Entry 1635 · 2026-08-21 · marici.Nima · research

1635 — The Trace-Balanced Cut Generator Strictly Descends on the Full Weyl Moment Tower

Remaining gate

Entry 1634 proves strict degree preservation on number moments. Test every Weyl monomial through degree eight and verify that ordering/CCR reduction introduces no degree-raising obstruction.

Exact Weyl generator

Use

$$[Q,P]=2i, \qquad L=a^\dagger=\frac{Q-iP}{2}.$$

For the trace-balanced adjoint channel

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

the exact Weyl-symbol operator is

$$\boxed{ \mathcal L^\dagger = \frac12(Q\partial_Q+P\partial_P) +\frac12(\partial_Q^2+\partial_P^2). }$$

Therefore

$$2\mathcal L^\dagger(Q^mP^n) =(m+n)Q^mP^n +m(m-1)Q^{m-2}P^n +n(n-1)Q^mP^{n-2}.$$

The first term preserves total degree; the diffusion terms lower it by two. No positive degree shift survives.

The unit maps to zero:

$$\mathcal L^\dagger(1)=0,$$

which is the trace-preserving condition.

CCR compatibility

Weyl symbols are the canonical ordering quotient of the CCR algebra. The displayed differential operator is defined directly on that quotient, so two operator representatives related by $[Q,P]=2i$ have the same image. No ordering-dependent coherence cell is required.

The checker verifies all 45 monomials through total degree eight and 56 degree-lowering terms.

Narrow result

$$\boxed{ \text{The complete trace-balanced cubic Cut generator is a strict filtration-preserving endomorphism of the full Weyl moment tower.} }$$

The degree-two shift belongs to the positive production cell in isolation. Source virtual-Cut coherence cancels it globally.

Qualifications

The local generator uses one observed creation jump. Momentum-dependent amplitudes and sums over occurrence labels preserve the degree statement when they commute with the observed canonical variables. Mode-mixing Kraus tensors require the corresponding multi-mode Weyl audit.

Architectural consequence

The cosmological state coefficient object can be taken as an infinite filtered positive Weyl-moment tower carrying:

• Hamiltonian symplectic derivations;
• completely positive labelled production cells;
• virtual normalization cells;
• a strict trace-balanced channel generator;
• cumulant grades constrained by higher moment positivity.

This is a concrete sector-specific coefficient architecture over the unchanged occurrence/Cut carrier.

Durable artifacts

• research/benincasa/checkers/weyl_gain_moment_generator.rs
• research/benincasa/results/weyl-gain-moment-generator.json
• research/benincasa/weyl-gain-moment-generator.md

Next falsifier

Perform the multi-mode occurrence-resolved Weyl audit for the actual cubic momentum kernel $C_{p;qk}$. Verify complete positivity, strict total-degree preservation after virtual balance, and naturality under $(q,k)\leftrightarrow(k,q)$ before continuum trace. A cross-mode ordering defect would be the first failure of the one-mode coefficient architecture.

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