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

Whether fermion masses, family replication, and the Koide relation can emerge from the internal closure structure of the reconstruction programme — and, where they cannot be forced, a systematic reduction of the obstruction to a small set of sharply identified structural locks. Status markers: E established, C conditional, N numerical consistency, O open. Setup and the question. A particle is modelled as an admissible reconstruction on a product space M × F, with F an internal fibre carrying a Fisher geometry and a spectral structure. The gauge group is the internal isometry group Isom(F, gF, μF), equivalently the automorphisms of the internal spectral triple. Masses are the eigenvalues of an internal Dirac operator D. The question is whether D, the family number, and the Koide pattern are inputs or selected configurations. The selection principle: free energy, not bare cost. D is not a free Yukawa input but is selected variationally — just as the external metric is selected rather than posited — by minimising the programme’s own maximum-caliber free energy F = Cint − λ Gint, the cost Cint measuring internal informational complexity and the gain Gint counting admissible closure multiplicities. Unlike a bare minimisation, which would favour a massless state, the free-energy competition generically favours non-zero masses. E (the selection sector is the right category to predict) Piece 1 — the vacuum: mass is generic. The leading expansion is a cusped double well, Veff(φ) = ½ IF φ² − λ g1 |φ|, so the massless point is always unstable and a non-zero mass is the stable closure. The Higgs field is the order parameter of internal closure. C Piece 2 — families as an index. The number of generations is the index of the admissible internal Dirac operator, predicting exactly three and forbidding a fourth. O Piece 3 — the hierarchy as a closure action. Mass hierarchies are read not as bare ratios but as exponential suppressions in a closure action, mn ∝ exp(−Sn); the charged-lepton ratios 1:207:3477 thereby reduce to action gaps of only about 5–8 nats. O Piece 4 — the pattern: Koide as a Z3 constraint. The familiar cosine representation is merely the Z3 Fourier basis of the three masses, hence vacuous on its own E; Koide cannot reside in the form, only in the amplitude ratio. The content splits in two. L1: cyclic Z3 invariance from three equivalent sheets fixes the singlet ⊕ doublet form and leaves the phase δ a flat modulus C — now pinned geometrically to the lens η-invariant, |η(S³/Z3)| = 2/9 C. L2: the value 2/3 itself is equivalent to equal singlet and doublet contributions (a 45° angle between √m and (1,1,1), B = √2 A), which the measured leptons satisfy to 0.001% (B/A = 1.41420 vs √2, Q = 0.666661). N The single lock and the demarcation. The four pieces reduce to a two-part lock: (L1) does sheet-equivalence make the closure action cyclically Z3-invariant? (L2) does the closure free energy minimise at the self-dual point B = √2 A? L1 yields the form and the flat phase but not the value; L2 is the whole of Koide. O The genuine excess content — statable without the individual masses — is the family integer 3, the Koide sum rule, and the neutrino sum rule. Absolute Yukawas are not predicted: the mechanism fixes ratios, structure, and the integer, while the overall scale rides on one caliber temperature and the internal Fisher scale. A standing guard-rail (re-description risk): replacing Yukawas by closure actions gains nothing unless Cint, Gint, and Sn are the programme’s canonical functionals, not chosen, and the Z3 form is derived from the fibre, not posited. The self-dual characterisation, and why the involution route is obstructed. Koide is the equality of the singlet and doublet norms — the signature of a fixed point of an amplitude duality D:(a,b) ↔ (b,a), whose fixed set is exactly Koide; a duality-invariant free energy with a unique interior minimum lands on it E. But the programme’s leading functionals are not D-invariant: the Fisher cost penalises the doublet while the caliber gain rewards the singlet, giving an asymmetric balance b² ∝ a, the wrong power. The precise open requirement is therefore a Kramers–Wannier-type self-duality of the closure whose critical point is 45° C/O. The