An exceptional four-dimensional unitary Hecke Yang–Baxter operator: a five-word Pauli–Clifford normal form

Abstract

A five-word real Pauli–Clifford formula gives an exact \(16\times16\) unitary Yang–Baxter operator in the exceptional class \([e^{i\pi/3},1/2,4]\). Scalar partial traces yield the \(\eta=1/2\) Markov trace and a faithful minimal localization of the \(\mathcal C(\mathfrak{sl}_3,6)\) Jones–Wenzl tower. Exact all-strand comparison with the independent concurrent Galindo–Rowell Family III construction transports its known finite-image and Clifford properties, while a direct scalar enhancement and skein calculation identify the trace invariant with \(2P_{\mathrm H}(L;i,i)\).

1. Result

Rowell and Wang introduced the ordinary-localization framework and conjecture. Galindo, Hong, and Rowell then studied this representation sequence in their 2012 case study. They ruled out ordinary local dimension two and supplied a two-dimensional unitary quasi-localization and a \((3,1)\)-generalized localization. In March 2026, Lechner's two-eigenvalue classification left \([e^{i\pi/3},1/2,2m]\) as its one unresolved existence family, identified it with this unknown GHR localization, and made base dimension four the first open case.

The construction proves the positive prediction of the Rowell–Wang localization conjecture, restated as GHR Conjecture 1.5, in this case: the simple tensor generator has \(\operatorname{FPdim}(X)^2=4\) and admits an ordinary unitary localization. This does not conflict with GHR Theorem 5.27, which rules out the dimension-two localization that a fiber functor would force; the present local dimension is four.

Note added. The complete explicit solution was publicly released in version 1.1.0 on 28 July 2026 at 04:10:58 UTC. C. Galindo and E. C. Rowell independently posted Unitary Yang–Baxter Operators: Towards a Classification, arXiv:2608.16865v1, on 17 August 2026 at 17:47:15 UTC and report earlier private work and circulation. Their Section 13 obtains the same existence and strict-localization conclusion, with dimension four smallest in Lechner's exceptional family. The works are treated as independent and concurrent; no claim is made about private discovery priority.

2. The exact construction

On \(V\otimes V\cong(\mathbb C^2)^{\otimes4}\), use the real Pauli–Clifford basis \(I,X,Z,J\), where \(J=XZ=-iY\), with a four-letter word denoting a tensor product. Set

The first four Pauli words combine into one reflection \[ M=\frac{-ZIZZ-ZIJJ-JIZJ+JIJZ}{2}, \] while \(E=XIXX\) is an anticommuting reflection. Thus \(H(\alpha,\beta)=\alpha M+\beta E\) is an involution on the circle \(\alpha^2+\beta^2=1\). The complete Pauli expansion shows that the braid relation restricts the circle to the four points \(\beta^2=1/3\), \(\alpha^2=2/3\). The displayed operator chooses \((\alpha,\beta)=(\sqrt{2/3},-1/\sqrt3)\). The sparse support came from the disclosed AI-assisted numerical search; the printed derivation and exact checks establish the final coefficients and identities.

These two identities immediately give the Yang–Baxter equation for \(R\).

3. Why this is a full localization

The \((-1)\)-spectral projection is \(P=(I-H)/2\). Exact calculation gives

For normalized matrix traces \(\tau_n=4^{-n}\operatorname{Tr}\), the partial trace is precisely the Markov step

With the conjugate-linear involution \(e_i^*=e_i\), the maps are unital \(*\)-representations and obey \(\rho_{n+1}(\iota_n(x))=\rho_n(x)\otimes I_4\). Therefore \(\tau_n\circ\rho_n\) is the Markov trace defining \(H_n(3,6)\). Faithfulness of ordinary matrix trace proves both inclusions between \(\ker\rho_n\) and the trace annihilator; the induced quotient injections commute with the tower maps. GHR identify \(H_n(3,6)\) with the entire categorical algebra generated by the braid image, so these are exactly the injections required for ordinary unitary localization.

Dimension one cannot carry the generator, and dimension two is excluded by the 2012 theorem. In dimension three, the only allowed Markov parameters force a Temperley–Lieb or complementary Temperley–Lieb quotient. The two corresponding relations are nonzero in \(H_3(3,6)\), each with squared \(L^2\)-norm \(1/18\) with respect to the exceptional trace, so neither candidate can be faithful. Hence four is minimal.

