Lepton masses as the spectrum of a Z3 flux ring

Lepton Masses as the Spectrum of a Z₃ Flux Ring This exploratory note proposes a boundary-operator model for the charged-lepton and neutrino mass spectra. It is explicitly presented as a speculative development rather than a central pillar of the reconstruction programme. Purpose The note investigates whether the three lepton families can be represented as the three sites of a Z₃ ring, with the square-root masses arising as the eigenvalues of a Hermitian Z₃-circulant mass operator. Main Construction The mass operator is written as a Z₃ flux-ring operator with an on-site term and a nearest-neighbour covering term. Its spectrum automatically takes the Koide cosine form: the cyclic structure explains the three-family pattern, while a flux phase controls the mass splittings. Koide as Equipartition The note shows that the Koide relation is equivalent to an exact equipartition condition: the Hilbert–Schmidt norm of the off-diagonal covering part of the operator equals the norm of the diagonal on-site part. In this form, Koide becomes a self-dual balance between identity and covering contributions rather than a numerical coincidence. Boundary Interpretation The relevant boundary object is identified as the FZZT boundary two-point function. The inter-brane two-point function is not circulant, but the Z₃ orbit amplitudes of a single brane do realise the required circulant structure and can reach the Koide value. However, the boundary two-point function contains the equipartition point without selecting it. Selection Principle Several internal selection candidates are tested and rejected: the metric orthogonality locus, the FZZT saddle condition, Liouville reflection, and Z₃ gauging self-duality. The note then proposes a programme-internal entropic selection mechanism: the on-site/covering split relaxes under the entropy-producing covering flow to maximum entropy, where the two sectors are equipartitioned. This is conditional on interpreting the off-diagonal operator as the covering relation of Article 19. Phenomenological Consequences Granting equipartition, the charged leptons fix a flux phase and reproduce the Koide relation to high precision. Applying the same operator to the neutrino sector with a shifted flux predicts a normal-ordered neutrino spectrum, including a lightest mass of approximately 0.37 meV and a total neutrino mass near 0.059 eV. Status The note separates clearly what is exact, what has been tested negatively, what remains conditional, and what is phenomenological. Its strongest result is the exact rewriting of Koide as a Z₃ flux-ring equipartition condition. Its main open point is the proposed entropic selection of that equipartition.