The Merkabit - A Ternary Computational Unit on the Eisenstein Lattice

We define the merkabit, a balanced ternary computational unit based on dual spinors (u, v) ∈ S³ × S³ with counter-rotating phase evolution, achieving coherence at π-lock. The hexagonal 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 single definition. The governing algebra E₆ enters via the McKay correspondence from the binary tetrahedral group P₂₄, determining every structural invariant with no free parameters. Two algebraically independent routes yield the integer α⁻¹ = 137; a three-order correction from the E₆ Coxeter spectrum gives α⁻¹ = 137.035999083, matching the measured fine structure constant to 0.005 parts per billion. Error correction operates at three nested levels — π-lock noise cancellation, pentachoric lattice complementarity, and the E₆ root system as structured syndrome space — unified by a single closure principle. The ouroboros cycle (period 12 = h(E₆)) constitutes a discrete time quasi-crystal with Z₂ symmetry-protected topological order. The gate set is computationally universal, containing qubit operations as a proper subset. Computational appendices (A–P, 46 simulations) provide numerical verification of every testable claim. Simulations confirm: 100% single-error detection on optimally assigned Eisenstein lattices; three-level composite error suppression of 3,500–6,200× at ε_raw = 10⁻³; a fault-tolerance threshold of ε ≈ 22% (far above the surface code's ~1%); and O(d) → O(1) reduction in long-range coupling cost via the F gate. Appendix P validates predictions P1–P5 under a realistic IBM Brisbane noise model (T₁/T₂ decoherence, depolarising gate errors, ZZ crosstalk) at both single-merkabit and 7-merkabit cell scale. All simulation code is included and requires only Python 3 and NumPy.