Toroid and Nanoparticle Models for Quantum-Gravitational Resonance via de Broglie–Schwarzschild Symmetry

⚠️ NOTICE – SUPERCEDED VERSION

This is Version 1 (July 23, 2025, DOI 10.5281/zenodo.16371538): please note that this version has been superseded by Version 2 (August 25, 2025) with updates and conceptual extensions:  https://doi.org/10.5281/zenodo.16449277 All Versions: https://doi.org/10.5281/zenodo.16371537  

This update retains the main toroidal graphene–tungsten model as the primary experimentally relevant design.

Toroid Models for Quantum–Gravitational Resonance via De Broglie–Schwarzschild (DBS) Resonance/Symmetry

Description:
We propose a conceptual framework for De Broglie–Schwarzschild (DBS) Resonance/Symmetry, in which the quantum de Broglie wavelength of a system matches the Schwarzschild radius of its gravitational mass:

λ = Rₛ

A crucial aspect of this approach is the separation into two interacting subsystems:

1. A mass subsystem generating the gravitational scale Rₛ via total mass.

2. A wave subsystem defining λ  via particle motion or collective quantum effects.

This dual-subsystem design makes the DBS resonance potentially accessible at mesoscopic scales in the laboratory, without invoking exotic particles or new physical laws.

The framework is conceptual first, highlighting the symmetry principle. Experimental implementations may include toroidal currents, graphene-coated nanoparticle ensembles, superconducting circuits, 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.

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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.

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Companion and foundational work:

Gravity Resonance (M.P.)

• A1b — Toroid Models for Quantum-Gravitational Resonance via de Broglie–Schwarzschild Symmetry DOI: https://doi.org/10.5281/zenodo.16449277
• A2 — Soft-Matter Toroid for Quantum–Gravitational Resonance: Toward a Tabletop Realization of de Broglie–Schwarzschild Symmetry DOI: https://doi.org/10.5281/zenodo.17541112
• A3 — De Broglie–Kerr Ergosphere Resonance in Rotating Coherent Systems: Listening to the Horizon DOI: https://doi.org/10.5281/zenodo.17594467
• A4 — Imaginary Kerr Horizon in Every Rotating Body Below One Solar Mass: A Universal Quantum-Gravity Boundary DOI: https://doi.org/10.5281/zenodo.17619734
• A5 — Quantum-Gravitational Double-Resonance Propulsion Rings: Spacecraft-Scale Architecture and Physical Basis DOI: https://doi.org/10.5281/zenodo.17634226
• A6 — Hybrid van der Waals Graphene–Hydrogel–Mass-Seed Resonator for Asymmetric Mechanical Response and Electromechanical Output DOI: https://doi.org/10.5281/zenodo.17684823
• A7 — Negative Effective Mass as a Universal Consequence of the Kerr Bound Applied to Ordinary Collective Modes DOI: https://doi.org/10.5281/zenodo.17731385
• A9: The Quantum Vacuum Granule: Three Complementary Mass Scales from the 1--5 Hz Universal Hum https://doi.org/10.5281/zenodo.18888582