slug: galindo-rowell-2026-pauli-rigidity-mixed-sign-refutation bibkey: galindorowell2026unitaryyb doi: null url: https://arxiv.org/abs/2608.16865 triage: theorem motivation_gids:

• D5/S3/Quantum/Algebra/CyclicYangBaxterPauliRigidityRefutation.result

A mixed-sign Gaussian refutes Galindo–Rowell Conjecture 10.6

Problem

C. Galindo and E. C. Rowell, Unitary Yang–Baxter Operators: Towards a Classification, arXiv:2608.16865v1, Conjecture 10.6 (§10.2.7), state:

Let $d\ge 2$ with $4\nmid d$, let $\alpha \in (\mathbb{Z}/d\mathbb{Z}{)}^{\times }$, and put $P_{\alpha }=X\otimes Z^{\alpha }$, ${\tau }_{\alpha }(s,t)=w^{\alpha st}$. If $a=(1,a_1,\dots ,a_{d-1})\in {\mathbb{T}}_d^{\mathrm{coef}}$, $R_{\alpha }(a)$ satisfies the Yang–Baxter equation, and $R_{\alpha }(a)$ is projectively unitary, then there is one sign $ϵ\in \{+1,-1\}$ with $a_{s+t}=a_s{a}_t{\tau }_{\alpha }(s,t{)}^{ϵ}$ for every $s,t\in \mathbb{Z}/d\mathbb{Z}$.

The Lean claim is this quantified statement with ZMod d, the displayed braid equation, and Mathlib projective unitarity. The theorem result proves its negation.

Motivation

The source records exact computations for dimensions $2,3,5,6,7,9,10,11$. Dimension $15$ admits a mixed-sign quadratic phase whose polarization exponent is $4$ modulo $15$, while the two conjectured signs have exponents $1$ and $14$. This supplies a concrete counterexample outside the listed computations.

Gap

Issue #11573 preregistered the named conjecture, its full quantifiers, the $d=15$ route, and a bounded literature screen before formalization. The source has arXiv version 1 only. Searches in the preregistration found no proof or refutation of Conjecture 10.6 in the searched scope; citing works not independently checked remain outside the settlement.

Route

Take $d=15$, $\alpha =1$, a primitive fifteenth root $w$, and $$a_t=w^{2t^2}.$$ The proof establishes the coefficient-to-operator bridge for the braid equation, translates the finite character sum to prove the coefficient identity, and uses character orthogonality to obtain $R{R}^{†}=15I$. Thus $(\sqrt{15}{)}^{-1}R$ is unitary. At $s=t=1$, the claimed signs would require $w^8=w^5$ or $w^8=w^3$, which contradicts primitivity.

Falsifier

A primitive fifteenth root satisfying one of the two displayed equalities would break the final contradiction. A failure of the coefficient identity, the braid equation, or $R{R}^{†}=15I$ would also invalidate the witness. The kernel-checked theorem establishes all of these obligations for the stated definitions.

Evidence

• Lean module: D5/S3/Quantum/Algebra/CyclicYangBaxterPauliRigidityRefutation.lean.
• Public declarations are the definitions required by the settlement (clockZ, P, R, BraidYBE, ProjectivelyUnitary), the definition claim, and the theorem result : ¬ claim.
• make lean and the direct module build complete with exit 0. The theorem’s #print axioms closure is [propext, Classical.choice, Quot.sound], the repository’s standard three axioms.
• The finite character-sum translation, the orthogonality calculation, and the sign contradiction are checked by the proof term in result; no numerical approximation is used by the theorem.

Triage

What the settlement shows

The Pauli-direction rigidity assertion is false for the stated generality: Yang–Baxter and projective unitarity do not force the coefficient polarization into the two signed Gaussian torsors. The failure mechanism is a mixed-sign Gaussian at $d=15$, where the polarization exponent $4$ is a nontrivial square root of $1$ modulo $15$. The source’s definitions of the shift, clock, cyclic operator, braid equation, and projective unitarity remain valid. The proposed restrictions excluding this $d=15$ mechanism and any revised torsor classifications are open; this delivery settles only the registered universal claim.

theorem: this dossier settles Conjecture 10.6 only in the quantified form encoded by claim; it does not settle the paper’s other conjectures or its equivalent classification sentence beyond the witness supplied here.

ASSUMED-UNVERIFIED

The literature statement is bounded by the searches recorded in issue #11573. Citing lists and inaccessible search surfaces were not independently verified, so exhaustive literature coverage and publication priority are ASSUMED-UNVERIFIED.

Registration is paused under CLAUDE.md §3.9 (信息逃逸登记暂缓).