The Merkabit - A Ternary Computational Unit on the Eisenstein Lattice
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2026
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| _version_ | 1866902308825071616 |
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| author | Stenberg, Selina |
| author_facet | Stenberg, Selina |
| contents | <p>The merkabit is a balanced ternary computational unit based on dual spinors (u, v) ∈ S³ × S³ with counter-rotating phase evolution at π-lock. The Eisenstein lattice ℤ[ω], balanced ternary logic {−1, 0, +1}, and the gate set {Rₓ, Rz, P, F, C-SWAP} are derived — not assumed — from this definition. The governing algebra E₆ enters via the McKay correspondence from the binary tetrahedral group P₂₄ with zero free parameters; E₆ is uniquely isolated from its Langlands companions B₆ and C₆ by the McKay constraint and the self-duality condition h = h∨ = 12.</p> <p>Three algebraically independent routes yield α⁻¹ = 137: Route A (phase-space counting: 7 × 24 − 31), Route B (Casimir–Eisenstein: N(12 + 5ω) + dim(so(8)) = 109 + 28), and Route C (zero-point Berry phase: −ln F/10, where the return fidelity F is transcendental with two exact rational layers capturing 99.999966% of −ln F). A three-order correction from the E₆ Coxeter spectrum gives α⁻¹ = 137.035999083, matching experiment to 0.005 ppb.</p> <p>Error correction is intrinsic and geometric at three nested levels (π-lock, pentachoric complementarity, E₆ root syndromes), with composite suppression of 3,500–6,200× at ε = 10⁻³. The fault-tolerance threshold is ε ≈ 22%; the Peierls bound on the intrinsic torus gives <span>ε_th ≈ 43</span> — both far above the surface code's ~1%. The ouroboros cycle (period 12) is a discrete time quasi-crystal with Z₂ topological order and Z₃ subharmonic character. The F gate reduces long-range coupling from O(d) to O(1). Appendices A–P (46 parameter-free simulations, Python 3 + NumPy only) verify every testable claim. <br><br><br></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18925475 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Merkabit - A Ternary Computational Unit on the Eisenstein Lattice Stenberg, Selina Quantum computers ternary logic merkabit dual spinor Eisenstein lattice hexagonal lattice E₆ Lie algebra McKay correspondence pentachoron quantum error correction fault tolerance fine structure constant balanced ternary computational universality gate architecture Floquet time crystal discrete time crystal Berry phase qutrit Hopf fibration topological order <p>The merkabit is a balanced ternary computational unit based on dual spinors (u, v) ∈ S³ × S³ with counter-rotating phase evolution at π-lock. The Eisenstein lattice ℤ[ω], balanced ternary logic {−1, 0, +1}, and the gate set {Rₓ, Rz, P, F, C-SWAP} are derived — not assumed — from this definition. The governing algebra E₆ enters via the McKay correspondence from the binary tetrahedral group P₂₄ with zero free parameters; E₆ is uniquely isolated from its Langlands companions B₆ and C₆ by the McKay constraint and the self-duality condition h = h∨ = 12.</p> <p>Three algebraically independent routes yield α⁻¹ = 137: Route A (phase-space counting: 7 × 24 − 31), Route B (Casimir–Eisenstein: N(12 + 5ω) + dim(so(8)) = 109 + 28), and Route C (zero-point Berry phase: −ln F/10, where the return fidelity F is transcendental with two exact rational layers capturing 99.999966% of −ln F). A three-order correction from the E₆ Coxeter spectrum gives α⁻¹ = 137.035999083, matching experiment to 0.005 ppb.</p> <p>Error correction is intrinsic and geometric at three nested levels (π-lock, pentachoric complementarity, E₆ root syndromes), with composite suppression of 3,500–6,200× at ε = 10⁻³. The fault-tolerance threshold is ε ≈ 22%; the Peierls bound on the intrinsic torus gives <span>ε_th ≈ 43</span> — both far above the surface code's ~1%. The ouroboros cycle (period 12) is a discrete time quasi-crystal with Z₂ topological order and Z₃ subharmonic character. The F gate reduces long-range coupling from O(d) to O(1). Appendices A–P (46 parameter-free simulations, Python 3 + NumPy only) verify every testable claim. <br><br><br></p> |
| title | The Merkabit - A Ternary Computational Unit on the Eisenstein Lattice |
| topic | Quantum computers ternary logic merkabit dual spinor Eisenstein lattice hexagonal lattice E₆ Lie algebra McKay correspondence pentachoron quantum error correction fault tolerance fine structure constant balanced ternary computational universality gate architecture Floquet time crystal discrete time crystal Berry phase qutrit Hopf fibration topological order |
| url | https://doi.org/10.5281/zenodo.18925475 |