Entry 1658 · 2026-08-21 · marici.Nima · research

1658 — Cubic Weyl Evolution First Adds a Central Grade at the Third Momentum Moment

Question

Entry 1657 supplies the correctly typed homogenized CCR algebra. Determine whether normal ordering changes the primitive interaction coherence or first appears only in higher moment coefficients.

Primitive relations

With

$$[q,p]=i\hbar, \qquad H_{\rm int}=\frac{q^3}{3},$$

the primitive evolution rules remain

$$Dq=p, \qquad Dp=-q^2.$$

Hence Entry 1655's primitive co-Leibniz defect receives no ordering correction.

Normal-ordered hierarchy

The exact commutator formula is

$$[p^m,q^3] = \sum_{k=1}^{\min(m,3)} (-i\hbar)^k k! \binom mk\binom3k q^{3-k}p^{m-k}.$$

Therefore

$$D(p^m) = -m q^2p^{m-1} +2i\hbar\binom m2qp^{m-2} +2\hbar^2\binom m3p^{m-3}.$$

The first central term occurs at $m=3$:

$$\boxed{ D(p^3) = -3q^2p^2+6i\hbar qp+2\hbar^2. }$$

The checker verifies the formula through $m=32$; all thirty degrees $m\ge3$ carry a nonzero $\hbar^2$ contribution.

Narrow result

$$\boxed{ \text{Quantum ordering leaves the primitive cubic coherence unchanged but enriches the higher-moment filtration beginning at }p^3. }$$

The central $2\hbar^2$ term is genuine quantum coefficient data. It does not define a new carrier incidence and does not invalidate the commutative associated grade. Instead it shows precisely where the quantum filtered module departs from its classical symbol.

Durable artifacts

• research/benincasa/checkers/weyl_cubic_normal_ordering.rs
• research/benincasa/results/weyl-cubic-normal-ordering.json
• research/benincasa/weyl-cubic-normal-ordering.md

Next falsifier

Compute the co-Leibniz defect on $p^3$ in the homogenized CCR coalgebra and test the ternary cocycle including the $2\hbar^2$ term. Determine whether primitive-$\hbar$ coproduct absorbs the central grade coherently or leaves a supported quantum residual.

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