A negative Clifford third-moment expectation at dimension eleven

H. Zhu, C. Mao and C. Yi, Third moments of qudit Clifford orbits and 3-designs based on magic orbits, arXiv:2410.13575v1 (Commun. Math. Phys. 407, 242 (2026)), Conjecture 2: for an odd prime $d$ and $|\Psi\rangle\in\mathcal H_d^{\otimes n}$, $0\le\kappa(\Psi,\mathcal T)\le1$ for every stochastic Lagrangian subspace $\mathcal T\in\Sigma(d)$, together with three aggregate inequalities that the authors derive from it. Here $\kappa(\Psi,\mathcal T)=\operatorname{tr}[R(\mathcal T)(|\Psi\rangle\langle\Psi|)^{\otimes3}]$, $R(\mathcal T)=r(\mathcal T)^{\otimes n}$ and $r(\mathcal T)=\sum_{(x;y)\in\mathcal T}|x\rangle\langle y|$. The verbatim statements are in the literature note.

Issue #14367 targets the pointwise clause for every odd prime $d$, every $n$, every normalized $\Psi$ and every $\mathcal T\in\Sigma(d)$.

The coefficients $\kappa(\Psi,\mathcal T)$ expand the third moment operator of a Clifford orbit in the commutant frame ${R(\mathcal T)}$; their nonnegativity would give the paper's conditional bounds on shadow norms and on the aggregate sums.

The authors prove $-1\le\kappa\le1$ (Lemma 20), the conjecture for $d=3$, $\kappa=1$ for stabilizer states and nonnegativity for their cubic-phase families. Issue #14367 records the literature check before any Lean: no later work exhibits a negative $\kappa$; not-found-in-searched-scope.

$d=11$, $n=1$, $\Psi=v/\sqrt{40}$ with $v=(-2,0,-2i,1-2i,-1-i,1+2i,-1-2i,-2,-2i,2-i,1-i)$, and $\mathcal T={(Oy;y)}$ with $O=\begin{pmatrix}7&8&8\8&7&8\8&8&7\end{pmatrix}$ over $\mathbb F_{11}$ ($O^{\mathsf T}O=I$, $O\mathbf 1=\mathbf 1$). Since $r(\mathcal T)$ maps $|y\rangle$ to $|Oy\rangle$, $\kappa(\Psi,\mathcal T)=40^{-3}\sum_y\prod_k v_{y_k}\overline{v_{(Oy)_k}}=-1196/64000=-299/16000$.

The refutation concerns the pointwise clause as printed (every $\mathcal T\in\Sigma(d)$); the aggregate inequalities are a different statement and are not refuted here.

The canonical source is D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.lean, with public claim and result : ¬ claim. The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry, native_decide or new axiom. The 1331-term Gaussian-integer sum is evaluated by kernel decide. The module statement is sha256:8532a0ef00a289544d8e4740ee71c4efe03920036ea636be7b7dca2fd58c007b, the result statement sha256:9600966d695b68a10866e2785508ddc3c245fc3c18eea03ea3890f4af1b6ec75 and the claim statement sha256:f77b58eed855b84858dd3983d641a85311fc5b754f5dca40843b4a8ca26b0e25. The Freeze event is sha256:6f1e7f1909bb5bd57dcbacf1b57dd272d84520dc0cf284c3de6f6235b4209edd; it has no project-level prerequisite (the module imports Mathlib only).

Tier 1 conjecture of an October 2024 paper (kept in the 2026 CMP version), preregistered in issue #14367 before any Lean. theorem; resolution refuted.

| declaration | proof_shape | escape_witness | admission_basis | | --- | --- | --- | --- | | result | bind-only | none | open-problem-resolution |

The private declarations (the subspace T, its membership criterion mem_T, stochastic_T, the state psi, norm_v, normalized_psi, gaussian_sum, kappa_T, kappa_psi) are each used on the proof of result; all are bind-only (instantiation, linear-algebra rewriting and kernel-evaluated arithmetic). The settlement is admitted as an external open-problem resolution.

Utility is kind=certified-instance; basis=refutes (the claim and its refutation). There is no digestion atom.

What the refutation shows

Proved by result: the pointwise clause of Conjecture 2 fails at $d=11$, already for one qudit: $\kappa(\Psi,\mathcal T)=-299/16000$.

Computed, not formalized (exact rational arithmetic over $\mathbb F_{11}$ and the Gaussian integers).

• Mechanism. For $\mathcal T=\mathcal T_O$ the operator $r(\mathcal T)$ is the permutation $|y\rangle\mapsto|Oy\rangle$ of $\mathcal H_d^{\otimes3}$, so $\kappa(\Psi,\mathcal T_O)=\langle\Psi^{\otimes3}|r(\mathcal T_O)|\Psi^{\otimes3}\rangle$. For $O\in S_3$ it permutes tensor factors and $\kappa=1$; for a non-permutation $O$ it mixes the three coordinates of the computational basis, and the overlap of $\Psi^{\otimes3}$ with its image can be negative once the amplitudes are unequal.
• All stochastic Lagrangian subspaces at $d=11$. The stochastic orthogonal group $O_3(11)$ has 24 elements, equal to $|\Sigma(11)|=2(11+1)$, so every $\mathcal T\in\Sigma(11)$ is a graph $\mathcal T_O$. For the state above the 24 values are $1$ (six permutations), $15763/64000$ (twelve) and $-299/16000$ (six); all are real.
• What survives. For this state $\kappa(\Psi,\Sigma(11))=28299/3200\approx8.84\in[6,24]$, $\kappa(\Psi,\mathscr T_{ns})=28299/3200-6\approx2.84\in[0,18]$ and $\kappa(\Psi,\mathscr T_{iso})=28299/3200\ge6$: the three aggregate inequalities of Conjecture 2 hold here, so the pointwise clause is not necessary for them.

Open. Whether the pointwise clause holds for one qudit at $d=5$ and $d=7$ is not determined here. The aggregate lower bounds fail for two qudits at $d=5$ (dossier), where the pointwise clause fails as well.

Effect on the paper. The derivation of the aggregate inequalities "thanks to Lemma 20" from the pointwise clause no longer applies. The aggregate upper bounds are proved by Lemma 20; the aggregate lower bounds fail for two qudits at $d=5$.

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.