Target Note M: Mass, families, and Koide as an internal closure selection

Mass, Families, and Koide as an Internal Closure Selection This note explores whether fermion masses, family replication, and the Koide relation can emerge from the internal closure structure of the reconstruction programme. Particles are modelled as admissible reconstructions on a product space M × F, where F is an internal fibre carrying both a Fisher geometry and a spectral structure. Purpose The central goal is to remove the arbitrariness of Yukawa couplings by replacing them with a variational selection principle acting on an internal Dirac operator. The proposal mirrors the role played by H5 in the external sector: masses are not inputs but selected configurations. Internal Selection Principle The note introduces an internal free-energy functional F = Cint − λGint, where the cost term measures internal informational complexity and the gain term counts admissible closure multiplicities. Unlike a bare minimisation, which would favour a massless state, the free-energy competition generically favours non-zero masses. Mass as Generic Closure The leading-order analysis produces a cusped double-well effective potential. The massless state is unstable, while non-zero mass corresponds to a stable closure configuration. Within this interpretation, the Higgs field becomes the order parameter of internal closure. Family Number as an Index The number of fermion generations is proposed to equal the index of the admissible internal Dirac operator. If established, this would explain the existence of exactly three families and forbid a fourth generation. Hierarchy as Closure Action The note argues that mass hierarchies should not be viewed directly as ratios of masses but as exponential suppressions generated by closure actions. The observed charged-lepton hierarchy is thereby reduced to action differences of only a few natural units. Koide as a Symmetry Problem A major part of the note analyses the Koide relation. It is shown that generic polynomial actions carry no Koide content and that the familiar cosine representation is merely the Z₃ Fourier decomposition of three masses. The physically relevant content of Koide reduces to a single condition: equal contribution of the Z₃-invariant and Z₃-breaking sectors. This corresponds to a 45° angle in generation space and to the observed value Q = 2/3. Negative Results Several natural explanations are tested and found insufficient: independent-sheet dynamics, Coulomb-gas closure ensembles, involution symmetries, larger discrete groups, and Legendre-duality constructions. The analysis identifies a common obstruction: no Z₃-symmetric internal geometry can naturally equate the one-dimensional singlet sector with the two-dimensional doublet sector. Koide as an Internal Light Cone The note introduces a signed form of signature (1,2), whose null cone is exactly the Koide locus. In this language, the observed lepton spectrum lies on an internal light cone. The problem of Koide is reformulated as the question of why the vacuum is lightlike with respect to this internal causal structure. Main Conclusion The note progressively transforms Koide from a numerical coincidence into a geometric and structural question. At the same time, it establishes a sequence of no-go results showing that Koide is not generated by generic local variational dynamics. The remaining viable route is a global or causal selection principle acting beyond the internal Z₃ geometry itself. Status The document combines established structural results, conditional arguments, numerical consistency checks, and explicit open problems. Its strongest contribution is not a derivation of Koide, but a systematic reduction of the problem to a small number of sharply identified structural locks and obstructions.