Entry 1634 · 2026-08-21 · marici.Nima · research

1634 — Virtual-Cut Balance Cancels the Production Moment-Filtration Shift

Apparent mismatch

Entry 1632 shows that the positive production Cut alone acts with degree shift two:

$$\mathbb M_{\le D+2}\to\mathbb M_{\le D}.$$

Entry 1633 adds the source virtual/no-production cell. Test whether their sum still requires a shifted tower or descends strictly at each moment degree.

Trace-balanced generator

For an observed creation jump $L=a^\dagger$, the adjoint generator is

$$\mathcal L^\dagger(O) = a O a^\dagger -\frac12\{aa^\dagger,O\}.$$

The two displayed terms separately have principal degree $D+2$ on a degree-$D$ observable. Their principal symbols agree and cancel.

For the number-moment subalgebra,

$$\boxed{ \mathcal L^\dagger f(N) =(N+1)\bigl[f(N+1)-f(N)\bigr]. }$$

If $\deg f=D$, then

$$\deg\mathcal L^\dagger f\le D.$$

For $D>0$, the leading term is

$$D N^D.$$

The checker verifies every degree through $32$.

Narrow result

$$\boxed{ \text{Production and virtual cells are individually shifted, but their source-balanced sum is a strict same-level moment map on number moments.} }$$

Consequently the restriction squares commute:

$$\operatorname{res}_{\le D} \circ\mathcal L_{D+2} = \mathcal L_D \circ\operatorname{res}_{\le D}.$$

No derived coherence cell is needed on this tested subalgebra.

Qualifications

The principal-symbol cancellation holds generally for a linear Kraus operator, but the exact lower-degree ordering terms have only been audited here on the number-moment basis. The full Weyl basis $Q^mP^n$ remains the next finite test.

Architectural consequence

This is a concrete example of why the complete comparison object matters:

$$\text{positive Cut cell alone} \neq \text{normalized physical channel}.$$

The full source pair restores both trace balance and strict filtration behavior. The carrier already contains the two typed cells; the cancellation is coefficient-level coherence.

Durable artifacts

• research/benincasa/checkers/trace_balanced_moment_tower.rs
• research/benincasa/results/trace-balanced-moment-tower.json
• research/benincasa/trace-balanced-moment-tower.md

Next falsifier

Compute $\mathcal L^\dagger$ on every Weyl monomial $Q^mP^n$ through total degree eight. Verify exact degree preservation and commutation with the CCR quotient. Any surviving degree-raising term is the first obstruction to a strict full phase-space moment tower.

View source on GitHub