Unified Formulation of the Pentadimensional (5D) Space-Time-Matter Framework: Inertial Quantization, Proper-Time Sampling, and Quantum-Gravitational Scales

We present a unified analytical formulation of the Space-Time-Matter framework on a non-compact, smooth five-dimensional manifold M5=M4×Rχ under a test-field approximation (RAB=0). Addressing prior theoretical gaps and operator constraints, we demonstrate that: (1) proper time is formulated via a Positive Operator-Valued Measure (POVM) framework defined over the continuous scattering mass spectrum [mthr,mmax], avoiding Pauli's theorem obstruction; (2) mass generation combines discrete bound states from attractive supersymmetric Pöschl-Teller potentials with an overarching continuous scattering spectrum starting at the threshold mthr=μℓ up to a UV cutoff mmax; (3) proper-time evolution over this band satisfies the generalized Shannon-Whittaker theorem with explicit phase demodulation at the central frequency ωc and orthogonal Nyquist sampling intervals Δτ=2π/B; (4) 5D spinor vielbein reductions induce standard Yukawa coupling structures; and (5) matching quantum fluctuations with the 5D gravitational radius defines the dimensional crossover mass scale M∗=(ℏ2/4G5)1/3. Finally, we discuss scalar sector consistency, radion mass screening conditions, and open foundational questions regarding measurement and the physical interpretation of proper time in a mass continuum.