Modal Triplet Theory: Foundations

Corrected eighth edition. This paper gives a functional-analytic foundation for Modal Triplet Theory using a Hilbert bundle, three compatible vertical structures, a joint coherent spectral projector, a stabilization flow, and separate hypotheses for spectral gap, invariance, existence, contraction, truncation, and admissibility. It distinguishes projected time-step fixed points from equilibria, stabilization time from physical time and RG scale, and Hilbert-space positivity from Lorentzian signature. It also adds a typed world-in-world/q79 geometry interface and an exact finite shared differential-line theorem, while keeping Boothby-Wang lens and nil circle bundles mathematically compatible but topologically distinct. Scope boundary. The framework does not automatically select physical spacetime, Lorentzian dynamics, the q79 carrier, or a global continuum intertwiner.