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

Note: there is a newer version 3. https://zenodo.org/records/20795446 Mathematical corrections and updates were made with the help of Claude Opus 4.7. Version 2 (revised 29 March 2026) This updated version clarifies the logical flow and strengthens the distinction between the geometric angular-duality wavelength λ_ad and the collective de Broglie wavelength. We introduce the angular-duality wavelength λ_ad ≡ J/(M c) as a purely geometric, mass-independent scale derived from rotational angular momentum in the Kerr framework. For ordinary orbital motion (J = M v R) this simplifies to λ_ad = v R / c. The Mass Cancellation Theorem shows that λ_ad depends only on geometry and motion, not on the test mass. Every participant in the same rotation — from nanoparticles to macroscopic bodies — shares the same rotational length scale. In the Kerr horizon formula, ordinary (non-collapsed) rotators generically produce complex roots. The magnitude of the imaginary part remains a real, measurable geometric scale set by λ_ad. While this regime is sometimes labelled unphysical in the context of collapsed objects, for extended ordinary matter it signals a change of regime: rotation generates a real curvature scale even when no classical horizon exists. De Broglie’s matter wavelength λ_dB = h/(M v) shares the same dimensional structure as λ_ad but is a distinct quantity. The two lengths are not equal in general. Equality holds only along a specific borderline in (J, v)-space (Jv = hc). This borderline represents an exceptional resonance condition explored across the A-series; it is not claimed as a universal identity. New calculations in Version 2 show that for tungsten nanoparticles (35–200 nm radius) and graphene nanodisks this borderline is reached at laboratory-accessible velocities (approximately 0.06–36 m/s). The quantum/particle estimations are left for possible refinement in a dedicated future article or by a specialized team. The paper maintains a conservative tone throughout and does not claim that material mass becomes negative or that a universal equality exists. It simply records the clean algebraic and geometric facts that emerge when the Kerr spin length is treated as a rotational scale for ordinary matter. Full A-series context and previous versions are available at https://zenodo.org/records/18888582