Toroid Models for Quantum–Gravitational Resonance via de Broglie–Schwarzschild Symmetry (A1-b)

Note: There is a new version 3 https://zenodo.org/records/20795392 Mathematical corrections and updates were made with the help of Claude Opus 4.7. This is the revised version A1b, superseding A1a. The earlier supercurrent model was abandoned in favour of the dual-subsystem tungsten-graphene architecture presented here. A1 is the first article of the series A linked here, for early and background models pls refer to the OSF proect. De Broglie–Schwarzschild (DBS) Symmetry Overview Where the de Broglie wavelength of a system resonates with its Schwarzschild radius: λ = Rₛ This framework separates into two subsystems: a mass subsystem, setting the gravitational scale Rₛ, a wave subsystem, defining λ through collective quantum effects. Such resonance naturally occurs at the Planck scale; here, we show it can be shifted upward by ~11 orders of magnitude through engineered systems (e.g. tungsten cores with graphene electron sheets). No exotic particles or new laws are required. Further improvement and new models are possible. The framework is conceptual first, highlighting the symmetry principle. Experimental implementations may include toroidal currents, graphene-coated nanoparticle ensembles, Cooper pairs, Bose–Einstein condensates, or other collective quantum systems. The collective de Broglie wavelength of the subsystem can significantly differ from single-particle wavelengths, allowing resonance even when individual particle λ would be too large. The Schwarzschild radius is used as a gravitational scale, not as a physical boundary; no singularities are implied. Such applications of Rₛ at small scales are consistent with theoretical treatments in quantum gravity, and Hawking’s proposals for quantum wormholes. This work lays the conceptual foundation for controllable quantum–gravitational interactions and is open to further research across quantum physics, nanotechnology, and materials science. Erratum to A1. Equation (4) contains an arithmetic slip of one order of magnitude: the stated result 1.49 × 10^-25 m should read 1.49 × 10^-26 m. All downstream tables and calculations should usethe corrected value. A corrected version 3 is in preparation. _________________________________________________________________________________ A note on coherence and the scope of the A-series: Early papers (A1–A6) propose engineered systems — toroids, hydrogel rings, nanoparticle arrays — designed to bring a macroscopic body into a resonance condition. The coherence properties of these systems, particularly the DNA-origami hydrogel toroid in A2, are not fully characterized. Whether such a system exhibits quantum coherence in the strict sense, mechanical coherence, or some intermediate biological quantum coherence of the kind observed in photosynthetic complexes remains an open experimental question. What A2 proposes is precisely to test this — the experiment itself would shed light on the coherence question. De Broglie's original formulation applied the wave relation to the center of mass of any system, not only to coherent quantum beams — empirically supported by Sagnac interferometry and SQUID measurements of macroscopic rotating bodies. Papers A7, A8, and A9 operate on entirely different ground. They do not propose any engineered body or coherence condition. Instead, they derive the mass and geometric scale of vacuum granules already present in the universe — from the observed 1–5 Hz noise floor common to all precision instruments, from the Kerr metric applied to any rotating body, and from Compton wavelength reasoning. The question of coherence does not arise in A7–A9: we are not creating a resonance condition, we are detecting one that already exists, at scales of 22 metres to 22+ kilometres — within the reach of LIGO, Virgo, the Einstein Telescope, and tabletop rotating systems. The A-series as a whole move from engineering proposals toward geometric discovery. The later papers stand independently of any coherence assumption. ____________________________________________________________________________________ Companion and foundational work: Gravity Resonance (M.P.) Full repository links available on A9 Zenodo page: https://doi.org/10.5281/zenodo.18888582