Modal Triplet Theory: Quantum Amplitudes from Modal Geometry

We present a complete and rigorous derivation of perturbative quantum field theory amplitudes from Modal Triplet Theory (MTT). Starting from a coherent fixed point sector of modal configuration space, we employ the MTT to quantum field theory projection to obtain a locally covariant algebraic quantum field theory on a globally hyperbolic four dimensional spacetime. Within this framework, causal locality and propagation are established via the Haag Kastler net structure and the time slice axiom. Interactions are constructed using perturbative algebraic quantum field theory, with renormalized time ordered products and the Bogoliubov map providing a state independent definition of interacting observables. Renormalization appears as a finite and local ambiguity constrained by covariance and symmetry, with dimensional regularization, minimal subtraction, and the covariant functional renormalization group recovered as concrete realizations. Standard Feynman rules, loop corrections, and renormalization group flow are derived as consequences of the algebraic construction rather than postulated. Scattering amplitudes are defined precisely and conditionally. Using Haag Ruelle and LSZ theory, amplitudes exist only in regimes admitting asymptotic particle states, such as asymptotically flat or stationary backgrounds. In generic curved or time dependent spacetimes, the same framework yields local or in in observables instead of a global S matrix. All effective couplings, masses, mixing parameters, and thresholds are fixed by bounded overlap integrals and curvature gap data intrinsic to MTT. When evaluated with identical low energy inputs, the resulting amplitudes coincide with those of the Standard Model. The construction provides a logically closed amplitudes layer for Modal Triplet Theory, completing the derivation of perturbative quantum field theory from coherent modal geometry while making explicit its domain of validity.