Bell Algebra without Bell Violation: CHSH Operators for Ising Anyons in the Kitaev Honeycomb Model

Abstract

We construct four spatially separated Majorana zero modes (MZMs) per party on a finite Kitaev honeycomb lattice in its non-Abelian phase, bound to well-separated Ising vortices, and use them to realize a local, noncommuting CHSH measurement algebra. Two static fusion-parity observables per party, Z_A=iγ_{A,1}γ_{A,2} and X_A=iγ_{A,2}γ_{A,3} (and analogously for Bob), commute exactly between parties (P₂=0) and fail to commute within a party (P₁=‖[Z_A,X_A]‖=2.000000); on the maximally entangled fusion-parity Bell state the CHSH expectation reaches the algebraic Tsirelson value S=2√2=2.828427. This value survives fermion-parity superselection exactly, because the Bell state is a definite-parity eigenstate with unit weight in its own sector — a statement specific to the present four-MZM, measurement-only construction, not to braid-then-fuse protocols, which are a different experiment. A commuting-only control setting returns K=√2 ≤ 2, and the finite-size Majorana footprint leaking across the bipartition decays monotonically with system size (isotropic scaling 17.0% → 10.2% → 2.3%). We stress the critical scope-limiting fact, following Howard and Vala [Phys. Rev. A 85, 022304 (2012)]: a physical Bell-inequality violation using only topologically protected Ising-anyon operations does not follow from this construction, since Ising braiding generates only the Clifford group; that step requires a non-Clifford ("magic") resource, the subject of a companion paper. What we report here is the local realization of the noncommuting Bell-operator algebra on a genuine microscopic lattice substrate, not a claim of Bell nonlocality. A loop-threading relative phase δ reproduces the analytic corollary S(δ)=2√2·cos²(δ/2) to a residual of 4.4×10⁻¹⁶, and an independent gauge-invariant flux-holonomy diagnostic confirms no adiabatic channel mixing at the quoted scale (‖[W,Z_A]‖_F=0.0132), with the sole open discrepancy being a 0.020-rad gap between the accumulated Berry phase at one full flux quantum and the target π, traced to genuine near-degeneracies along the loop rather than to a construction error. About this series: This record is part of a series of related works from my independent research on Fibonacci anyons, with Ising anyons as their natural counterpart. I started in April 2026, and it has been a long and insightful journey in which I learned a lot; the work uses different methods and stays within verifiable, nonspeculative physics. The common thread of the series is a split: Ising anyons are limited to Clifford operations, while Fibonacci anyons are computationally universal, and across the series I map what standard witnesses of nonclassicality can and cannot certify on such systems. I consider Fibonacci anyons a serious candidate for topological quantum computing, given their universality and their topological protection against local noise. A hybrid approach with Ising is conceivable, but problems such as instability and certification would have to be solved first, and each needs research of its own. This paper asks whether a real microscopic lattice can carry the Bell structure, and constructs four Majorana modes per party on a Kitaev honeycomb lattice in its non-Abelian phase, realizing the full CHSH measurement algebra locally, with observables that commute between the parties and fail to commute within a party; the algebra holds, reaching the algebraic Tsirelson value and surviving the fermion-parity superselection, but a physical Bell violation does not follow, because topologically protected Ising operations generate only Clifford gates and the missing step is a non-Clifford resource, which is the subject of a companion paper; within the series it is the substrate test, where algebra on real material is not yet a violation. Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, including cross-checks with an unrelated AI model from a different company, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication. ------------------- Version notes (v1.1 -> v1.2). This version adds material; it does not correct any value, claim, or citation printed in v1.1. • Setup made explicit: the Setup section now states that the bare nearest-neighbor Kitaev model is gapless in the B phase, and that the gapped, non-Abelian phase used throughout this paper is opened by a time-reversal-symmetry-breaking perturbation -- the chiral three-spin term (equivalently, a next-nearest-neighbor Majorana hopping of strength kappa), following Kitaev's original construction. No value, method, figure, or numerical result changed; kappa's numerical value in the deposited code is unchanged. • Bibliography completed: the Karzig et al. entry now lists all 12 authors (adding Charles M. Marcus and Michael H. Freedman), replacing the first ten followed by "et al." • Release date: the deposited build is dated 2026-09-22; the Zenodo publication date is set at upload and may differ. Version notes (v1.0 -> v1.1), condensed. The complete notes for that version remain in the v1.1 record: https://doi.org/10.5281/zenodo.22698091 • No previously reported value changed. Three corrections were made, to printed form and to attribution: Eq. (4) is now written in terms of Bob's rotated settings, which is the CHSH functional actually evaluated -- the printed form had been written in terms of his unrotated parities; the resource that Brennen et al. require in addition to Ising braiding is called a non-stabilizer rather than a non-Clifford resource, in the three places where the requirement is stated; and the abstract now attributes the scope-limiting argument to Brennen et al., sharpened into a precise no-go by Howard and Vala, matching the discussion section. • Scope stated rather than assumed, in four places: the construction uses only Clifford combinations of static fusion-parity observables, and the Howard-Vala condition additionally requires that the measurements be stabilizer operations; B and B' are treated at the level of the ideal logical Majorana algebra, and their realization as individually protected measurements is not constructed here; the parity observables are gauge-invariant only when dressed with a Wilson line, which in the fixed flux sector used throughout is a number +-1, so every reported value is unchanged (reference added); and the basis invariance of W is the Wilczek-Zee freedom of the degenerate subspace, not the Z2 gauge redundancy (reference added). • Measurements made explicit: the finite-size quantity is renamed to the mean cross-party support weight of Alice's modes on Bob's region, and its scan is now given with its lattice sizes (24x12, 32x16 and 44x22) and inter-cluster separations (d = 18, 24 and 33); both finite-size quantities vanish as L grows and neither enters a Bell value. Orthonormality is an identity of the construction rather than a numerical finding, holding to 2.3e-14 before any orthonormalization step. The delta-corollary now carries its qualifier in the summary passage as well, together with the lattice size (28x16) on which it is reproduced. • Licensing changed: paper, figures and data are released under CC BY 4.0, the deposited source code under the Apache License, Version 2.0, replacing the MIT license of version 1.0. Licenses are not retroactive, so the files published as version 1.0 remain under MIT. • Claim strength reduced: the Tsirelson value is stated to be fixed by the algebra -- what the lattice establishes is that this algebra is realized by eight well-separated vortex-bound modes of a genuine microscopic Hamiltonian. • Added: the numerical driver for the four locked targets (p5b_m0_numeric.py) and the finite-size scan it produces (p5b_m0_finite_size.json), so that the rep