Kerr Geometry and the Angular-Duality Wavelength: Mass Cancellation and Borderline Resonance from Nanoparticle to Galaxy

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________________________________________________________________________________________ Version 4 (revised August 27, 2026) The Kerr spin length a = J/(M c), introduced in 1963 as a parameter of rotating spacetimes, is defined here for any system carrying angular momentum, whether collapsed or not. Here we examine what follows when this length is taken seriously for ordinary rotating matter across all scales. For any body in orbital motion, J = MvR, so a = vR/c. The mass cancels exactly. We call this the Mass Cancellation Theorem: the angular-duality wavelength λad ≡ vR/c depends only on geometry and motion, not on the test mass. Every participant in the same rotation — from a small rotor to a planet — shares this rotational length scale. In Kerr’s horizon formula, any non-collapsed rotating body satisfies a ≫ Mg ≡GM/c2, placing it in the regime where the horizon roots become complex. The magnitude of the imaginary part equals a, providing a geometric scale even when no classical horizon exists. De Broglie’s matter wavelength λdB = h/(Mv) shares the same dimensional structure as λad but is a distinct quantity, equal only on the exceptional borderline Jv = hc. New calculations in this version show that for tungsten nanoparticles (10–200 nm radius) and graphene nanodisks this borderline is reached at laboratory velocities of approximately 0.06–36 m/s. Extending the a parameter to ordinary rotating matter, rather than restricting it to collapsed objects, follows the precedent set by Carter (1968), whose Kerr–Newman treatment of the electron with mass, charge and spin reproduces the measured g = 2, and by Burinskii’s Dirac–Kerr–Newman program placing the electron’s ring structure at the Compton scale. Both treatments operate in the super-extremal regime (a ≫ Mg) with complex-valued horizon roots. The extension considered here is scope-limited to the a parameter itself; no claim is made about the full highermultipole structure of the Kerr metric, which by no-hair is specific to black holes. The present work derives and interprets λad on its own geometric terms, while keeping the deep dimensional kinship with de Broglie clearly visible. Full A-series context and previous versions are available at https://zenodo.org/records/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