Abstract

The charged-lepton pole masses exhibit the well-known Koide relation, in which the square-root geometry reduces the three-mass pattern to a one-angle problem. This paper first gives a self-contained algebraic formulation of that geometry: the Koide condition is equivalent to equality between the democratic and orthogonal components of the charged-lepton root vector, and the resulting spectrum lies on a cone around the democratic direction. The geometric equivalence and the one-angle parameterization are exact. A reconstruction framework is then summarized to motivate a neutral-parent interpretation of the root space. In this framework, premetric equivalence gives equal structural weighting, the Indefinite Reconstruction Stability Principle motivates minimal saturation, and persistent charged readouts are associated with carrier-supported codimension-two holonomy structure. The remaining Koide cone angle is assigned by a phenomenological weak-closure selector constructed from the electron, proton, and neutron masses and the low-energy fine-structure constant. No continuous coefficient is optimized against the muon or tau mass, but the selector was formulated retrospectively rather than selected from a prespecified finite hypothesis class. The retained baseline selector gives a muon mass output of approximately 105.6565 MeV and a tau mass output of approximately 1776.94 MeV, with relative deviations of approximately 0.0017 percent below and 0.00061 percent above the adopted pole-mass values. Accordingly, the numerical agreement is reported descriptively and is not assigned a look-elsewhere-corrected statistical significance. The selector is an ansatz motivated by the reconstruction picture; its full form and coefficients are not yet derived from a complete premetric calculus or matched to a post-readout effective action. Therefore, the result is presented as reproducible pole mass phenomenology and as a concrete target for future formal reconstruction rather than as a derivation of running Standard Model Yukawa couplings.

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