Prime Numbers as Resonant Escapes: A Wave-Mechanical Characterisation via Harmonic Filtering in Nested Gabriel's Horns

We present a wave-mechanical characterisation of the prime numbers as the unique frequencies that propagate freely through an infinite cascade of nested resonant horn shells. The physical foundation is multiplicative closure: wave systems with coupled resonators naturally generate sum frequencies, and any frequency set closed under this combination rule exhibits a prime structure under the cascade -- a subset of frequencies unreachable by combining existing shells, which must therefore open new ones. The positive integers are the canonical such set; their primes are what the cascade selects.

The cascade is scale-invariant: the prime positions (1st, 2nd, 3rd, 5th, 7th... escaping frequency) depend only on the multiplicative structure of the frequency set, not on the absolute size of the horns. Any physical realisation of a multiplicatively closed resonant cascade produces new shells at exactly the prime-numbered positions.

The natural geometry for this cascade is Gabriel's Horn (r = 1/x). Nested horns of radius 1/(nx) have volumes pi/n^2, and their shell gaps carry odd-integer numerators 2n+1. The resonance lattice of k prime shells -- the additive monoid generated by their log-frequencies -- is proved equivalent to the set of log-frequencies of integers whose prime factors all lie among the existing shells. We prove by strong induction (without invoking the Fundamental Theorem of Arithmetic) that every composite integer is trapped by this lattice before it could open a new shell, and that every prime escapes all existing shells. The Fundamental Theorem of Arithmetic is recovered as a consequence: completeness of trapping corresponds to the existence of a prime factorisation, and the mutual incommensurability of prime logarithms (proved independently) corresponds to uniqueness.

The Bessel cutoff frequencies of a physical cylindrical horn share the no-integer-harmonic property, grounding the abstract cascade in a physically realisable geometry.