The Exponential Horn as Formal Cause: Structural Isomorphism Between Acoustic Waveguides and Cosmological Expansion, and the Geometric Necessity of Inflation

The Exponential Horn as Formal Cause We demonstrate a structural isomorphism between the Webster horn equation governing sound propagation in acoustic waveguides and the cosmological perturbation equation governing density fluctuations in the expanding universe. The isomorphism A complete, systematic equation-level mapping (13 parameters) connects: Horn flare rate m → Hubble parameter 2H Horn cutoff frequency → sound horizon at recombination Horn mouth radiation → baryon acoustic oscillations (BAO) Horn standing waves → CMB acoustic peaks Compression driver → primordial perturbation Horn body → radiation-dominated era The formal-causal argument Building on this isomorphism, we argue that exponential expansion is not a contingent outcome of particular initial conditions, as standard inflationary cosmology implies, but a formal-causal constraint: it is the unique expansion geometry that preserves wavefront phase coherence across cosmological interfaces. Key result Uniqueness theorem: Among all monotonically increasing area functions, the exponential is the unique function for which the wave equation has a constant friction coefficient — and therefore the unique geometry preserving wavefront coherence. Framework Form-Cause Processing Principle (FCPP): Reinstates Aristotelian formal causation as a mode of physical explanation complementary to efficient causation. Interface Universality Principle (IUP): The physics of waves at interfaces constitutes a unifying structural foundation operative from loudspeaker horns (10⁻¹ m) to the observable universe (10²⁶ m). Dispersion compensation: Identified as a universal engineering problem solved independently in every domain (optics, fiber optics, RF, electronics, acoustics, cosmology). Related work FCPP: doi:10.5281/zenodo.18786296 Acoustic Chromatic Aberration: doi:10.5281/zenodo.19334407 Keywords Webster equation · Friedmann equation · formal causation · cosmological inflation · baryon acoustic oscillations · interface physics · FCPP · wavefront coherence · structural isomorphism