Horn–Nozzle Correspondence as Impedance-Gradient Optimisation

Acoustic horns and aerodynamic intake nozzles exhibit geometric similarity because both solve the same class of optimisation problem: minimising reflection loss in a duct with spatially varying impedance structure. This research note establishes a horn–nozzle correspondence principle through three propositions. Proposition 1 identifies both systems as members of a common impedance-structure optimisation class. Proposition 2, developed through two stages of pre-registered numerical testing, shows that band-averaged reflection is primarily controlled by RMS log-area curvature (κ_rms), with boundary impedance coupling entering as a significant interaction term rather than an independent predictor (adj. R² = 0.957, interaction p = 0.012; sensitivity analysis under exclusion of influential points: adj. R² = 0.980, p = 0.001). The original monotone-smoothness hypothesis was rejected by a pre-registered pilot (Spearman r_s = 0.32); the two-factor interaction model emerged from systematic diagnostic analysis. Proposition 3 delineates the boundaries of the correspondence, including transverse mode onset, sonic transition, and the irreducibility of the boundary interaction term. The geometry kernel governing reflection quality is not a scalar smoothness measure but the RMS curvature of the log-area profile, modulated by the boundary radiation environment. A conjectural extension is proposed: when quasi-1D control fails (transverse modes in acoustics, shock formation in compressible flow), both systems converge on the same class of repair response — centerbody insertion (phase plug / inlet spike) that restores effective one-dimensional geometry. This suggests a three-layer theoretical structure: operator correspondence, optimisation law, and repair law. The first two layers are established; the third is recorded as an outlook. All judgment criteria were fixed before computation; exclusion of influential data points strengthens rather than weakens the results. Version Note (v0.3) Version 0.3 establishes the horn–nozzle correspondence on a quantitatively robust footing: the geometry–boundary interaction model remains strongly supported under all sensitivity conditions, and exclusion of influential points strengthens rather than weakens the result. A three-layer theoretical structure is proposed — operator correspondence (established), optimisation law (established), repair law (conjectural) — with the observation that phase plugs and inlet spikes are homologous repair devices for quasi-1D breakdown. This is the first publishable-stable version. Changelog v0.3 (current) Two-factor regression (M1/M2/M3) with pre-registered thresholds Sensitivity analysis: all 5 exclusion conditions pass STRONG/sig Proposition 2 revised to geometry–boundary coupled optimality Evidence chain summary table added (§4.1) B main effect note added (latent under noise removal) Interaction term confirmed as stable core finding Remark on repair-beyond-breakdown (phase plug / inlet spike homology) Three-layer theoretical structure formalised (§4.5) Next Step 5 added: repair-device homology investigation v0.2 Original monotone hypothesis (κ_sup predicts R) rejected by pre-registered pilot (r_s = 0.32) Diagnostic analysis revealed two-factor structure Family G (boundary-matched profiles) introduced Proposition 2 revised from smoothness-alone to two-factor model v0.1 Initial formulation: three propositions Common optimisation class (Prop. 1) Log-area curvature boundedness (Prop. 2, original) Isomorphism boundaries (Prop. 3) Keywords acoustic horn; aerodynamic nozzle; impedance matching; Webster equation; reflection minimisation; log-area curvature; boundary coupling; horn design; pre-registered testing; variational optimisation; phase plug; inlet spike; repair geometry Related Identifiers DPIB Horn programme: Zenodo concept DOI 10.5281/zenodo.19350570 (isRelatedTo) Structural transposition manuscript (Webster–nozzle deformation parameter): JSV submission (isContinuedBy) License Creative Commons Attribution 4.0 International (CC BY 4.0) Language English Subjects Acoustics Fluid dynamics Engineering design optimisation Notes This note is part of the DPIB Horn research programme. The central finding — that the geometry kernel is RMS log-area curvature modulated by boundary coupling — connects to the broader formal-causation framework developed by the author across cosmology, acoustic engineering, and AI/LLM research. All numerical computations used the transfer matrix method (2000 segments, 196 frequency points per case, 16 total profiles across two families). All judgment criteria were fixed before computation. Python source code for the pilot, regression, and sensitivity analysis is available upon request. Suggested Citation Takagi, T. (2026). Horn–Nozzle Correspondence as Impedance-Gradient Optimisation (v0.3). Zenodo. https://doi.org/10.5281/zenodo.19427924 Files to Upload horn_nozzle_correspondence_v03.md — Main research note two_factor_prereg.md — Pre-registration sheet prop2_pilot_design_v02.md — Pilot design document prop2_pilot_results.png — Pilot results figure prop2_extended_diagnostic.png — Diagnostic analysis figure two_factor_results.png — Two-factor regression figure sensitivity_results.png — Sensitivity analysis figure two_factor_results.json — Numerical results (machine-readable)