KAEL: a self-certifying lattice medium whose transport generator turns out to be curved-space Dirac evolution

Abstract

KAEL is a single, dependency-free Python file (~1,800 lines, stdlib only) that builds an exact algebraic core and, on top of it, a small 2D lattice spinor medium. It runs its own 16-check self-certification on load. Everything below was independently re-derived and measured against the unmodified engine, not read off its own printout. **THE EXACT PART** Everything downstream comes from one weighted arrow c: 1→0 and its adjoint d = c†. The four combinations P=dc+cd, Q=c+d, Γ=dc−cd, N=d−c reproduce the Pauli algebra exactly (Jordan–Wigner for one mode, written as incidence arrows rather than assumed). Tensoring these across sites gives the general Clifford relation {e_i,e_j}=2δ_ij·I for any dimension, checked to zero residual for d=1..6, with the "Fourier symbol" identity S²=(Σi²)I falling out as a pure corollary of anticommutation. One closed-form identity — exp(t(μI+X)) for X²=σρ²I — is reused, unmodified, for four different physical roles: the local metric frame, spin/gauge rotations, the transport gate, and the connection-relaxation flow. **THE LOAD-BEARING THEOREM** The two-site transport gate was previously just asserted unitary. It's provable: for any frame, any spin frame, any SU(2) link, the edge generator satisfies cc†=c†c=ρ²I exactly, because in 2×2 the anticommutator of two traceless Hermitian matrices is always a scalar. I reran this independently over 2000 random trials: max|cc†−ρ²I| = 5.7×10⁻¹⁴. The gate is an exact exponential, not a small-step approximation — unitarity is structural, not asymptotic. **THE MEASURED PHYSICS** - Flat dispersion: ω±(k) = ±√(sin²k_xh+sin²k_yh)/h, matching the engine's own one-step operator to machine precision, with the palindromic edge sweep confirmed second-order in Δt (ratio 4.00 under step-halving). - Four massless cones, not one — the flat medium doubles like any centered-difference Dirac operator. Stated plainly rather than hidden: ω=0 at all four zone corners, measured to 10⁻²⁹. - The local frame is a genuine Riemannian metric: E=exp(uI+rΓ+sQ) is symmetric positive-definite with eigenvalues e^(u∓ρ) — I diagonalized it directly and it matches the engine's own independently-computed min/max propagation speed to float precision. - The spinor measure isn't a free choice. Fixing it to what the engine already conserves exactly forces the transport generator to be the massless curved-space Dirac Hamiltonian in the density-rescaled variable χ=g^(1/4)ψ, with the spin connection appearing as the engine's existing symmetrization term (half-divergence), not something missing. Measured against an independently-solved torsion-free spin connection: residual ~10⁻¹⁴. - The gauge coupling is forced, not chosen: because KAEL's SU(2) gauge group and its Clifford generators act on the same 2D space, minimal coupling −γ·D isn't even self-adjoint here. The symmetrized coupling −½{γ,D} is the unique Hermitian completion. - Plaquette holonomy converges to non-Abelian field strength at O(h³), gauge covariance of the full coupled step holds to ≤10⁻¹⁵, and matter norm is conserved exactly while connection energy descends monotonically — a driven-dissipative gauge medium, not a closed Hamiltonian field theory, by design. **WHAT THIS DOESN'T CLAIM** No lapse or spacetime extension (curved space, not curved spacetime — no time-time metric component). No Einstein-sourced geometry — matter writes the frame through a relaxation rule, not a field equation. No conserved total energy for the coupled system. No solitons or bound states demonstrated. No continuum limit at arbitrary lattice size or parameter range — every "measured" claim above is pinned to the specific grids and parameters it was run on. There's also an explicit falsifier list in the full writeup: eight concrete, cheap numerical tests that would kill specific claims above if they failed (e.g., any frame/link where cc†≠ρ²I beyond roundoff kills the gate theorem outright). Included on purpose — this should be easy to break if it's wrong. Full exposition with every derivation, every measured residual, and a blind reconstruction experiment (recovering the frame metric purely from transport-operator tomography, converging at the expected second order) located in the PDF.