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 ($f_n = f_0 2^{(n-n_0)/12}$). When applied to physical lattices exhibiting fourth-order Euler-Bernoulli flexural dispersion ($B_n > 0$) and non-diagonal bridge compliance ($K_{ij} \neq 0$), 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 $B_n > 0$ and $K_{ij} \neq 0$, 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 $\mathcal{O}(B N^2)$. Second, we demonstrate that anchoring a central temperament octave via a symmetric three-oscillator frustrated triad ($\cos\theta = -1/3$) and propagating $(2,1)$ and $(4,2)$ partial coincidences outward under coupled gradient relaxation converges in $\mathcal{O}(N)$ passes to the empirical S-shaped Railsback stretch curve ($0.19 \pm 0.05\text{ Hz}$ mean beat error across $N=88$ oscillators, compared with $4.82 \pm 1.94\text{ Hz}$ for equal temperament). Finally, we model dissipative frequency locking via a Lindblad master equation and prove that maintaining a fractional phase-slip damping floor $\eta_{\text{eff}} = 3/16$ regularizes the high-frequency bridge termination impedance singularity at the cutoff node $\nu_{23}$, bounding the peak impedance at $\vert{}Z(\nu_{23})\vert{} = \frac{4}{\sqrt{3}}Z_0$.