Topological phonon gaps in Fibonacci isotope nanoribbons of hexagonal boron nitride

We show that a Fibonacci ¹⁰BN/¹¹BN isotope modulation in a strictly single-mode hexagonal boron nitride nanoribbon opens topologically labelled phonon band gaps, and that these gaps survive realistic edge roughness — because the roughness itself is far more transparent than ray-limit theory predicts. Main results Born-transparency law. In the single-mode wave limit (λ >> W) the coherence length obeys the zero-parameter law ξedge = 2W²/[q²Sw(2q)], and the Ziman ray formula underestimates it by a median factor of 10.8. All three scaling exponents are pinned numerically (−1.91±0.06, +1.87±0.08, −1.384±0.095 against the predicted −2, +2, −1.38), and the absolute magnitude to 3.5%, with the residual confirmed as a second-order correction by two independent routes agreeing to 10⁻³. Edge-flatness tolerance relaxes from atomic (≲0.13 nm) to electron-beam-lithography scale (≲1 nm). Falsifiable gap labeling. Twelve gaps carry gap-labeling-theorem indices with median IDOS residual 3×10⁻⁶, against 4×10⁻³ for matched random controls — a 1385× separation, p = 1.4×10⁻³. Full counter-propagating phason pumping. Per-edge spectral flow ±q verified for all twelve gaps up to |q| = 11, including a documented silent-failure mode of eigenvector-overlap branch tracking at in-gap avoided crossings. Working point and experiment. Operating gap at 0.250 THz (W = 30 nm, d = 24.8 nm, L = 20 μm); topology costs 2.20× in localization length relative to a periodic superlattice at matched frequency. An ASOPS acoustic-cavity measurement is proposed, with a built-in opposite-Chern-number control gap at 0.154 THz in the same device. Methods. Hermitian variational boundary-conforming recursive Green's-function transport, validated by three gates (grid convergence, transmission-eigenvalue unitarity, analytic single-bump Born anchor); transfer-matrix Lyapunov spectra; Sturm-count integrated density of states. Three numerical failure modes and the gate that catches each are documented in the text. Scope. Computations use a scalar single-polarization model with h-BN parameters; a scope section in the paper details which conclusions are polarization-generic and which numbers await a full elastic treatment. Companion paper (Paper 1 of the series): Hyperuniform isotope arrangements and the scaling of phonon scattering in hexagonal boron nitride, doi:10.5281/zenodo.21981511.