Projective Braid Group Representations from a Microscopic Long-Range Kitaev Chain

#1. Introduction

Topological phases of matter support exotic quantum statistics: when quasiparticles are exchanged adiabatically, the wave function transforms by a unitary matrix, and for non-Abelian anyons these matrices form a representation of the braid group B_N. A recurring theme across exactly soluble field theories is that such representations are generically projective: the unitaries satisfy the group relations only up to phases, because the phase ambiguity is the fingerprint of a nontrivial central extension or 2-cocycle. Canonical Chern–Simons theory on a Riemann surface provides the canonical example: its wave functions carry projective representations whose cocycles depend on the (possibly rational) Chern–Simons level [1, 2], and the symplectic approach to the Wess–Zumino–Witten model yields projective representations of the loop group by the same mechanism [4].

The purpose of this paper is to derive such a projective braid representation not from a topological field theory but from a microscopic lattice model: the Kitaev chain, generalized to include long-range (power-law decaying) hopping and pairing. The motivation is threefold.

First, microscopic derivations anchor the abstract theory. Chern–Simons derivations [1, 2] are continuum and topological; a lattice derivation shows precisely which microscopic operators generate the braid action and where the projective phase comes from — it comes from the Clifford algebra {γ_i, γ_j} = 2δ_ij, i.e., from fermionic anticommutation, not from any gauge field.

Second, long-range hopping changes the locality structure. In the short-range Kitaev chain the MZMs are exponentially localized; with hopping t_ℓ ~ t₀/ℓ^α, localization and splitting scale differently, and for α ≤ 1 the zero modes can fail to be localized at all. We quantify this and show that the algebraic structure of the braid representation is independent of α, while its physical realizability is not.

Third, there is a live question about what braid representations do and do not determine. Recent analyses indicate that braid group representations of non-Abelian anyons need not uniquely fix the modular data (S and T matrices) of the underlying topological order, with the Ising theory as the canonical test case [12, 13]. A microscopic lattice construction of the braid representation, decoupled from any modular tensor category, is a useful control experiment for such uniqueness questions.

Our main results are: (i) an explicit verification that the Majorana braid matrices satisfy the braid relation exactly and the relation σ⁴ = 1 only projectively, with U⁴ = −𝟙 in the fixed-parity sector; (ii) explicit numerical estimates of zero-mode localization, gap, and cumulative phase drift for representative long-range parameters, showing class stability for α > 2; and (iii) a roadmap for completing the derivation, including the marginal regime α ∈ (1, 2] where the standard short-range arguments require modification.

#2. Background and Related Work

Chern–Simons braid representations. The canonical route to anyon statistics is canonical quantization of Chern–Simons theory coupled to non-dynamical matter. In [1], the quantization is generalized to rational Chern–Simons coefficient k, arbitrary genus, and arbitrary total charge, and the question is posed of which braid group representations on a Riemann surface the wave functions carry. In [2], an explicit solution of the Schrödinger equation for Chern–Simons theory coupled to charged particles is given for rational k and zero total charge, and the wave functions are shown to carry a projective representation of the group of large gauge transformations. These works establish the template our lattice construction mirrors: a quantization ambiguity (rational k there, fermion parity here) produces a projective rather than honest representation.

Symplectic and loop-group perspectives. The symplectic approach to the Wess–Zumino–Witten model [4] obtains the quantum theory as a projective representation of the loop group, with Gauss constraints imposed at a finite set of points. This is conceptually parallel to our construction: the Majorana zero modes are precisely "Gauss constraints at points" (defects), and the projectivity arises from quantizing a symplectic space whose prequantum line bundle has nonzero curvature. The Kashaev quantization of universal Teichmüller space similarly yields projective representations of the Ptolemy–Thompson group and associated dilogarithmic central extensions [5]; the structural lesson — that quantization of a classical symmetry group naturally produces central extensions — is the same one we instantiate for B_N.

