A qutrit SIC fiducial that is not a Clifford-stabilizer state

M. Erew and M. Goldstein, Extremizing Measures of Magic on Pure States by Clifford-stabilizer States, arXiv:2512.19657v2, prove that states uniquely stabilized by a subgroup of the finite eigenphase-extended Clifford group extremize several magic measures, and state:

Conjecture 1 (Stabilizer nature of SIC fiducials). Every SIC-POVM fiducial state is a Clifford-stabilizer state.

The verbatim definitions are in the literature note. Issue #14173 reads it for every prime $d$ and every normalized single-qudit fiducial.

The conjecture would explain the extremality of SIC fiducials for $L^p$-norm magic and stabilizer Rényi entropies as a consequence of a stabilizer symmetry, placing them among the Clifford-stabilizer states covered by the paper's main theorem.

Issue #14173 records the literature check before any Lean: two arXiv versions; the only citing work found (arXiv:2607.07197) does not discuss the conjecture; arXiv, web and Zenodo searches found no settlement. not-found-in-searched-scope.

The eigenphase-extended Clifford group $\mathcal C'_{1,3}$ is finite, so there are only finitely many Clifford-stabilizer states modulo phase. The qutrit fiducials $\psi_z=(0,1,-z)/\sqrt2$, $|z|=1$, form a continuum of pairwise non-proportional states. This is the classical continuous family of qutrit SIC fiducials (Tabia and Appleby 2013 state it for $z=e^{2it}$, $t\in[0,\pi/6]$); the module proves the SIC condition for every unit $z$ directly. Hence some fiducial is not a Clifford-stabilizer state.

The refutation is for the paper's definitions (single qudit, the eigenphase-extended Clifford group, uniqueness of the stabilized line); a weaker reading, for example "an eigenstate of some Clifford unitary", is not addressed by result.

The canonical source is D5/S3/Quantum/Magic/ErewGoldsteinSICStabilizerRefutation.lean, with public zeta, pauliGroup, cliffordGroup, specialClifford, Lambda, eigenphaseClifford, invariantSubspace, IsNormalized, IsCliffordStabilizerState, IsSICFiducial, qutritFiducial, claim and result : ¬ claim. It reuses the frozen Weyl words WeylDisplacement.displacement and the frozen commutant theorem WindowRegister.window_commutant_eq_scalars (the clock and shift commute only with scalars). 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 sha256:97d15233dd8ee9b1ac6ddc959dc2a90dc9177583ce8197b03aa278178cd50c78, the result statement sha256:c45352b51f54bba6cd4e2556959b8debb0b0fc32d95f87ee5295dedd4acb7624 and the claim statement sha256:fc36430abb9941ca4716ffe380ddde4ac06a4d1f7a5f5db9cdfb41e74403d232. The Freeze event is sha256:5dea10ff83cd1d3917616e488df413626188ddb09e2c0f65ffd46b8933d33fc2; its project-level prerequisite is the frozen WeylDisplacement module.

Tier 1 conjecture of a December 2025 paper, preregistered in issue #14173 before any Lean. theorem; resolution refuted.

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

The other theorems of the module (finiteness of the special and eigenphase-extended Clifford groups and of the stabilizer projectors at $d=3$, the fiducial property and injectivity of the qutrit family) lie on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」). 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 universal claim fails at $d=3$: there is a normalized qutrit SIC fiducial that spans no one-dimensional invariant subspace of any subset of the eigenphase-extended Clifford group $\mathcal C'_{1,3}$.

Established inside the proof, for general data. $\tilde{\mathcal C}{1,3}$, $\Lambda$ and $\mathcal C'{1,3}$ are finite; hence the set of rank-one projectors of Clifford-stabilizer states is finite. Every $\psi_z=(0,1,-z)/\sqrt2$ with $|z|=1$ is a SIC fiducial, and $z\mapsto\psi_z\psi_z^\dagger$ is injective.

Argued and checked numerically, not formalized.

• Mechanism. For three sampled values of $z$, a numerical enumeration of the 216-element projective Clifford group finds exactly three elements fixing the ray of $\psi_z$, and after rescaling to fix $\psi_z$ their common invariant space is two-dimensional; for those samples the fiducial is an eigenvector of Clifford elements but not the unique invariant line. That the same holds for every generic $z$ is ASSUMED-UNVERIFIED. Independently of the samples, the fiducials that are Clifford-stabilizer states form a finite set of rays inside the continuous family (the finiteness proved inside result).
• What survives. The argument says nothing about dimensions in which the fiducials are isolated (finitely many up to Clifford equivalence); there the conjecture may still hold, and the weaker statement "every SIC fiducial is an eigenvector of some Clifford unitary" is not addressed.

Open. Whether every SIC fiducial in a prime dimension $d\ne3$ is a Clifford-stabilizer state, and which members of the qutrit family are, are not settled here.

Effect on the paper. Conjecture 1 is false as stated; the explanation it proposes for the extremality of SIC fiducials does not cover the continuous qutrit family.

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