A Minimal Arc-Geometric Reconstruction of Single-Electron Double-Slit Interference: From the R = DA Constraint to de Broglie Wavelength and Projection Density

This paper presents a minimal arc-geometric reconstruction of single-electron double-slit interference within the conceptual framework of Arc Theory. The central aim is not to reject the empirical success of standard quantum mechanics, but to show that a nonstandard geometric ontology can be expressed in a traditional mathematical-physical form, with explicit definitions, axioms, propositions, derivations, and stated limitations. Starting from the proposed arc-geometric constraint R = DA, where R denotes the curvature radius of the external zero-weight ring and DA denotes the intrinsic arc-radius of the electronic arc-helix, the paper derives the de Broglie relation lambdaA = h/p through phase closure and action quantization. The double-slit condition d sin(theta) = m lambdaA is then reconstructed as a consequence of arc-branch phase closure under boundary-induced splitting. The paper further formulates the double-slit process in terms of four structural operators: the arc-branching operator SAlpha, the inter-branch coherence kernel SigmaAlphaBeta, the arc-difference field DeltaAlphaBeta, and the measurement-localization operator LambdaD. In this formulation, the Born density is interpreted not as a primitive probabilistic substance, but as the observable projection density of an underlying high-dimensional arc trajectory onto a lower-dimensional detection interface. In compact form, the proposed interpretation may be written as: rhoArc(x) = |A(x)|^2 where A(x) denotes the projected arc amplitude and rhoArc(x) denotes the observable arc-projection density. A central part of the work is the quantitative formulation of topological friction. The proposed self-decoherence rate GammaSelf is expressed as a functional of curvature mismatch, torsion mismatch, and deviation from the R = DA constraint: GammaSelf = Integral over GammaA of [ gammaKappa * (DeltaKappaA)^2 + gammaTau * (DeltaTauA)^2 + gammaR * ((R - DA)/R)^2 ] dmuA This gives a precise geometric meaning to internal self-decoherence and separates it from ordinary external environmental decoherence. The paper also demystifies observational collapse by representing measurement as environment-induced arc localization. In the proposed formalism, the disappearance of interference is described by the suppression of off-diagonal coherence terms: rhoAlphaBeta -> muAlphaBeta * rhoAlphaBeta with muAlphaBeta -> 0 corresponding to complete which-path distinguishability or environmental trace formation. The scope of the paper is deliberately limited. It does not claim to derive the numerical values of Planck's reduced constant, electron mass, electric charge, magnetic moment, or the electron g-factor. Rather, it provides a mathematically disciplined reconstruction of the kinematics of single-electron double-slit interference and proposes a possible geometric foundation beneath the standard probabilistic formalism. The paper is intended as a preprint-level contribution to the foundations of quantum theory, mathematical physics, and alternative geometric models of microscopic phenomena.