Projective statistics beyond braiding. The analysis in [3] of non-Abelian braid statistics versus projective permutation statistics clarifies that non-Abelian projective representations of the permutation group can serve as statistics valid in any dimension, but that these are strictly different structures from braid statistics. This distinction matters for us: a strictly 1D chain can at best realize the projective-permutation sector unless the chain is embedded in a network that permits genuine exchange. In a related direction, [8] analyzes the three-dimensional model of Teo and Kane, in which defect lines carry Majorana zero modes whose exchange and twist implement a "ghostly" remnant of 2D Ising-anyon braiding; the governing group T_{2n} (projective ribbon permutation group) is computed there. Our work is the 1D-chain analogue: the Kitaev chain's MZMs, moved around a network, implement braid-like unitaries whose projectivity is the same remnant phenomenon.

Algebraic machinery. The Drinfeld associator machinery [6] associates to representations of a certain Hopf algebra braid group representations over a Laurent-series field and studies their dependence on the associator for GT-rigid representations; this rigidity is the algebraic shadow of our physical claim that the cohomology class of the statistics cocycle cannot change without a phase transition. The cluster-geometric construction of [7] endows moduli spaces of G-local systems on decorated surfaces with a cluster Poisson structure equivariant under the mapping class group, providing a geometric arena in which braid-type actions arise from quantization of cluster varieties. Braid group actions on representation categories also appear in [11], where an action of the braid group of a simple Lie algebra on imaginary roots of a quantum affine algebra determines cyclicity of tensor products and zeros of R-matrices; and in [9], where brane quantization connects Lagrangian branes on the SL(2,ℂ) character variety to representations of the spherical double affine Hecke algebra of C^∨C₁, with an affine braid group action on the category. Both illustrate that braid actions on zero-mode/defect Hilbert spaces are a robust algebraic phenomenon.

Modular data and uniqueness. Two recent analyses [12, 13] address whether the braid group representation carried by N non-Abelian anyons uniquely determines the modular data of the underlying topological order, with the Ising/Majorana fusion rules as the central test case; the emerging picture is one of non-uniqueness. Our microscopic construction is directly relevant: since the Majorana braid representation we derive in Section 4 is fixed entirely by the Clifford algebra, any "modular data" attributed to it must come from extra structure, not from the braid representation alone. Finally, [14] studies ultrametric relaxation dynamics in topological quantum memory, which bears on the stability of any memory built from our zero modes: the hierarchical relaxation barriers are a plausible model for error processes in a Majorana-based qubit whose protection is weakened by long-range hopping.

We note that reference [10] (a QCD collider-physics report) is part of the provided bibliography but is topically unrelated to this work and is not cited substantively.

#3. Methods

#3.1 The long-range Kitaev chain

The standard Kitaev chain on L sites is H₀ = −Σⱼ [ μ a†ⱼaⱼ + t(a†ⱼaⱼ₊₁ + h.c.) + Δ(aⱼaⱼ₊₁ + h.c.) ], with t, Δ real. We generalize to long-range hopping and pairing with algebraic decay: H_α = −Σⱼ Σ_{ℓ=1}^{L−j} [ (t/ℓ^α)(a†ⱼaⱼ₊ℓ + h.c.) + (Δ/ℓ^α)(aⱼaⱼ₊ℓ + h.c.) ] − μ Σⱼ a†ⱼaⱼ, with power-law exponent α > 0. The model is in symmetry class BDI (or D with a phase on Δ); in the topological regime |μ| < 2t Σ_ℓ ℓ^{−α} = 2t ζ(α) (valid for α > 1, where Σ ℓ^{−α} = ζ(α)), the open chain hosts unpaired MZMs γ₁, γ₂ at its ends, with γ₁ = Σⱼ v_α(j)(aⱼ + a†ⱼ), γ₂ = Σⱼ (−1)ⱼ v_α(j)(aⱼ + a†ⱼ), where v_α is a normalized Majorana wave function whose decay is controlled by α. For α > 1 the decay is exponential with a length ξ(α) that grows as α → 1⁺; for α ≤ 1 the zero modes delocalize and topological protection is lost. We treat ξ(α) as a quantity to be bounded, not assumed.

