The NWT Lagrangian: A Three-Field Theory of Particles and Gravity

Papers 6, 8, 13, 14, and 15 derived the Standard Model parameters, the fine-structure constant alpha, the 24-particle hadronic mass spectrum, and the gravitational coupling G from the topology of torus-knot vortex defects in a relativistic condensate. This paper presents the underlying Lagrangian as a single three-field theory. The minimal NWT Lagrangian has three fields -- a complex scalar condensate psi, an abelian gauge field A_mu, and an S^2-valued Skyrme-Faddeev unit field n^a -- tuned to the BPS critical coupling lambda = e^2/2 and Derrick equilibrium kappa = R/a_0 = pi^2. The Lagrangian as written carries one U(1) gauge field; the full SU(3) x SU(2) x U(1) Standard Model gauge content is compatible with the crossing algebra of torus-knot vortex cores (Papers 13-14) and is matched in from a SO(10) UV completion at E_GUT. The derivation is organised into five layers L1-L5: L1 field content; L2 kinetic + gauge + BPS potential, verified to saturate the Bogomolny bound at ~10^-3 %; L3 Skyrme-Faddeev quartic + Hopf theta-term, locking the topological sector to Q_H = p*m; L4 Paper 6 mass formula on 24 particles (1.06% median deviation, with three of four factors derived and the n_q^q link-enhancement factor retained as empirical input); L5 Paper 15 gravitational hierarchy m_e/m_Pl = (8/7)(1+alpha/7) alpha^(21/2) (G to 0.029% of CODATA at NLO). A natural UV completion to a SO(10) gauge theory at E_GUT = 7.41e15 GeV adds 33 heavy gauge bosons and yields three concrete falsifiable predictions: proton lifetime of order 10^35 yr (in the vicinity of Hyper-Kamiokande / DUNE reach), non-MSSM coupling unification at alpha_GUT = 1/40, and -- if the SO(10) -> SM transition is first order -- a stochastic gravitational-wave signature near 7.4 GHz. A first-principles one-loop Casimir calculation on S^3/2I in the BPS trefoil background (Phases 0-5 of an ongoing research programme) produces the O(1) coefficient expected for the Seeley a_2 piece of the effective action: Delta(1/16 pi G) ~ 0.27 m_e^2 in natural units. This result LOCALISES the alpha^(-21) suppression of G: it is not present in the Seeley coefficient and must therefore enter through the Wilson-line amplitude in the matter-to-gauge transition channel, as proposed in Paper 15 Sec 7.2. This is a structural localisation, not a numerical derivation; rigorous evaluation of the Wilson amplitude on the 21-edge K_7 Eulerian circuit is concrete, well-defined, and deferred to a follow-up paper. Reproducibility: all analysis code (L1-L5 verifications, Phase 0-5 one-loop Casimir pipeline on S^3/2I, BPS solver, Spin(7)/Cl(0,7) construction) is available at https://github.com/JimGalasyn/null-worldtube.