**CAT’S Theory Capstone (Nov2025-Mar2026):** Complete zero-parameter derivation of the Standard Model lepton sector from topological self-confinement of EM radiation on $S^3$. **Key Results:** - Charged leptons: muon 0.13%, tau 0.02% PDG2024 (post-QED dressing) - PMNS matrix: all angles $≤1σ$, $delta_{CP}=202.5^circ$ ($0.2σ$) - Neutrino sum: $sum m_ u approx 59.7$ meV (factor 2 below bounds) - Quark charges: $2/3$, $-1/3$ from $T(3,q)$ writhe (no color invented) **Core Math:** - Triadic axiom $mathcal{R}=Psi\timesmathcal{I}\timesmathcal{P} eq0$ → E-L coupling - Mass exponent $n=26/7$ via **4 routes**: spectral zeta (Bar/APS), triadic dims, bosonic $D_mathrm{crit}$, QCD $\beta$ ($b_1/b_0$) - Hopf solitons $T(p,q)$: helicity/dual-gate stable (Cho/Niemi/Witten) **Falsifiables (8):** 1. Normal hierarchy only (DUNE) 2. $m_{ u_e}=0.82$ meV (KATRIN) 3. $delta_{CP}=202.5^circ$ (HyperK) 4. GW proxy 8.37 Hz (LIGO) [... full list in paper] **Full registry** Appendix tracks every claim to proof/DOI. Builds: [10.5281/zenodo.18905443](https://doi.org/10.5281/zenodo.18905443), [10.5281/zenodo.18911645](https://doi.org/10.5281/zenodo.18911645). **Code/CSVs** in repo. hep-ph/arXiv ready.