Toroid Models for Quantum–Gravitational Resonance via de Broglie–Schwarzschild Symmetry (A1 v4)

Note: This is the corrected version A1 v4 (October 2026), superseding earlier versions. It corrects several numbers, tables and figures, adds references to Carr’s earlier work, states the main limitation of the graphene design, and adds a new Section 11 with matching speeds for any mass. Corrections were made with the help of Claude Opus 5.5 (Anthropic).A1 is the first article of the A-series linked here. For early and background models, please refer to the OSF project.DOI for all versions: 10.5281/zenodo.16371537 De Broglie–Schwarzschild (DBS) Symmetry Overview Where the de Broglie wavelength of a system matches its Schwarzschild radius:λ = RₛThis framework separates into two subsystems: a mass subsystem, setting the gravitational scale Rₛ, a wave subsystem, defining λ through collective quantum effects. In a single body the match occurs at the Planck mass near the speed of light, and for heavier bodies at lower speeds. Here we propose shifting the scale 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 differ significantly from single-particle wavelengths, allowing a match even when the 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 use of Rₛ for small masses is common in the literature, from Hawking’s small primordial black holes to the mass–size chart of Carr and Rees. This is a work in progress, together with later directions such as the Compton lock. It is open to further research across quantum physics, nanotechnology and materials science. For collaboration, please contact the originator: [e-mail] ____________________________________________________________________________________ New paper A11: The Compton Lock in a Superconducting Ring - Quantum circulation, Kerr geometry, and the case for a laboratory test New paper A12 The Proton-to-Planck Window – Proposed rotational and superconducting tests of quantum-scale matching - a companion paper to A11 is online with the newest, including reference to DIA released papers on vacuum energy Full repository links available on A9 Zenodo page: The Quantum Vacuum Granule: Three Complementary Mass Scales from the 1--5 Hz Universal Hum https://doi.org/10.5281/zenodo.18888582 Claude Opus 4.7 evaluation of A series and experiments of Ligo Grok 4.3 Review of A-Series Grok note on de Broglie's waves coherence and Cosmic censorship of Penrose