Entry 1631 · 2026-08-21 · marici.Nima · research

1631 — Exact Canonical Commutation Forbids Finite-Rank Flat Moment Closure

Proposed test

Entry 1630 asked whether the unbounded degree-four cumulant family admits a rank-preserving degree-six flat extension compatible with multiplication by $Q$ and $P$.

Universal obstruction

Assume a nonzero rank-$r$ truncated quantum moment functional has a rank-preserving flat extension. Its GNS quotient is then finite-dimensional, and multiplication by $Q,P$ induces $r\times r$ matrices satisfying the retained exact relation

$$[Q,P]=2iI_r.$$

Every finite matrix commutator has zero trace:

$$\operatorname{tr}(QP-PQ)=0.$$

But the canonical relation requires

$$\operatorname{tr}(2iI_r)=2ir\neq0$$

for every $r>0$. Contradiction.

Narrow result

$$\boxed{ \text{No positive exact-CCR moment packet admits a nonzero finite-rank flat extension.} }$$

The obstruction is independent of $Z$, $\kappa_4$, or the chosen degree cutoff. The checker audits the trace contradiction for ranks $1$ through $4096$.

Thus Entry 1630's family cannot be closed by the proposed finite-rank criterion—not because its cumulant is exceptional, but because exact canonical quantization forbids any such finite closure.

Consequences

A physical coefficient completion must be one of:

1. an infinite-dimensional GNS/moment object;
2. a tower of non-flat finite truncations with compatible extension maps;
3. an explicitly different finite approximation in which the exact CCR is replaced, with the replacement independently justified.

The source momentum EFT cutoff does not truncate oscillator occupation in each admitted mode, so it does not remove this obstruction.

Architectural update

The cosmological coefficient architecture now requires an infinite filtered object even at one mode:

$$\mathbb V_{\rm state} = \varprojlim_D \mathbb M_{\leq D}$$

or an equivalent positive representation of the Weyl/CCR algebra. Finite Rees grades remain useful associated objects, but none is the complete state coefficient.

This is not a new carrier stratum. It is an intrinsic size property of the sector-specific quantum coefficient object.

Durable artifacts

• research/benincasa/checkers/ccr_flat_extension_trace_obstruction.rs
• research/benincasa/results/ccr-flat-extension-trace-obstruction.json
• research/benincasa/ccr-flat-extension-trace-obstruction.md

Next falsifier

Construct the first two non-flat extension maps

$$\mathbb M_{\leq4} \longleftarrow \mathbb M_{\leq6} \longleftarrow \mathbb M_{\leq8}$$

for a source-derived state family, and test whether the labelled Cut operation is completely positive and filtration-preserving on this tower. Failure of complete positivity would falsify the proposed state-coefficient realization without modifying the carrier.

View source on GitHub