Dual-Coordinate Engine — Φ: ℤ² ↔ ℤ³

The reversible mapping between the 2D flavor lattice and the cubic harmonic space. Formalizes the relationship between the six extensions (lattice model) and the C-MANN neural net (cubic model). Five interconnected frameworks: (1) DCE — interactive coordinate transform for all six quarks, invariants Q/Π/ω preserved across both systems. (2) Harmonic Resonance Spectrum — ω(q) = ζ(−1)·a + ζ(−3)·b, baryon resonance Ω(B) as a new conserved quantity, resonance parity as a decay-selection rule. (3) Flavor-Charge Jacobian — J = ∂(Q,Π)/∂(a,b) = [[+2/3,−1/3],[+1,−1]], det = −1/3. Volume element 1/3 = one quark baryon contribution. (4) Multiplicative Gauge Group G× — generators φ_u (flavor transition), φ_gen (generation step / CKM), φ_color (SU(3)). Conserved currents J_Q, J_Π, J_ω, baryon number B. (5) C-MANN Forward Pass as Physical Pipeline — input 2D lattice → DCE Φ → SU(3) color triplet → singlet gate (confinement) → observables.

Keywords: dual, coordinate, engine, DCE, flavor, lattice, cubic, harmonic, space, bijection, mapping, phi, transform, quark, coordinate, invariant, charge, polarity, resonance, jacobian, differential, geometry, curvature, gauge, group, symmetry, generator, CKM, Cabibbo, angle, color, SU3, baryon, collapse, manifold, stable, manifold, basin, attraction, pipeline, C-MANN, forward, pass, singlet, gate, Beck, physics, multiplicative, algebra, bilingual