Horn–Nozzle Correspondence as Impedance-Gradient Optimisation (v1.0): Generalised Curvature Law and Family-Expansion Diagnostics

This deposit extends the horn–nozzle impedance-gradient correspondence (v0.3) through systematic generalisation testing. The v0.3 note established that acoustic horns and aerodynamic nozzles solve a common impedance-structure optimisation problem, and identified a two-factor regression (adj. R² = 0.957) linking band-averaged reflection R to both RMS log-area curvature κrms and a curvature–boundary interaction κ × B. The present v1.0 subjects that two-factor structure to two independent generalisation tests. Design-space expansion Area-ratio sweep. The area ratio m = (r2/r1)² was varied over {4, 8, 16, 32, 64} with fixed outlet radius (r2 = 50 mm) and duct length (L = 200 mm). Within the original two families, the interaction term β₃ was consistently negative but decayed monotonically, losing significance at m ≥ 32. Diagnostic analysis confirmed this as a genuine regime shift rather than a scale artifact. Family expansion. Two new profile families were introduced alongside the original polynomial (C) and boundary-matched polynomial (G): Family T (sinusoidal perturbation): h(ξ) = ξ + ε sin(nπξ), introducing oscillatory internal curvature. Family H (centred hyperbolic tangent): h(ξ) = [tanh(α(ξ − ½)) + tanh(α/2)] / [2 tanh(α/2)], producing a monotone saturating transition with curvature sign reversal. With four families (30 profiles per area ratio), the interaction term collapsed entirely: partial ΔR²(M3 − M2) was negative at every m, while the single-factor model M1 (κrms alone) maintained adj. R² = 0.89–0.96 across m = 8–64. Principal findings The curvature law generalises. Band-averaged reflection is primarily governed by κrms (adj. R² = 0.89–0.96 across m = 8–64, four families, 30 profiles per condition). The κ coefficient β₁ is positive and highly significant (p < 10⁻¹⁰) at every area ratio, and increases monotonically with m, consistent with the physical expectation that larger ln m amplifies curvature sensitivity. The interaction term does not. The boundary mismatch B and interaction κ × B identified in v0.3 do not survive family expansion. A family-confounding diagnostic showed that when family identity is controlled for, B loses all explanatory power (p(B|family) > 0.6 at both m = 8 and 16). In the restricted C/G space, B effectively proxied for family identity (G has B = 0 by construction; C has B > 0), creating an apparent second factor. A quadratic correction κ² provided only marginal improvement (Δadj. R² ≤ 0.011), confirming that the linear M1 form is sufficient. The law is not a solver artifact. 2D axisymmetric Helmholtz FEM (960-node structured mesh, bilinear quadrilateral elements) confirms the rank structure of the κrms–R relationship (rank Spearman rs = 0.976, 6/8 exact rank matches) and approximate magnitude (Pearson r = 0.980, MAE = 0.018). Systematic high-κ overestimation by TMM (up to 20% for κrms > 1700) is attributable to the plane-wave approximation's known limitation for rapidly varying ducts, and does not affect rank structure. Revised Proposition 2 Across expanded profile families and area ratios, band-averaged reflection is primarily controlled by κrms. The curvature–boundary interaction identified in v0.3 reflects family-structured coupling rather than a universal second factor. The universal component was not the full two-factor structure, but the curvature term it contained. Contents Research note (v1.0 manuscript, Markdown) Five computation scripts (Python): area-ratio sweep, β₃ decay diagnostic, family expansion, M1 universality / confounding test, FEM validation Numerical results (JSON) and diagnostic figures (PNG) for each step Zenodo metadata reference (Markdown) Relationship to v0.3 This is a new version of 10.5281/zenodo.19427924. The v0.3 result is not retracted but reclassified: it captured a restricted-family law within the C/G design space, while v1.0 identifies the generalised law embedded within it.