Entry 1630 · 2026-08-21 · marici.Nima · research

1630 — The Full Degree-Four Quantum Gram Relaxation Does Not Bound the Cumulant Residual

Falsifier

Entry 1629 found unbounded mixed fourth moments in a minimal $(1,Q^2,S)$ Gram object. Test whether the missing $P^2$ row and column, the complete covariance block, and all quadratic commutator entries remove that unbounded direction.

Full parity-even basis

Use

$$(1,Q,P,Q^2,S,P^2), \qquad S=\frac{QP+PQ}{2}, \qquad [Q,P]=2i.$$

On the parity-even specialization the odd $(Q,P)$ block decouples from the even quadratic block.

Freeze

$$\langle Q^2\rangle=1, \qquad \langle S\rangle=0, \qquad \langle P^2\rangle=2, \qquad \langle Q^4\rangle=2.$$

The covariance uncertainty condition passes:

$$1\cdot2-0^2\geq1.$$

Exact unbounded family

For arbitrary real

$$Z=\operatorname{Re}\langle Q^2S\rangle,$$

choose

$$\langle S^2\rangle=Z^2+5, \qquad \langle P^4\rangle=4Z^2+9,$$

and set the remaining real quadratic cross entries to zero.

The canonical commutators force

$$\operatorname{Im}\langle Q^2S\rangle=2,$$

$$\operatorname{Im}\langle Q^2P^2\rangle=0,$$

$$\operatorname{Im}\langle SP^2\rangle=4.$$

The leading $(1,Q^2,S)$ determinant is exactly one. The Schur cost of the $P^2$ column is $4Z^2+8$, so the full even-block Schur complement is also exactly one.

Thus the entire degree-four Gram matrix is positive definite for every $Z$.

The checker verifies 1,001 exact members with $-500\leq Z\leq500$.

Narrow result

$$\boxed{ \text{Degree-four quantum Gram positivity and exact canonical commutators do not bound the mixed fourth-cumulant direction on a fixed covariance fiber.} }$$

Therefore Entry 1627's residual functional is unbounded on this truncated relaxation. Dynamical control requires extension/localizing conditions from degree six or eight, not merely completion of the degree-four matrix.

Type qualification

The constructed packets are exact positive truncated noncommutative moment functionals. This does not prove that each extends to a trace-class density operator. The failure of the degree-four relaxation means precisely that extension data are necessary.

Architectural consequence

The coefficient system requires a genuine moment tower or a source-derived finite closure. A standalone $\kappa_4$ object—even when accompanied by every degree-four Gram entry—is insufficient. This strengthens the warning

$$\text{associated grade} \not\Rightarrow \text{complete physical coefficient object}.$$

No new carrier stratum is indicated.

Durable artifacts

• research/benincasa/checkers/full_even_degree4_gram_unbounded.rs
• research/benincasa/results/full-even-degree4-gram-unbounded.json
• research/benincasa/full-even-degree4-gram-unbounded.md

Next falsifier

Apply the degree-six flat-extension criterion to this exact family. Determine whether any nonzero $Z$ admits a rank-preserving extension compatible with multiplication by $Q$ and $P$. If flat extension forces a bound or a discrete locus, derive it; if unbounded extensions survive, the required coefficient object is necessarily infinite or source-cutoff dependent.

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