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 not presented as a central pillar of the reconstruction programme, but as a structured satellite result: exact where the algebra closes, negative where internal selection candidates fail, and conditional where a programme-internal selection mechanism is proposed. 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 operator M = α1 + βg + β̄g2. In this reading, the cyclic structure accounts for the three-family pattern, while a compact flux phase controls the mass splittings. Main Exact Results The first exact result is spectral: every Hermitian Z3-circulant operator has eigenvalues of Koide cosine form, √mk = α + 2|β| cos(φ + 2πk/3). Thus the cyclic structure alone fixes the cosine parametrisation of the lepton square-root masses. The second exact result is structural: the Koide relation Q = 2/3 is equivalent to operator equipartition, ||Moff||HS = ||Mdiag||HS, or |β|/α = 1/√2. In this form, Koide is not treated as a numerical accident but as the self-dual balance between the on-site identity contribution and the off-diagonal covering contribution. The value 1/√2 is also the fixed point of the duality ρ → 1/ρ induced by the Z3 Fourier transform, equivalently the geometric mean of the two Fourier-self-dual sectors. Boundary Interpretation The relevant boundary object is identified explicitly as the FZZT boundary two-point function in its double-sine form. The inter-brane two-point function is shown not to be translation-invariant, and therefore not to realise the desired circulant operator directly. However, the Z3 orbit amplitudes of a single brane do realise the required circulant structure and can reach the Koide value. The boundary kernel therefore contains the equipartition point, but does not select it by itself. Eliminated Selection Routes Four natural internal candidates for selecting the Koide point are tested and rejected: the raw FZZT metric orthogonality locus, the FZZT saddle condition P* = 1/(2π), Liouville reflection, and Z3 gauging self-duality. These negative results are part of the note's content: they show that the equipartition point is present in the boundary structure but is not fixed by these simpler mechanisms. Proposed Selection Principle The note then proposes a programme-internal entropic selection mechanism. The mass operator is decomposed into an on-site sector and an off-diagonal covering sector. Within the covering interpretation, this split is binary rather than ternary: g2 = g−1 is the converse of the same covering relation, not an independent backward channel. This binarity follows from the antisymmetry of the causal order ≺ together with Hermiticity. Under the entropy-producing covering flow of Causal Succession by Segments, the binary on-site/covering content relaxes toward maximum entropy. Its stable equilibrium is precisely r = 1, i.e. the Koide equipartition point. This proposed selection is conditional on identifying the off-diagonal part of M with the elementary covering relation and on the associated relaxation model. Phenomenological Consequences Granting equipartition, the charged-lepton sector fixes a flux phase φℓ = 2/9 and reproduces the observed Koide value with high precision. Applying the same operator to the neutrino sector with the shifted flux φν = 2/9 + π/12 predicts a normal-ordered neutrino spectrum, with a lightest mass around 0.37 meV and a total neutrino mass near 0.059 eV. The overall neutrino scale is fixed from the atmospheric splitting; the model predicts the ratios and the resulting spectrum. Status The exact content of the note is the reconstruction of the Koide cosine form from a Hermitian Z3-circulant operator, the equivalence between Koide and Hilbert–Schmidt equipartition, and the identification of the Fourier self-dual structure behind the value 1/√2. The boundary analysis is exact in its identification of the FZZT two-point kernel and negative in its elimination of four selection routes. The proposed entropic selection remains conditional. The flux values 2/9 and π/12 remain phenomenological inputs, and the neutrino prediction is falsifiable as oscillation and cosmological data improve.