Dynamic Fermion Mass Spectrum and Standard Model Lagrangian from the HQIV Discrete Octonion Action: Nielsen TUFT Hopf Shells and Inner–Outer Casimir Realization

From two discrete axioms (light-cone combinatorics + informational monogamy)and concrete octonion tables, Lean certifies (zero sorry) a sector-by-sector bridge Shqiv ⇒ Lsm inwhich masses and Yukawa weights follow the inner–outer Casimir chart κ6(ξ, Φ, t) = ηlocal(ξ) γ C2(ξ, Φ, t).At the reference lock-in slice ξlock = 5 (Φ = t = 0), charged leptons, electroweak bosons, neutrinos,and selected hadrons match PDG centrals to ∼ 0.2%, ∼ 0.4%, Σmν ≈ 6.6 × 10−3 eV (∼ 5% of thecosmological cap), and ∼ 1% respectively, without per-flavour fitting.PDG values are comparison layers only. Under the default proton lock-in scale witness,a single local measurement mp = 938.272 MeV fixes the entire dimensionful chart and downstreamcosmological outputs—apparent age ≈ 13.8 Gyr, wall-clock age ≈ 51.2 Gyr, and CMB temperature≈ 2.7255 K—with no simultaneous fit to ηobs, Tcmb, or other observables.Status. Proved on the accessible patch: Hopf shells, κ6 factorisation, five sector equalities, Yukawa linearity, finite SM anomaly-trace cancellation, topological-sector discharge, fibre-torsion subleading sector zeta, light-cone so(8) holonomy with weak/colour non-abelian charts, shell-to-harmonic mode limits, and rapidity-phase Lorentz closure. Not claimed: Nielsen’s external smooth principal-bundle and contact-manifold theorems, continuum path-integral measure choice on a separate C∞ base, or CKM/PMNS unitary discharge (scaffold only). Partial: rotated-frame Haar measure and general discrete-action Poincaré.