing sector dynamics.

Durable artifacts

• research/benincasa/checkers/derivative_interaction_ordering.rs
• research/benincasa/results/derivative-interaction-ordering.json
• research/benincasa/derivative-interaction-ordering.md

Next falsifier

Freeze Weyl ordering and test whether the resulting derivative interaction preserves the positivity/process-tensor architecture of Entries 1641–1646. In particular, determine whether its momentum-dependent Cut loading remains completely positive after internal pushforward or requires additional source-derived contact terms.

View source on GitHub