Entry 1629 · 2026-08-21 · marici.Nima · research

1629 — The Mixed Fourth Cumulant Requires a Canonical Quantum Gram Companion

Objective

Construct the smallest quantum phase-space positivity object that contains the two fourth moments entering Entry 1627's residual and retains the canonical commutator before taking an associated grade.

Operator basis

Scale the canonical pair by

$$[Q,P]=2i$$

and define

$$S=\frac{QP+PQ}{2}.$$

Use the operator column

$$\mathcal O=(1,Q^2,S)^T.$$

For every positive density matrix, its Gram object

$$M=\langle\mathcal O^\dagger\mathcal O\rangle$$

is positive semidefinite.

Exact commutator entry

The canonical relation gives

$$[Q^2,S]=4iQ^2.$$

Writing

$$Q_2=\langle Q^2\rangle, \quad R=\langle S\rangle, \quad U=\langle Q^4\rangle, \quad Z=\operatorname{Re}\langle Q^2S\rangle, \quad W=\langle S^2\rangle,$$

the Gram matrix is

$$\boxed{ M= \begin{pmatrix} 1&Q_2&R\\ Q_2&U&Z+2iQ_2\\ R&Z-2iQ_2&W \end{pmatrix} \succeq0. }$$

The imaginary entry is forced; omitting it would forget the quantum normal direction.

Determinant inequality

The Schur complement gives

$$\boxed{ \operatorname{Var}(Q^2)\operatorname{Var}(S) \geq \operatorname{Cov}(Q^2,S)^2+4Q_2^2. }$$

The real mixed entry $Z$ contains the Weyl-ordered $pqqq$ fourth moment needed by Entry 1627. It is controlled only together with $W=\langle S^2\rangle$.

Finite unboundedness test

At the fixed lower data

$$Q_2=1, \qquad R=0, \qquad U=2,$$

every real $Z$ lies on the determinant boundary after choosing

$$W=Z^2+4.$$

The checker verifies 5,670 positive-semidefinite integer packets and 201 such unbounded mixed-moment witnesses.

Narrow result

$$\boxed{ \text{The mixed fourth cumulant is not dynamically controlled by covariance and }Q^4\text{ alone; the canonical companion }\langle S^2\rangle\text{ is mandatory.} }$$

Positive semidefiniteness of this truncated Gram object is necessary, not sufficient, for extension to a full quantum state.

Architectural consequence

The coefficient object is intrinsically multi-grade and noncommutative:

• the fourth-cumulant coordinate;
• its companion fourth moment;
• the commutator-fixed imaginary extension;
• the lower covariance block.

Projecting to $\kappa_{pqqq}$ alone destroys the positivity constraint. This is coefficient complexity over the existing labelled carrier, not evidence for a new carrier cell.

Durable artifacts

• research/benincasa/checkers/quantum_fourth_moment_gram.rs
• research/benincasa/results/quantum-fourth-moment-gram.json
• research/benincasa/quantum-fourth-moment-gram.md

Next falsifier

Test extension of this $3\times3$ Gram packet to the full degree-four Weyl moment matrix with basis

$$(1,Q,P,Q^2,S,P^2).$$

Compute whether the Entry 1627 residual functional remains unbounded on fixed covariance fibers after all degree-four commutator and positivity constraints are imposed. If it does, degree-six/eight moments are required; if bounded, derive the sharp bound without fitting.

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