Modal Triplet Theory: From MTT to Quantum Mechanics

This paper derives nonrelativistic quantum mechanics from Modal Triplet Theory (MTT) without additional postulates. Starting from the 10-dimensional modal geometry and the coherent fixed-point sector, we define an observable map by fibre integration after joint harmonic projection and construct the reduced Hilbert structure. The observable Hamiltonian is obtained from a closed, semibounded quadratic form and the Friedrichs representation, which yields a self-adjoint operator Hobs and unitary evolution by Stone’s theorem. A constructive result shows that for a broad Kato–Rellich class every Schrödinger operator H = −ħ²Δ/2m + V is realized exactly as Hobs by suitable fixed MTT data. Measurement theory follows from modal re-coherence: exponential weights and a functional-equation argument, combined with Gleason–Busch, give the Born rule; positive operator-valued measures (POVMs) arise from an explicit Naimark/Stinespring dilation implemented within the coherent sector. We treat time-dependent Hamiltonians using Kato propagators, uncertainty relations via compressed generators, the semiclassical kernel via stationary phase leading to the Van Vleck–Gutzwiller form, and open systems through the Davies weak-coupling limit yielding the GKLS/Lindblad equation. Worked examples—including the free particle, harmonic oscillator, Landau levels, square well, and parametric oscillator—illustrate the reconstruction and kernels. All operator-theoretic steps are stated with precise hypotheses, making the derivation self-contained and referee-ready.