#3.2 From zero modes to a braid action

Given 2n well-separated MZMs γ₁, …, γ_{2n} with {γ_i, γ_j} = 2δ_ij, an adiabatic exchange of modes i and j implements, on the zero-mode Hilbert space, the unitary U_ij = exp((π/4) γ_i γ_j), whose action on the Majorana operators we verify directly in Section 4. The generators σ_i:= U_{i,i+1} are the candidate braid generators. The Hilbert space of 2n Majoranas is the Fock space of n complex fermions, with 2ⁿ states; fixing total parity (superselection) gives a representation of dimension 2^{n−1}.

#3.3 Exchange protocol in a chain

In a strictly 1D chain, "exchange" is not a native motion; we assume the standard network implementation: the chain is patterned into a T-junction network in which Majoranas are moved by tuning local chemical potentials, or exchanges are implemented by parity measurements. The braid unitary is the same either way; only the adiabaticity condition changes.

#3.4 Cocycle extraction and stability criterion

For a loop ℓ in the ground-state manifold representing a braid σ_i, the geometric phase φ(ℓ) defines a normalized 2-cocycle c(σ_i, σ_j) = φ(σ_iσ_j) − φ(σ_i) − φ(σ_j) mod 2π on B_N with values in U(1); its class in H²(B_N, U(1)) ≅ ℤ₂ (for N ≥ 5, since H₂(B_N; ℤ) ≅ ℤ₂ and H¹(B_N) = 0, so the universal-coefficient theorem gives H²(B_N; U(1)) ≅ Hom(H₂(B_N), U(1)) ≅ ℤ₂) is the invariant of interest. The Ising statistics corresponds to the nontrivial element of this ℤ₂; the projective phase e^{iπ/8} is a representative-level quantity, with the class fixed only by the parity of the cocycle. The stability criterion is: the class is invariant under deformation of the Hamiltonian provided the bulk gap never closes along the path. For the long-range chain, the couplings t/ℓ^α are perturbatively irrelevant for the zero-mode sector when α > 2 (the analogue of the familiar locality threshold α > d + 1 in dimension d = 1), so the gap stays open and the class is protected; only the cocycle representative can drift, by an amount controlled by the zero-mode overlap.

#4. Analysis

All numbers in this section are computed explicitly from stated inputs.

#4.1 The braid unitary and its action on Majoranas

Let A = γ_iγ_j with i ≠ j. Then A² = γ_iγ_jγ_iγ_j = −γ_i²γ_j² = −1, so A has eigenvalues ±i, and U_ij = e^{(π/4)A} = cos(π/4) 𝟙 + sin(π/4) A = (1/√2)(𝟙 + γ_iγ_j). Using the adjoint action: [A, γ_i] = Aγ_i − γ_iA = −γ_j − γ_j = −2γ_j and [A, γ_j] = γ_i − (−γ_i) = 2γ_i. The series e^{θ·ad_A}γ_i = γ_i cos 2θ − γ_j sin 2θ then gives, at θ = π/4: U_ij γ_i U_ij† = −γ_j, U_ij γ_j U_ij† = γ_i. The exchange acts as i ↔ j up to a sign convention (either sign is a valid orientation convention for the braid generator).

#4.2 Explicit matrix computation: braid relation and projective lift

Take three Majoranas γ₁, γ₂, γ₃; fix the parity sector spanned by {|0⟩, γ₃|0⟩}, where |0⟩ is the vacuum of the complex fermion f = (γ₁ + iγ₂)/2. In this basis, choosing phases consistently, γ₁γ₂ = iσ₃ and γ₂γ₃ = iσ₁. With θ = π/4: U₁₂ = e^{(π/4)(iσ₃)} = diag(e^{iπ/4}, e^{−iπ/4}), U₂₃ = e^{(π/4)(iσ₁)} = (1/√2)[[1, i], [i, 1]].

Fourth power (projectivity). U₁₂⁴ = e^{iπσ₃} = diag(e^{iπ}, e^{−iπ}) = −𝟙. So the braid relation σ⁴ = 1 is satisfied only up to the phase −1: the representation is projective. Equivalently, U₁₂² = iσ₃ = iγ₁γ₂: the square of an exchange is the fermion parity operator, and four exchanges differ from the identity by the fermion parity (−1)^F — the well-known "fermion parity anomaly" of Majorana braids, here derived rather than assumed.

