The Grown Sieve Universe: Fine Structure and Three Generations from First Principles

Deriving the fine-structure constant and the lepton generation pattern from first principles is a long-standing challenge for discrete, emergent-spacetime frameworks. Parts I and II of this programme derived spacetime kinematics and the fermion mass spectrum from a growing prime lattice, and sealed two quantitative accounts — the fine-structure constant α and the charged-lepton generation ratios — behind explicitly registered falsification conditions. This paper reports that both resurrection conditions have now been met by direct measurement on the grown sieve (the "h-universe"). Three results carry the weight. First, a measurement ontology is established and verified: the electric charge of a defect is the flux of the elastic re-radiation field through the unit core box, an observable that is strictly linear in the occupancy number (ratio 2.000000 for double occupancy), invariant to six decimal places under far-field medium permutation, and stable to ±0.2 % across all simulated box sizes. Second, the α account closes: with the original calibration chain untouched, the core-box convention gives α −1 = 136.7 against the required 137.036 (0.2 % remainder), while the previously used (2R/Lx ) 3 extrapolation is shown analytically to be a form error — not a size error — that manufactured an apparent 17 % thermodynamic-limit gap. Third, the three lepton generations are re-founded on a screening-polarity spectrum: the midfield flux obeys an exponential law Φ(R) = Φ0e −R/ξ whose sign is set by the asymmetry of the defect's ±2 stiffness clamps, yielding a leaking electron (ξ ≈ 74–116), a neutral locked muon (ξ → ∞), and an anti-screened locked tau whose core flux is exactly 0.7221 of the electron's. The generation mass bridge narrows to a single undetermined ratio ξe /ξμ = 5.748. A shuffled-medium control (identical stiffness multiset, randomized arithmetic assignment) destroys α outright (α −1 = 1.7 and 0.9 in two seeds), closing the "any elastic medium would do" alternative at a joint probability far below 10 −3 . Four falsifiable predictions are registered, and the remaining open items — an a priori derivation of ξ and of the stiffness-to-potential map — are stated plainly. All results are obtained under a strict pre-registration audit paradigm, with no post-hoc parameter tuning.