The Standard Model permits charged-lepton masses through Yukawa couplings but does not derive the observed electron, muon, and tau mass ratios. The Koide charged-lepton relation suggests that the square-root mass vector occupies a special three-component cone, but the relation alone does not explain why the phase in a cyclic three-family parameterization should take a particular value. This paper studies a restricted mathematical and phenomenological question: if a primitive three-family orbit is cyclic and the charged-lepton mass phase is relational rather than one-indexed, what is the natural finite domain on which that phase should be defined? Let F be a primitive three-element orbit under a C3 action. A one-index family datum has domain F and therefore three primitive locations. An orientation-sensitive relational datum compares an ordered source family with an ordered target family and therefore has domain F × F. The ordered-pair phase domain contains nine source-target cells. Under the induced diagonal C3 action, those cells decompose into three orbits representing diagonal, forward-cyclic, and backward-cyclic relations. The paper proves that an orientation-sensitive relational phase on a primitive three-family orbit cannot be faithfully reduced to a one-index domain without losing source-target information. Its natural ordered finite phase domain is therefore F × F, with cardinality nine. This ordered-pair denominator is the principal mathematical result. Under the additional Quantized Dimensional Cell hypothesis that the charged-lepton mass-access sector receives two primitive frequency-recurrence units, the normalized phase coordinate is θℓ = 2/9. The existence of two QDC frequency channels is structurally inherited from the L3F2 cell signature, but their identification with the numerator of the charged-lepton phase remains conditional. When θℓ = 2/9 is inserted into the standard C3-symmetric Koide-cone parameterization, the charged-lepton mass ratios are fixed up to one common radial scale. The resulting values are approximately mμ/me = 206.7703159727 and mτ/me = 3477.4728371046. Relative to representative observed ratios, the muon/electron residual is approximately 0.0020329851 and the tau/electron residual is approximately 0.1075705028. The tau/electron residual is within the scale set by the tau-mass uncertainty, whereas the muon/electron residual is experimentally significant and requires a radiative, mass-convention, phase-correction, or other independently derived explanation. The ordered-pair domain also has a standard field-theoretic interpretation. Before diagonalization, a charged-lepton Yukawa matrix carries ordered left-family and right-family indices and therefore naturally has the domain F × F. A C3-equivariant toy decomposition separates diagonal, forward-cyclic, and backward-cyclic matrix sectors and illustrates how the ordered-pair structure can enter a family-space construction. The paper does not derive a complete Yukawa texture or establish basis-independent physical phase data. The work does not claim a complete derivation of the Standard Model, proof of the QDL or QDC substrate, absolute charged-lepton masses, quark or neutrino masses, radiative stability, renormalization-group invariance, or a complete theory of flavor. It supplies a finite ordered-pair phase-domain theorem, a conditional QDC phase assignment, and a reproducible charged-lepton ratio consequence.
James D. Bourassa (Fri,) studied this question.