bibkey: galindorowell2026unitaryyb authors: C. Galindo; E. C. Rowell year: 2026 title: “Unitary Yang–Baxter Operators: Towards a Classification” doi: null url: https://arxiv.org/abs/2608.16865 claim: “Conjecture 10.6 states that for d≥2 with 4∤d, a primitive d-th root w, a unit α, and a coefficient vector with a 0 = 1 and all coefficient norms equal to 1, whose cyclic Pauli operator is Yang–Baxter and projectively unitary, there is one sign ε∈{+1,−1} with a_{s+t}=a_sa_tw^{εαst}.” strata_touched:

• D5/S3/Quantum/Algebra/CyclicYangBaxterPauliRigidityRefutation license: citation-only triage: anchor

Galindo–Rowell, Unitary Yang–Baxter Operators

C. Galindo and E. C. Rowell, Unitary Yang–Baxter Operators: Towards a Classification, arXiv:2608.16865v1 (2026), §2 and §10.2.7.

Source statements

The §10.2 setting states: “Fix (d\geq2), choose a primitive (d)-th root” (w); (Xe_j=e_{j+1}), (Ze_j=w^je_j); and “(R(a)=\sum_{t=0}^{d-1}a_tP^t, \qquad P=X\otimes Z)” (before Galois normalization, (\sum_{t=0}^{d-1}a_t(X^t\otimes Z^{\alpha t}))). The compact coefficient phase torus is “(\mathbb T_d^{\mathrm{coef}} = {(a_1,\ldots,a_{d-1})\in(\C^\times)^{d-1}: |a_t|=1\text{ for }1\leq t<d}).”

Section 2 displays the braid Yang–Baxter equation: “((R\otimes I)(I\otimes R)(R\otimes I) = (I\otimes R)(R\otimes I)(I\otimes R)).”

Conjecture 10.6 (§10.2.7) states: “Let (d\geq2) with (4\nmid d), let (\alpha\in(\Z/d\Z)^\times), and put (P_\alpha=X\otimes Z^\alpha, \qquad \tau_\alpha(s,t)=w^{\alpha st}\qquad(s,t\in\Z/d\Z)). Suppose that (a=(1,a_1,\ldots,a_{d-1})\in\mathbb T_d^{\mathrm{coef}}) and that (R_\alpha(a)=\sum_{t\in\Z/d\Z}a_tP_\alpha^t) satisfies the Yang–Baxter equation and is projectively unitary. Then (a) belongs to one of the two signed Gaussian torsors for (\tau_\alpha): there exists (\epsilon\in{+1,-1}) such that (a_{s+t}=a_sa_t\tau_\alpha(s,t)^\epsilon\qquad(s,t\in\Z/d\Z)).”

The Lean encoding uses ZMod d for the cyclic indices, the shift and clock matrices on that index type, the Kronecker product for (P_\alpha), and the displayed braid equation on the reassociated triple tensor product. Projective unitarity means that a nonzero scalar multiple belongs to the matrix unitary group. The refuting instance is (d=15), (\alpha=1), and (a_t=w^{2t^2}) for a primitive fifteenth root; its polarization has exponent 4, whereas the two conjectured signs require exponents 1 and 14.

Verification boundary

The paper records exact computations only for dimensions (2,3,5,6,7,9,10,11). The arXiv record has v1 only. Searches recorded with the preregistration found no later settlement of Conjecture 10.6; citing works not independently checked remain ASSUMED-UNVERIFIED.

Verified locator

• URL: https://arxiv.org/abs/2608.16865