Overview: This manuscript provides a complete, self-contained proof of the Riemann Hypothesis. Departing from traditional analytic methods, we establish this proof by constructing the Vector Descriptor Space ($\Omega$-Space). Within this framework, prime numbers are no longer treated as discrete arithmetic values, but as the exact spectral eigenvalues of a continuous, unitary operator system governed by the symplectic geometry of the Sieve of Eratosthenes. By defining the Metric Operator $M_p$ as a local spectral probe and the Memory Matrix $V_{\text{mem}}$ as a Rank-1 stabilizing projector, we transform the analytic problem of the Zeta zeros into a problem of variational stability. We demonstrate that restricting the operator to the arithmetic bulk subspace induces a strict energy flux balance (Renormalized Coercivity) that forces the state vector to evolve dynamically unless the spectral parameter strictly satisfies $\Re(s) = 1/2$. Furthermore, any deviation from the critical line induces a non-integrable power singularity at the origin (The Hard Wall), topologically excluding unstable states from the Hilbert space. Key Mathematical Innovations: The $\Omega$-Space Manifold: A polynomial Hilbert space that acts as an exact algebraic dual to the integer field, preserving the Unique Factorization property via linear independence (the Vandermonde Barrier). The Rigid Cage Identity: The derivation of the Canonical Commutation Relation $[\Xi, H] = I$ as the fundamental geometric invariant of the prime distribution. Spectral Locking & Transverse Drag: A deterministic mechanism where the "Memory" of the sieve stabilizes specific high-frequency resonances (primes) via exact phase-cancellation, while violently rejecting composite numbers via multi-dimensional Transverse Drag. Resolution of Historic Conjectures (Extended Applications): Because the critical line is maintained by the preservation of symplectic area, the apparent "chaos" of the primes is shown to be a projection of a rigid, higher-dimensional symmetry. This allows the framework to resolve several historic number-theoretic problems: The Diophantine Limit (Appendix K): We derive the famous 26-variable Jones-Sato prime-generating polynomial as the exact discrete lattice projection of the $\Omega$-Manifold's continuous spectral stress function. Odd Perfect Numbers (Appendix F): We prove their non-existence via the "Flux-Parity Barrier," demonstrating that the unique scaling flux of the prime $2$ is a topological requirement for volumetric stability. Mersenne & Sophie Germain Primes (Appendices D & E): These sequences, alongside Twin Primes and Goldbach pairs, are reframed as joint eigenstates and topological necessities of ergodic recurrence on the Bohr Torus. Implications for Cryptography: While the $\Omega$-Theory reformulates integer factorization as a continuous variational optimization problem on a differentiable manifold, it formally proves the physical security of RSA. We demonstrate that evaluating a massive cryptographic composite against a sparse mathematical history yields a perfectly flat "zero-gradient" landscape, establishing that cryptographic hardness is a strict thermodynamic boundary condition enforced by the conservation of geometric information. Subjects: Spectral Geometry, Number Theory, Operator Theory, Functional Analysis, Quantum Chaos, Cryptography. Keywords: Riemann Hypothesis, Spectral Geometry, Operator Theory, Prime Sieve, Heisenberg Algebra, Symplectic Topology, Integer Factorization.