Symmetry-Forbidden Aperture as Analog Circuit: A Resonator-Bank Test of Geometry-Induced Partitioning
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2026
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| _version_ | 1866901948172599296 |
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| author | De Jesus, Elias |
| author_facet | De Jesus, Elias |
| contents | <p>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 <span>n = \{1,2,3,4,6\}</span>, 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.</p> <p>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,</p> <p><span>\gamma = \frac{NQ - S}{S},</span></p> <p>where <span>N</span> is the total number of harmonic orders spanned, <span>S</span> is the sum of permitted orders, and <span>Q</span> is a quality factor fixed independently by a coherence corridor <span>[\sqrt{e},\,7/4]</span>. For the crystallographic set <span>\{1,2,3,4,6\}</span>, this yields <span>\gamma \approx 5.29</span>–<span>5.38</span>, in numerical agreement with the Planck 2018 cosmic mean dark-to-baryon ratio.</p> <p>A decisive causal test is performed by restoring the forbidden fifth resonator, which shifts <span>\gamma</span> 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.</p> <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18558274 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Symmetry-Forbidden Aperture as Analog Circuit: A Resonator-Bank Test of Geometry-Induced Partitioning De Jesus, Elias crystallographic restriction theorem; symmetry-forbidden aperture; analog circuit experiment; resonator bank; hard vs. soft aperture; geometric partitioning; harmonic filtering; quality factor; power-ratio invariants; falsifiable analog models; dark-to-baryon ratio (numerical correspondence) <p>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 <span>n = \{1,2,3,4,6\}</span>, 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.</p> <p>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,</p> <p><span>\gamma = \frac{NQ - S}{S},</span></p> <p>where <span>N</span> is the total number of harmonic orders spanned, <span>S</span> is the sum of permitted orders, and <span>Q</span> is a quality factor fixed independently by a coherence corridor <span>[\sqrt{e},\,7/4]</span>. For the crystallographic set <span>\{1,2,3,4,6\}</span>, this yields <span>\gamma \approx 5.29</span>–<span>5.38</span>, in numerical agreement with the Planck 2018 cosmic mean dark-to-baryon ratio.</p> <p>A decisive causal test is performed by restoring the forbidden fifth resonator, which shifts <span>\gamma</span> 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.</p> <p>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.</p> |
| title | Symmetry-Forbidden Aperture as Analog Circuit: A Resonator-Bank Test of Geometry-Induced Partitioning |
| topic | crystallographic restriction theorem; symmetry-forbidden aperture; analog circuit experiment; resonator bank; hard vs. soft aperture; geometric partitioning; harmonic filtering; quality factor; power-ratio invariants; falsifiable analog models; dark-to-baryon ratio (numerical correspondence) |
| url | https://doi.org/10.5281/zenodo.18558274 |