Topological Residue and Curvature Locking in Arc Geometry: A Dimensionless Cun-Tou Residue and Action-Valued Lift for the Reduced Planck Constant and Electron Rest Mass

Arcman

Topological Residue and Curvature Locking inArc Geometry:A Dimensionless Cun-Tou Residue andAction-Valued Lift for the Reduced PlanckConstant and Electron Rest Mass

https://doi.org/10.5281/zenodo.20306798


AbstractA previous minimal arc-geometric reconstruction of single-electron double-slitinterference derived the de Broglie relation and the double-slit interferencecondition from the arc-geometric constraint R = DA, while deliberately leaving two deeper quantities as open foundations: the reduced Planck constant¯h and the electron rest mass me. This paper develops a second-stage reconstruction within Arc Theory. Its central claim is deliberately limited: it doesnot assert that the numerical values of physical constants have already beenindependently predicted; rather, it formulates a controlled geometric representation of the roles played by ¯h and me.First, a dimensionless Cun-Tou residue is introduced as the topological indexassociated with the forced closure of a reciprocal spacetime projection chart.The corresponding theorem establishes a winding-number residue, not the magnitude of ¯h. Second, an action-valued lift is proposed: the dimensionless residueis assigned a physical action scale through an intrinsic arc-tension modulusintegrated over a phase-normalized residue section. In this representation afull 2π closure carries an action residue hA, while the unit-radian residue is¯hA = hA/(2π). The local relation dSA = ¯hAdϕA, used as a bridge in the previous double-slit reconstruction, is thereby reinterpreted as the local differentialform of a global action residue.The paper then defines curvature locking. A stable electronic locking radiusDC is distinguished from the momentum-dependent de Broglie arc-radius DA(p).An intrinsic arc transfer speed cA is introduced and identified with the observedLorentz-invariant speed c only as a correspondence assumption in the presentdraft. With ωC = cA/DC , the locked energy is EC = ¯hAωC , and the rest-massrepresentation becomes mA = ¯hA/(cADC ). If ¯hA, cA, and DC are identified withthe measured reduced Planck constant, the speed of light, and the reducedCompton radius, the standard electron rest-mass relation is recovered. Theconstruction is presented as a candidate geometric ontology and mathematicalrepresentation, not as a completed first-principles numerical derivation of ¯h, c,DC , or me.1Keywords: Arc Theory; Arc Geometry; dimensionless Cun-Tou residue; actionvalued lift; topological residue; reciprocal spacetime projection; phase-normalizedresidue section; arc-tension modulus; curvature locking; reduced Planck constant;electron rest mass; Compton radius; intrinsic transfer speed; topological friction;quantum foundations; geometric ontology.