Braid relation, exact. Direct 2×2 multiplication, first ordering: U₁₂U₂₃ = (1/√2)[[e^{iπ/4}, i e^{iπ/4}], [i e^{−iπ/4}, e^{−iπ/4}]]; right-multiplication by the diagonal U₁₂ rescales the columns by e^{iπ/4} and e^{−iπ/4}, giving U₁₂U₂₃U₁₂ = (1/√2)[[e^{iπ/2}, i], [i, e^{−iπ/2}]] = (1/√2)[[i, i], [i, −i]] = (i/√2)[[1, 1], [1, −1]] = (i/√2)(σ₁ + σ₃). Second ordering: U₂₃U₁₂ = (1/√2)[[e^{iπ/4}, i e^{−iπ/4}], [i e^{iπ/4}, e^{−iπ/4}]]; left-multiplication by U₂₃ gives entrywise (1,1) = (1/2)(e^{iπ/4} − e^{−iπ/4}) = i/√2, (1,2) = (i/2)(e^{iπ/4} + e^{−iπ/4}) = i/√2, (2,1) = i/√2, (2,2) = (1/2)(e^{−iπ/4} − e^{iπ/4}) = −i/√2, so U₂₃U₁₂U₂₃ = (i/√2)[[1, 1], [1, −1]] = (i/√2)(σ₁ + σ₃). Both orderings therefore agree exactly, U₁₂U₂₃U₁₂ = U₂₃U₁₂U₂₃ = (i/√2)[[1, 1], [1, −1]], using the same orientation convention (γ₁γ₂ = iσ₃, γ₂γ₃ = iσ₁, forcing γ₁γ₃ = −iσ₂) for both. The braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ therefore holds exactly (not merely projectively) in this representation, while σ⁴ = 1 holds only projectively. This is precisely the Ising anyon structure: the exchange phases in the two fusion channels are R^{σσ}₁ = e^{−iπ/8} and R^{σσ}_ψ = e^{3iπ/8} (consistent with the Ising test case of [12]), and the per-exchange projective phase relative to a bosonic baseline is φ_Ising = π/8 = 3.14159/8 = 0.3927 rad.

#4.3 Input numbers

#4.4 Localization and drift bounds

Localization parameter. For a semi-infinite chain with t(r) = t₀r^{−α}, the MZM decay factor λ satisfies, at small μ, λ ≈ μ/(2t₀ζ(α)). At α = 3, μ/t₀ = 0.1: λ = 0.1/(2 × 1.2020569) = 0.1/2.4041138 = 0.0416. The localization length is ξ = 1/|ln λ| = 1/3.1792 = 0.3145 sites — essentially on-site localized — so the overlap of MZMs separated by s sites is ~ λ^s.

Drift projection at α = 3 (assumption-labeled). We assume, without derivation, that the long-range correction to the exchange phase scales as ε ~ λ^s · s^{−α}. No Berry-connection calculation or perturbative derivation of the overlap-to-phase coupling is given here, and no prefactor is fixed; this is a projection, not a proved bound, and is subject to the same assumption-auditing as the α = 2.1 estimate below. For s = 10, α = 3: λ¹⁰ = e^{10 ln 0.0416} = e^{−31.79} = 1.46 × 10⁻¹⁴; ε = 1.46 × 10⁻¹⁴ × 10⁻³ = 1.46 × 10⁻¹⁷. The per-braid phase drift is bounded by |Δφ| ≤ 2πε; over N_pairs = 10⁶ braids, |Δφ| ≤ 2π × 1.46 × 10⁻¹⁷ × 10⁶ = 9.2 × 10⁻¹¹ rad — negligible under the stated assumption. Applying the same factor-of-3 uncertainty treatment as for the α = 2.1 projection, the cumulative drift lies in [3.1, 27.6] × 10⁻¹¹ rad. The Ising projective class is expected to be stable in this regime, contingent on the assumed scaling ε ~ λ^s · s^{−α}.

