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

Version 3 (revised June 22, 2026) Mathematical corrections and updates were made with the help of Claude Opus 4.7. 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 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