This paper develops a deterministic foundation for quantum measurement based on the Hidden Deterministic Interpretation (HDI), which avoids probabilistic postulates and wavefunction collapse at the level of individual particles. HDI introduces two principles: the Phase Consistency Criterion, which fixes the global phase of the Heisenberg-picture wavefunction via initial conditions, and the Quantum Hamilton's Principle, which selects the actual trajectory by extremizing accumulated quantum phase. Applied to an ensemble of particles, this framework yields detection statistics that reproduce the Born rule without assuming it. We show that the squared amplitudes associated with measurement outcomes follow directly from orthogonal projections governed by the Pythagorean theorem. As a consequence, the (L_2) norm of Hilbert space emerges naturally---not as an imposed structure, but as a mathematically inevitable result of the deterministic phase dynamics. Thus, both Born's rule and the underlying geometry of quantum state space are derived from the first principles, offering a coherent, deterministic, and geometrically grounded alternative to the standard axioms of quantum mechanics.