Gap. At μ = 0, α = 3: Δ_gap = 2Δ₀ζ(3) = 2 × 1.2020569 Δ₀ = 2.4041 Δ₀ > 0, confirming an open gap along the deformation path from α = ∞ to α = 3 at these parameters.

Dangerous regime (projection, clearly labeled). The above bound degrades as α → 2⁺ and μ → μ_c = 2t₀ζ(α), where ξ diverges. Projection with stated assumptions: at α = 2.1, μ/μ_c = 0.9, take ξ ≈ 10 sites (from ξ ~ ξ₀/(1 − μ/μ_c) with ξ₀ ~ 1) and ε ~ e^{−s/ξ} s^{−α} = e^{−1} × 10^{−2.1} = 0.368 × 0.00794 = 2.92 × 10⁻³. Per-braid drift ≈ 2π × 2.92 × 10⁻³ ≈ 1.8 × 10⁻² rad; over 10³ braids, cumulative drift ~ 18 rad, destroying gate fidelity. Assumptions: (a) exponential localization with the stated ξ; (b) the Berry-curvature correction is first order in ε; (c) no gap closing. Each could fail by factors of order unity; we quote the per-braid drift as 1.8 × 10⁻² rad with a factor-of-3 uncertainty band, [0.6, 5.4] × 10⁻² rad.

#4.5 Cohomological conclusion

In the short-range limit the cocycle representative takes values generated by e^{iπ/8}; the Ising statistics corresponds to the nontrivial element of H²(B_N, U(1)) ≅ ℤ₂ (N ≥ 5). The drift computed above changes the representative but not the class, provided the deformation path never closes the gap. Note that the phase e^{iπ/8} itself is a representative-level quantity: the cohomology class distinguishes only the parity of the cocycle, i.e., Ising versus trivial statistics. Since α > 2 keeps the long-range couplings perturbatively irrelevant and the gap open (Δ_gap = 2.4041 Δ₀ at α = 3, μ = 0), the class is invariant — the microscopic counterpart of the GT-rigidity observed algebraically in [6].

#5. Results

1. Braid unitary: U_ij = exp((π/4)γ_iγ_j) exchanges γ_i ↔ ∓γ_j; verified by explicit adjoint-action computation.
2. Projectivity: U² = iγ_iγ_j (fermion parity); U⁴ = −𝟙 in the fixed-parity sector; braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ holds exactly. Cocycle value ω(σ,σ,σ,σ) = −1.
3. Anchor phase: φ_Ising = π/8 = 0.3927 rad per exchange, from the Ising R-matrix values e^{−iπ/8}, e^{3iπ/8}.
4. Representation dimension: 2^{n−1} on 2n Majoranas at fixed parity.
5. Localization at α = 3, μ/t₀ = 0.1: λ = 0.0416, ξ = 0.3145 sites.
6. Drift projection at α = 3, s = 10 (assumed scaling ε ~ λ^s · s^{−α}, not derived): ε = 1.46 × 10⁻¹⁷ per braid; cumulative over 10⁶ braids ≤ 9.2 × 10⁻¹¹ rad, with factor-of-3 band [3.1, 27.6] × 10⁻¹¹ rad. The nontrivial ℤ₂ projective class (Ising) is expected to be stable under these assumptions.
7. Gap at μ = 0, α = 3: Δ_gap = 2.4041 Δ₀.
8. Projection (α = 2.1, μ/μ_c = 0.9): per-braid drift ≈ 1.8 × 10⁻² rad, band [0.6, 5.4] × 10⁻² rad; ~18 rad over 10³ braids — loss of topological protection near the marginal regime.

