The Riemann Hypothesis As A Theorem of Admissible Spectral Realization: A Self-Adjoint, Variational, Weyl-Titchmarsh, Paley-Wiener, and Determinant-Theoretic Resolution through Admissibility Exhaustion Introduction The Riemann Hypothesis has traditionally been approached as a problem of localization. Since Riemann's original memoir in 1859, mathematicians have sought to determine where the nontrivial zeros of the zeta function lie. The work of Hardy, Littlewood, Selberg, Levinson, Conrey, Montgomery, Odlyzko, Katz–Sarnak, Tao, and many others has progressively constrained the location, density, correlation, and statistical behavior of the zeros. Yet despite these advances, the central question has remained unchanged: why should the critical line be unique? Existing approaches largely address where zeros may occur; they do not provide a complete criterion determining which spectral states are capable of realization in the first place. The present work approaches the problem from a different perspective. Rather than treating the nontrivial zeros as isolated analytic points whose locations must be estimated, we interpret them as realized spectral states generated by an admissible self-adjoint geometry. The guiding question is therefore not where a zero lies, but what allows a zero to belong to the spectrum at all. The distinction is subtle but profound. A point may exist formally within the critical strip and yet fail to possess the structure required for realization. Just as a singular point may exist without belonging to a smooth manifold, a candidate spectral state may exist without belonging to a realized spectrum. To investigate this idea, we construct a self-adjoint admissibility operator T=T∗ together with its associated variational structure, Sturm–Liouville realization, Weyl–Titchmarsh geometry, Herglotz spectral measure, Paley–Wiener arithmetic confinement, and spectral determinant. Beginning from Euler–Lagrange stationarity, Rayleigh–Ritz admissibility, and Courant–Fischer spectral selection, we derive the realized eigenstate condition (T−λ)u=0, which serves as the local realization law of the theory. We then show that realized states are equivalently characterized through positive spectral support, λ∈supp⁡(dρ) through arithmetic confinement, S={mlog⁡p: m≥1, p prime}, and through determinant realization, DT(s)=C ξ(s). These structures are unified into a single admissibility hierarchy. The principal mathematical result of the manuscript is the construction of the admissible spectral manifold Madm=R−1(0)∩supp(dρ)∩S∩Gdet, which we prove is the compressed representation of the full realization hierarchy generated by the operator. We further establish an admissibility exhaustion theorem showing that every admissible host for spectral mass is contained within this hierarchy and that no additional admissible realization mechanism exists outside it. The proof then proceeds through two complementary results. First, we establish realization completeness, Z(ξ)⊆Madm, showing that every nontrivial zero belongs to the admissible manifold. Second, we establish residual exclusion, Z(ξ)∩Rres=∅, showing that no nontrivial zero belongs to the residual locus. Together these results establish a complete dichotomy between realized spectral states and unrealized spectral potentials. A zeta zero is a realized spectral state. A residual is an unrealized spectral potential lacking the variational, spectral, arithmetic, and determinant structure required for realization. In this framework, the Riemann Hypothesis is not fundamentally a statement about localization. It is a statement about form. The nontrivial zeros are not isolated analytic points scattered throughout the critical strip. They are realized spectral states inhabiting a geometry capable of supporting them. The proof therefore proceeds not by asking where zeros may occur, but by determining which geometries can host them and proving that every admissible host is exhausted. Once realization is understood, localization follows as a consequence. In this sense, the manuscript returns to the idea from which it began. Form is not decoration imposed upon existence. Form is the condition that permits existence to become real. The same principle that governs the star, the tree, the flower, and the atom appears here in a different language. It appears as admissibility, realization, and geometry. The pages that follow are an attempt to express that principle mathematically.