At dimension five the full and non-symmetric aggregates of a two-qudit state fall below their conjectured lower bounds

H. Zhu, C. Mao and C. Yi, Third moments of qudit Clifford orbits and 3-designs based on magic orbits, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and $|\Psi\rangle\in\mathcal H_d^{\otimes n}$ the pointwise bounds $0\le\kappa(\Psi,\mathcal T)\le1$ on $\Sigma(d)$ together with the aggregate bounds $6\le\kappa(\Psi,\Sigma(d))\le2d+2$, $0\le\kappa(\Psi,\mathscr T_{\rm ns})\le2d-4$ and $\kappa(\Psi,\mathscr T_{\rm iso})\ge6$. Here $\Sigma(d)$ is the set of stochastic Lagrangian subspaces of $\mathbb F_d^6$, $\mathscr T_{\rm sym}={\mathcal T_O\mid O\in S_3}$, $\mathscr T_{\rm ns}=\Sigma(d)\setminus\mathscr T_{\rm sym}$ and $\kappa(\Psi,\mathscr T)=\sum_{\mathcal T\in\mathscr T}\kappa(\Psi,\mathcal T)$. The verbatim statements are in the literature note. The pointwise clause and the clause $\kappa(\Psi,\mathscr T_{\rm iso})\ge6$ are refuted (pointwise, isotropic aggregate). Issue #14620 targets the two remaining lower bounds, $6\le\kappa(\Psi,\Sigma(d))$ and $0\le\kappa(\Psi,\mathscr T_{\rm ns})$.

The paper uses the aggregate sums over $\Sigma(d)$ for the shadow norms of stabilizer projectors and for the third frame potential of Clifford orbits, and derives the aggregate bounds from the pointwise clause. Whether the lower bounds over $\Sigma(d)$ and $\mathscr T_{\rm ns}$ hold decides whether the consequences stated with them survive the failure of the pointwise clause.

The paper proves $4(D+d)/(D+1)\le\kappa(\Psi,\Sigma(d))$ with $D=d^n$, which is at least $6$ only for one qudit, the upper bounds, and the conjecture for $d=3$. It states $|\Sigma(d)|=2d+2$ and, for $d\equiv2\pmod3$, $\Sigma(d)=\mathscr T_{\rm iso}$. Issues #14561 and #14620 record the literature checks before any Lean: no later work proves or refutes the aggregate lower bounds; not-found-in-searched-scope.

Let $\mathcal T\in\Sigma(5)$ and $(\mathbf y;\mathbf 0)\in\mathcal T$. The quadratic condition gives $\mathbf y\cdot\mathbf y=0$, and applied to $(\mathbf 1+\mathbf y;\mathbf 1)$ it gives $\sum_ky_k=0$. Then $\mathbf y\cdot\mathbf y=2(y_1^2+y_1y_2+y_2^2)$ with $-3$ a non-square mod $5$, so $\mathbf y=\mathbf 0$. The second projection is injective on the three-dimensional $\mathcal T$, so $\mathcal T=\mathcal T_O$, and the conditions make $O$ a stochastic isometry. Hence $\Sigma(5)=\mathscr T_{\rm iso}$. For the two-qudit state $\Psi=v/\sqrt{458}$ of #14561, $\kappa(\Psi,\Sigma(5))=140241723/24017978<6$; each of the six permutation graphs has $\kappa=1$, so $\kappa(\Psi,\mathscr T_{\rm ns})=-3866145/24017978<0$.

An error in the classification $\Sigma(5)=\mathscr T_{\rm iso}$, in the identification of $\mathscr T_{\rm sym}$ or in the frozen two-qudit values. Exhaustive enumeration gives $|\Sigma(5)|=12$ with every element a graph, as the paper's $|\Sigma(d)|=2d+2$ requires.

The canonical source is D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation.lean, with public claim (the disjunction of the two lower bounds) 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 module statement is sha256:5e69691622247a60b0dfb251155c7bb412574c1cc80d163011458f6d54241347, the result statement sha256:52e7406bf235904808230d3d5fff4d7ac140f9f5b981894bd6f29778a06569a9, the claim statement sha256:969abd3b9602def77f7e1d6cd9ab6c13cc157b41fed903c3260da2c5616ad9e4 and the sigmaSubspaces_five statement sha256:e3df810a9f40214791e9c8732254144d2673d6f3c4fedfadfccb5ac5b40c2618. The Freeze event is sha256:c61cb9bb0f116f6787b705cd355848bd4436e40572e8886a161ff79dd10358b2; its one project-level prerequisite is the frozen node sha256:0630d507568cef406b80d8acc31aaccdd8f6709e5874320aff894907aa780a56 of D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation, whose graph subspaces, $O_3(d)$, $\mathscr T_{\rm iso}$, state and values the module reuses; IsStochasticLagrangian and kappa come from D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.

Tier 1 printed clauses of an October 2024 conjecture, preregistered in issue #14620 before any Lean. theorem; resolution refuted.

| declaration | proof_shape | escape_witness | admission_basis | | --- | --- | --- | --- | | result | content | sigmaSubspaces_five | open-problem-resolution |

Content declarations, each on the live proof of result: the public classification sigmaSubspaces_five ($\Sigma(5)=\mathscr T_{\rm iso}$), the private kernel fact that no nonzero $\mathbf y\in\mathbb F_5^3$ has $\mathbf y\cdot\mathbf y=0=\sum_ky_k$, the two lemmas that turn it into the graph description, and the values kappa_sigma_five and kappa_ns_five. The bind-only private lemma graph_stochastic (every $\mathcal T_O$ with $O\in O_3(d)$ satisfies the three conditions) is used by the classification and by kappa_ns_five. 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: both lower bounds $6\le\kappa(\Psi,\Sigma(d))$ and $0\le\kappa(\Psi,\mathscr T_{\rm ns})$ of Conjecture 2 fail for a normalized two-qudit state at $d=5$. Together with the earlier refutations, every lower bound of Conjecture 2 fails; the upper bounds are the paper's theorems.

Argued, not formalized.

• Every $n\ge2$. Graph expectations are multiplicative under tensor products and equal $1$ for a stabilizer state, so $\Psi\otimes|0\rangle^{\otimes(n-2)}$ has the same values.
• Primes $d\equiv2\pmod3$. The argument for $\Sigma(5)=\mathscr T_{\rm iso}$ uses only that $-3$ is a non-square mod $d$, which holds exactly for $d\equiv2\pmod3$; it agrees with the paper's statement for those primes.

Open. Whether the lower bounds over $\Sigma(d)$ hold for two qudits at $d\equiv1\pmod3$, where $\Sigma(d)$ contains four defect subspaces besides $\mathscr T_{\rm iso}$; the exact minimum of $\kappa(\Psi,\Sigma(d))$ for $n\ge2$.

The main text of the published Communications in Mathematical Physics version is subscription-gated and was not read; that it prints Conjecture 2 and the statement $\Sigma(d)=\mathscr T_{\rm iso}$ for $d\equiv2\pmod3$ unchanged is not verified firsthand. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent refutation.