#6. Discussion

Limitations. First, the drift bound in the localized regime relies on the self-consistency equation for λ, which is exact only for the pure power-law chain at small μ; realistic devices have chemical-potential disorder that can resonantly enhance zero-mode overlaps by orders of magnitude, invalidating the 10⁻¹⁷ estimate. Second, the cohomological invariance argument assumes the deformation path never closes the bulk gap; long-range chains are known to exhibit re-entrant gap closings at intermediate α, and a single gap closing would allow the class to jump between the two elements of ℤ₂. Third, the analytic skeleton presented here should be checked by exact diagonalization (the model is quadratic, so exact) for finite chains, 2n ≤ 10, before the marginal regime α ∈ (1, 2] is trusted. Fourth, the drift bound may miss nonperturbative accumulation from the nonlocal Majorana–Majorana coupling H_eff ~ iδγ₁γ₂ with δ ∝ s^{−α}, which is only algebraically small just above α = 2.

Failure modes and falsifiability. The central claim — class stability for α > 2 — would be falsified by: (i) an exact-diagonalization computation of the exchange unitary for a finite long-range chain showing a phase outside {e^{−iπ/8}, e^{3iπ/8}} up to smooth drift, at parameters where the gap is provably open; (ii) a demonstration that the long-range chain realizes a different central extension, e.g., the dilogarithmic extension of [5]; (iii) a demonstration that the relevant group is not B_N at all but the projective permutation group of [3] — a serious possibility for a strictly 1D chain, in which case the 3D remnant group T_{2n} of [8] is the correct target if the chain is embedded in a higher-dimensional defect network.

Non-uniqueness warnings. The analysis [12] shows that even a complete braid representation does not uniquely fix modular data; hence our cocycle, even if derived perfectly, underdetermines any putative 2D topological order adjacent to the chain. The partial classification of MZM fusion rules in [13] means we cannot exclude exotic fusion channels beyond the Ising set from braid data alone. The relaxation analysis of [14] suggests that even correct statistics can be operationally swamped by ultrametric error dynamics in the encoded memory.

Open questions. (1) Does the long-range chain in the marginal regime 1 < α ≤ 2 realize a continuous family of cocycle representatives, and can the class jump only via gap closings? (2) Can the brane–DAHA correspondence of [9] be made literal for the long-range chain? (3) What does the tensor-product cyclicity criterion of [11] imply about which fusion channels of N long-range-coupled MZMs are reachable? (4) How do disorder-induced overlap enhancements modify the drift bounds quantitatively?

#7. Conclusion

We have constructed an explicit projective representation of the braid group from the microscopic Hilbert space of a long-range Kitaev chain. The projectivity originates entirely in the Clifford algebra of Majorana zero modes — the fermion parity anomaly U⁴ = −𝟙 — and defines the nontrivial element of H²(B_N, U(1)) ≅ ℤ₂ (N ≥ 5), i.e., the Ising projective class. Long-range hopping with exponent α > 2 leaves this class invariant, with projected numerically negligible drift of the cocycle representative (≤ 9.2 × 10⁻¹¹ rad over 10⁶ braids at α = 3, under the assumed scaling ε ~ λ^s · s^{−α}), while the marginal regime α → 2⁺ near the phase boundary destroys protection. The construction bridges microscopic lattice Hamiltonians and the abstract projective representations of Chern–Simons theory, WZW models, and higher-dimensional Majorana defect constructions, and provides a clean control experiment for questions of modular-data uniqueness. Future work should address exact finite-size verification, disorder effects, and the marginal regime.

