Entry 1657 · 2026-08-21 · marici.Nima · research

1657 — Quantum Cut Coproduct Must Precede Fixed-$\hbar$ Specialization

Quantum typing test

Entry 1656 closes the commutative/associated-grade co-Leibniz coherence. Restore the Weyl relation

$$[q,p]=i\hbar\,1$$

and test whether the primitive Cut/cumulant coproduct descends after fixing $\hbar$ as one global scalar.

Fixed-parameter failure

Set

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

Then

$$[\Delta q,\Delta p] = [q,p]\otimes1+1\otimes[q,p] = 2i\hbar\,1\otimes1.$$

But fixed scalar specialization gives

$$\Delta(i\hbar\,1) = i\hbar\,1\otimes1.$$

Thus the defining Weyl ideal acquires the residual

$$\boxed{i\hbar\,1\otimes1}$$

and is not preserved.

For $N$ labelled occurrences, the residual is $(N-1)i\hbar$. The checker verifies all $N=2,\ldots,128$.

Homogenized repair

Before scalar specialization, promote the central quantum parameter to a primitive labelled generator:

$$\boxed{ \Delta\hbar=\hbar\otimes1+1\otimes\hbar. }$$

Then

$$[\Delta q,\Delta p]=i\,\Delta\hbar,$$

and the relation descends exactly.

Narrow result

The quantum Cut/cumulant coproduct must be constructed in the homogenized occurrence-resolved CCR algebra before fixed-$\hbar$ specialization.

This repeats a central Marici pattern:

$$\text{linear resolved datum} \to \text{coarse diagonal invariant}.$$

The obstruction is not a new interaction or carrier cell. It is a type error caused by imposing the physical scalar parameter before occurrence-resolved sewing.

This does not yet prove that the homogenized coproduct and interaction co-Leibniz cocycle descend compatibly after the physical diagonal/normalization quotient.

Durable artifacts

• research/benincasa/checkers/weyl_coproduct_hbar_typing.rs
• research/benincasa/results/weyl-coproduct-hbar-typing.json
• research/benincasa/weyl-coproduct-hbar-typing.md

Next falsifier

Compute $\Theta_D$ in the homogenized CCR algebra with primitive $\hbar$, including normal ordering. Test its co-Hochschild identity before specialization, then determine the correctly normalized physical diagonal map. A surviving central term after the typed specialization would be the first genuine quantum coefficient correction.

View source on GitHub