Symmetry-Forbidden Aperture as Analog Circuit: A Resonator-Bank Test of Geometry-Induced Partitioning

This paper presents a physical realization of a symmetry-forbidden partition arising from the crystallographic restriction theorem, implemented as an analog resonator-bank circuit. The theorem limits periodic rotational symmetries in two dimensions to orders n = \{1,2,3,4,6\}, forbidding five-fold symmetry. We map this discrete constraint onto a harmonic frequency spectrum by constructing bandpass resonators at the permitted orders and deliberately excluding the forbidden fifth harmonic. Under broadband excitation, the circuit separates input power into two channels: a captured channel corresponding to permitted symmetries and a rejected channel corresponding to forbidden and uncoupled modes. We show that, under rectangular passband assumptions, the ratio of rejected to captured power obeys an exact algebraic identity, \gamma = \frac{NQ - S}{S}, where N is the total number of harmonic orders spanned, S is the sum of permitted orders, and Q is a quality factor fixed independently by a coherence corridor [\sqrt{e},\,7/4]. For the crystallographic set \{1,2,3,4,6\}, this yields \gamma \approx 5.29–5.38, in numerical agreement with the Planck 2018 cosmic mean dark-to-baryon ratio. A decisive causal test is performed by restoring the forbidden fifth resonator, which shifts \gamma by approximately 28%, demonstrating that the symmetry gap itself determines the ratio. We further show that filter topology controls convergence toward the analytical limit: sharper (hard-aperture) filters approach the predicted ratio, while softer filters do not. The circuit is inexpensive, buildable on a laboratory bench, and accompanied by pre-registered predictions that allow direct falsification. The work does not model cosmology or dark matter dynamics. Instead, it establishes that a discrete geometric constraint can be physically instantiated in an analog system and that its induced partition ratio is robust, causal, and experimentally testable.