Relational Beat Cancellation on Coupled Inharmonic Lattices: Non-Convex Frustration, Dissipative Boundary Regularization, and the Railsback Manifold

Closed-loop numerical optimizers and uncoupled tuning models frequently approximate mechanoacoustic oscillator arrays as separable equal-temperament grids (fₙ = f₀ 2(n-n₀)/12). When applied to physical lattices exhibiting fourth-order Euler-Bernoulli flexural dispersion (Bₙ > 0) and non-diagonal bridge compliance (Kᵢⱼ ≠ 0), this separable ansatz induces severe combinatorial frustration. Here we formulate the coupled inharmonic tuning problem as a frustrated non-convex lattice optimization functional. First, we prove that for any dispersive lattice with Bₙ > 0 and Kᵢⱼ ≠ 0, the zero-beat coincidence manifold is strictly empty and single-pass open-loop equal temperament incurs a positive lower bound on partial-interference beat error scaling as O(B N²). Second, we demonstrate that anchoring a central temperament octave via a symmetric three-oscillator frustrated triad (cosθ = -1/3) and propagating $(2,1)$ and $(4,2)$ partial coincidences outward under coupled gradient relaxation converges in O(N) passes to the empirical S-shaped Railsback stretch curve (0.19 ± 0.05 Hz mean beat error across $N=88$ oscillators, compared with 4.82 ± 1.94 Hz for equal temperament). Finally, we model dissipative frequency locking via a Lindblad master equation and prove that maintaining a fractional phase-slip damping floor ηeff = 3/16 regularizes the high-frequency bridge termination impedance singularity at the cutoff node ν₂₃, bounding the peak impedance at Z(ν₂₃) = 4√3Z₀.