realisation by an internal involution is blocked outright: no (anti)linear involution of the generation space R³ can have the Koide cone as fixed set, because it would have to equate the norm of the one-dimensional singlet with that of the two-dimensional doublet — a dimension obstruction E; the grading γ (linear) and the conjugation J (antilinear) both fail (J acts only on the doublet phase δ, and J-reality of a real-mass vector is vacuous) E; and larger groups (A4) place their distinguished directions at Q = 1/3 and Q = 1, bracketing but missing 45° E. The internal lightcone: the obstruction lifts. The correct object is not a map but a signed form: the Koide locus is the null cone of B = Ps − Pd of signature (1,2), since ⟨v, Bv⟩ = ∥Psv∥² − ∥Pdv∥² = ∥v∥² cos 2α, null iff α = 45°. A (1,2) form has a genuine two-dimensional null cone, so the dimension obstruction — which forbids involutions — does not apply to forms. Read as an internal Lorentzian metric (singlet timelike, doublet spacelike), Koide states that the √m vacuum is internally lightlike (null to about nine parts in a million), with the phase δ its celestial-circle coordinate and Z3 acting as rotation by 2π/3. The cone is plausibly not added structure but the degeneration locus of the programme’s own description-to-reconstruction quotient; the cone half-angle is 45° iff the two sectors are equally weighted, which is the Koide condition itself, so “45°” is equivalent to Koide, not a derivation. E/O The variational no-go. No local, Markovian, variational Z3 closure forces Q = 2/3: under additivity the spectrum is degenerate at Q = 1/3, and otherwise Q(t) = 1/3 + t²/6 sweeps smoothly through 2/3 only at the tuned value t = √2, a codimension-one locus with no stability change and no symmetry enhancement. A forced Koide must come from non-variational structure. E The surviving routes. Two non-variational routes remain, both immune to the no-go O: (1) that the internal (1,2) form is the shadow of the external causal order ≺ — Koide as the causal lightcone — the programme-native place to seek “why null”; and (2) that an admissible reconstruction is the Legendre-self-dual, minimal-sufficiency fixed point (unit weight, c = 1) of the description-to-reconstruction quotient, which is Legendre-structured by Kähler reduction (a single Hessian / dually-flat structure). The closure action Sn is to be taken from the WKB closure cost of the segment structure of Causal Succession by Segments: Tunnel Effect, Superposition, and Measurement in the Reconstructive Order. An honest caveat: Q is not renormalisation-group-invariant (a pole-mass relation), which disfavours a purely local origin. E The decisive computation (in order of priority): (1) why is the vacuum null with respect to B — sought in the causal layer’s lightcone, not the free energy; (2) cyclic Z3 invariance of the closure action from the index-3 operator’s three equivalent sheets; (3) Sn from the segment structure, checking the 5–8 nat gaps; (4) the dominant power of Gint near zero (linear vs quadratic), deciding cusped vs smooth breaking, and whether λ is forced. Status ledger. Claim Status Selection sector is the right category to predict E F = Cint − λGint is the maximum-caliber functional E Cusped double well; mass generic; Higgs = order parameter C Fermion/gauge dichotomy; no massless fermions C Families = index D = 3 (forbids a fourth) O Hierarchy mn ∝ exp(−Sn), gaps ~5–8 nats O Quadratic action carries no Koide content N Cosine form is the Z3 Fourier basis (vacuous alone) E L1: cyclic Z3 invariance ⇒ form + flat δ C Flat phase pinned: δ = |η(S³/Z3)| = 2/9 C L2: Koide ⇔ B = √2 A (equal singlet/doublet) N Generic free energy (sheets / Coulomb gas) gives Q ≠ 2/3 N Koide = self-dual fixed point of D: a ↔ b E γ (linear) and J (antilinear) fail to realise D E Dimension obstruction: no R³ involution fixes the Koide cone E A4: distinguished points at Q = 1/3, 1 bracket but miss 2/3 E Koide = null cone of signed (1,2) form B = Ps − Pd E No-go: variational closure never forces 2/3 E Candidate: Koide = Legendre-self-dual point (c = 1) of the quotient O Q not RG-invariant (pole-mass) disfavours the local route E L2 core: why is the vacuum null w.r.t. B (causal lightcone?) O Measured Koide Q = 0.666661 (B/A = √2 to 0.001%) N Absolute Yukawas not predicted