#References

[1] Canonical Chern-Simons Theory and the Braid Group on a Riemann Surface. arXiv:hep-th/9306050v1. https://arxiv.org/abs/hep-th/9306050v1 [2] Canonical Chern-Simons Theory and the Braid Group on a Riemann Surface. arXiv:hep-th/9301036v1. https://arxiv.org/abs/hep-th/9301036v1 [3] Nonabelian braid statistics versus projective permutation statistics. arXiv:hep-th/0201240v2. https://arxiv.org/abs/hep-th/0201240v2 [4] Symplectic Approach of Wess-Zumino-Witten Model and Gauge Field Theories. arXiv:dg-ga/9504001v1. https://arxiv.org/abs/dg-ga/9504001v1 [5] The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization. arXiv:1211.4300v4. https://arxiv.org/abs/1211.4300v4 [6] Caracteres de rigidite du groupe de Grothendieck-Teichmuller. arXiv:math/0502117v1. https://arxiv.org/abs/math/0502117v1 [7] Quantum geometry of moduli spaces of local systems and representation theory. arXiv:1904.10491v4. https://arxiv.org/abs/1904.10491v4 [8] Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions. arXiv:1005.0583v4. https://arxiv.org/abs/1005.0583v4 [9] Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category. arXiv:2412.19647v3. https://arxiv.org/abs/2412.19647v3 [10] Tevatron-for-LHC Report of the QCD Working Group. arXiv:hep-ph/0610012v1. https://arxiv.org/abs/hep-ph/0610012v1 [11] Braid Group Actions and Tensor Products. arXiv:math/0106241v1. https://arxiv.org/abs/math/0106241v1 [12] DOI 10.5281/zenodo.23087164. QNFO: Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case. [13] DOI 10.5281/zenodo.22739626. QNFO: Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors. [14] DOI 10.5281/zenodo.18640261. QNFO: Ultrametric Relaxation Dynamics in Topological Quantum Memory.

#Appendix A. Divergence report

D1. Value and provenance of the exchange phase (C2).

• Draft A: φ = (π/2)(t₂/t₁) = 0.15π ≈ 0.471 rad, obtained by postulating a linear interpolation φ(r) = φ₀r with φ₀ = π/2 and r = t₂/t₁ = 0.3 (t₁ = 1, t₂ = 0.3 attributed to the parameter regime of [8]). A itself flags this as a postulate, not a derived result.
• Draft B: anchor phase φ_Ising = π/8 = 0.3927 rad, from the standard Ising R-matrix values e^{−iπ/8}, e^{3iπ/8} (consistent with [12]).
• Draft C: generator convention U = exp((π/4)γγ), giving U² = parity and U⁴ = −𝟙; the per-channel exchange phases are the Ising e^{±iπ/8} values.
• Underlying disagreement: A assumes the projective phase is a continuous function of the microscopic hopping ratio (linear interpolation convention); B and C assume the standard Ising/Clifford convention in which the phase is topologically quantized and cannot vary continuously without a phase transition. These conventions are incompatible: a continuous deformation of the phase at fixed class is impossible for the nontrivial isolated class in H²(B_N, U(1)) ≅ ℤ₂ unless the representative (not the class) drifts.
• Resolution adopted: the main text uses the B/C convention (Ising anchor π/8, generator exp((π/4)γγ)), which is derived from the Clifford algebra rather than postulated. A's linear-interpolation proposal is not adopted; it is reported here as an unverified conjecture that would, if true, imply either a class change (requiring a gap closing) or a representative drift far larger than the bounds computed in Section 4.4.

D2. Effect of long-range hopping on the representation (C3).

• Draft A: long-range hopping "deforms the usual Ising anyon representation into a continuous family of projective representations" parameterized by t₂/t₁.
• Drafts B, C: the algebraic structure of the braid representation is independent of α; long-range hopping controls only localization, gap, and the drift of the cocycle representative; the class is stable for α > 2 with open gap.
• Underlying disagreement: A treats the phase as a microscopic, continuously tunable quantity; B/C treat it as topologically protected. B/C's position is supported by the cohomological argument (class in ℤ₂, invariant under gap-preserving deformations) and by explicit drift bounds.
• Resolution adopted: main text adopts the B/C position (class stability); A's continuous-family claim is documented here and listed as falsifiable (an exact-diagonalization demonstration of a continuously varying phase at open gap would overturn it).

D3. Magnitude of the short-range phase (π/2 vs π/8 vs π/4). A uses φ₀ = π/2; B uses π/8; C uses π/4 in the generator exponent. These are partially convention-dependent: π/4 is the generator angle (U = exp((π/4)γγ)), π/2 is the phase of U² relative to identity in a particular channel convention, and π/8 is the Ising topological spin per exchange relative to a bosonic baseline. The main text states all three relations explicitly (Section 4.2) so the conventions are unambiguous; no physical contradiction remains once the conventions are fixed.

#Appendix B. Claim attribution