4. Quaternionic factorization and concurrent comparison

Set \(\mathsf A=-i\sqrt2M\), \(\mathsf B=iE\), \(U_{\mathrm K}=(\mathsf A+\mathsf A\mathsf B)/2\), and \(V_{\mathrm K}=(\mathsf A-\mathsf A\mathsf B)/2\). Then

This puts the sparse representative in the same abstract quaternionic Family III form as Galindo–Rowell. For their literal Section 13 operator \(R_{\mathrm{GR}}\), the paper displays a unitary \(S\) and proves exactly

where \(\Sigma\) reverses the two four-dimensional sites. Thus the exhibited comparison is local equivalence to the opposite operator; it does not decide whether direct local equivalence without the opposite also exists. The new \(R_{\mathrm{GR}}\) comparison is separate from the older 8-by-8 \(K_{\mathrm{GHR}}^{\mathrm{gen}}\) generalized-localization comparison.

5. Global braid and link consequences

Tensor-site reversal and conjugation by the Garside half twist extend the local comparison to a same-word unitary equivalence of the full braid representations for every strand number. The quaternionic source results therefore transfer finite image and Clifford structure to the five-word representation, with the Clifford statement made in a fixed conjugated Pauli frame.

The scalar partial traces of \(R^{\pm1}\) prove this is a Turaev enhancement. The Hecke relation yields \(R-qR^{-1}=\kappa I\) and hence

Here \(d_2(L)\) is the mod-2 first-homology dimension of the oriented triple cyclic cover branched over \(L\). The exact sign and factor were checked directly against the original Lickorish–Millett convention. The invariant is mirror-insensitive and exactly computable by the deterministic polynomial-time Family III algorithm. Also \(R^3=-I\) and \(R^6=I\), giving three-twist sign change and six-twist periodicity in local twist families.

6. Exact verification

• verify_exact.py — standard-library arithmetic in \(\mathbb Q(\sqrt2,\sqrt3,i)\); checks the complete finite identities exactly.
• verify_tensor_words.py — no matrices; checks all 18 surviving generic six-letter Pauli words and their polynomial coefficients.
• verify_supplied.py — hardened, optimization-safe SymPy implementation; the original discovery-era checker is retained separately for provenance.
• verify_concurrent_equivalence.py — independent exact encoding of both local formulas, the site swap, the displayed unitary, and the intrinsic quaternionic identities.
• verify_braid_link.py — exact intrinsic/comparison, enhancement, skein, two- and three-strand link, standard-frame witness, Clifford, reversal, and Garside checks.
• test_failure_modes.py — rejects optimized execution and deliberate mutations of every supported route.
• verification_output.txt — frozen output from the successful reference run.

Minimal dependency-free check

cd exceptional_ybe_d4
python3 verify_exact.py
python3 verify_tensor_words.py
python3 verify_concurrent_equivalence.py
python3 verify_braid_link.py

After swapping the two qubits inside every ququart, one qubit is a global spectator and the active \(8\times8\) operator satisfies the standard \((3,2)\)-generalized Yang–Baxter equation. The global conjugacy shows that this generalized representation has the same kernel as the faithful ordinary one. An exact comparison gives unnormalized squared Frobenius residuals \((0,48)\) for the displayed GHR operator at shifts \((1,2)\), versus \((24,0)\) here. This distinguishes the displayed \((3,1)\) and \((3,2)\) tensor structures, not the bare matrices after tensor structure is forgotten.

7. Related work, chronology, and limitations

C. Galindo and E. C. Rowell, in Unitary Yang–Baxter Operators: Towards a Classification, arXiv:2608.16865v1, independently prove the same existence and strict-localization conclusion, with dimension four smallest in Lechner's exceptional family, in a substantially broader classification framework. The present work contributes the sparse Pauli–Clifford representative, elementary 18-word certificate, direct trace and dimension-three arguments, active \((3,2)\) form, exact opposite/basis-change and all-strand comparisons, and the explicit enhancement normalization. The finite-image, Clifford, tower, and classical link-evaluation phenomena are credited to the existing sources.

Tensoring with identities gives examples in every base dimension \(4m\). The result does not settle dimensions \(6,10,14,\ldots\), classify all solutions, or prove uniqueness within the Pauli ansatz.

AI-assistance and verification disclosure

OpenAI GPT-5.6 Sol in Pro mode through ChatGPT at chatgpt.com and OpenAI GPT-5.6 Sol in Ultra mode through Codex in the ChatGPT desktop application were used for research support, review, and composition; Anthropic Claude through claude.ai was used for review. Alec Kriebel reviewed and edited the content and takes full responsibility for it. Model output was not treated as evidence. The sparse support arose from an AI-assisted numerical search whose original code and seeds were not retained; no mathematical claim relies on that search.

Suggested citation

Alec Kriebel, “An exceptional four-dimensional unitary Hecke Yang–Baxter operator: a five-word Pauli–Clifford normal form,” version 1.2.0, preprint and exact-verification package, Zenodo (2026), doi:10.5281/zenodo.22013710. The preceding v1.1.3 edition remains at doi:10.5281/zenodo.21971507.