Entry 1632 · 2026-08-21 · marici.Nima · research

1632 — The Labelled Cubic Cut Is Completely Positive with a Forced Moment-Filtration Shift

Objective

Entry 1631 requires an infinite non-flat moment tower. Test whether the source cubic labelled Cut defines a positive map on that tower and determine its exact filtration variance.

Kraus form

After retaining and then tracing the internal occurrence label $r=(q,k)$, the production contribution has the form

$$\boxed{ \Phi_{\rm Cut}(\rho) = \sum_r C_r\rho C_r^\dagger. }$$

For every auxiliary system and every $X\succeq0$,

$$(\Phi_{\rm Cut}\otimes\operatorname{id})(X) = \sum_r(C_r\otimes I)X(C_r^\dagger\otimes I) \succeq0.$$

Thus the occurrence-resolved Cut map is completely positive before forgetting labels. Positive integration over the symmetric finite-EFT measure of Entry 1618 preserves this property.

Filtration variance

For cubic vacuum production, each $C_r$ contains one observed creation operator. The adjoint map is

$$\Phi_{\rm Cut}^\dagger(O) = \sum_r C_r^\dagger O C_r.$$

If $O$ has observed polynomial degree $D$, then $C_r^\dagger O C_r$ has degree $D+2$. Therefore the finite truncation is typed as

$$\boxed{ \Phi_D: \mathbb M_{\leq D+2} \longrightarrow \mathbb M_{\leq D}. }$$

It is not a same-level endomorphism of $\mathbb M_{\leq D}$.

Finite check

The checker realizes the observed creation incidence on a finite occupation basis and verifies positive quadratic forms for rank-one input states and probes. It also verifies the forced degree-two shift.

Narrow result

$$\boxed{ \text{The labelled cubic Cut is completely positive and filtration-preserving only as a degree-two shifted tower map.} }$$

This corrects the tempting but mistyped same-level formulation.

The map need not be trace-preserving by itself: it is the positive production contribution, not the complete normalized time-evolution channel. Source normalization and virtual terms belong to the full Dyson comparison.

Architectural consequence

The state coefficient tower and Cut calculus fit naturally as a filtered pro-object:

$$\cdots \to\mathbb M_{\leq D+2} \xrightarrow{\Phi_D} \mathbb M_{\leq D} \to\cdots.$$

Occurrence labels index Kraus legs; forgetting them performs the physical partial trace. This supplies a concrete quantum-information realization of the existing labelled Cut carrier without changing its incidence geometry.

Durable artifacts

• research/benincasa/checkers/cut_kraus_shifted_moment_tower.rs
• research/benincasa/results/cut-kraus-shifted-moment-tower.json
• research/benincasa/cut-kraus-shifted-moment-tower.md

Next falsifier

Add the source virtual/Hamiltonian Dyson term and normalization cell. Test the trace-preserving identity

$$\sum_r C_r^\dagger C_r -i(H_{\rm eff}-H_{\rm eff}^\dagger) =0$$

at the same shifted filtration degree. The production Cut alone is CP; the complete finite-time channel must satisfy the optical-theorem normalization without a fitted anti-Hermitian term.

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