We develop moment and prime-trace formulations of the Riemann hypothesis and prove two classes of spectral exclusions. The analytic results use explicitly stated entire-function, moment, and operator hypotheses; a separate TCGS--SEQUENTION application formulates the source-to-readout realization problem. Writing by its entire power series, we establish a self-contained chain between the Riemann hypothesis, positivity of every Hankel matrix generated by , and a positive trace-class Fredholm determinant. The moment formulation is situated in the classical Grommer--Hamburger real-zero theory, with its reciprocal-square indexing made explicit. Its connections to typed constitutive readout and selector covariance are conditional realization statements. A Gram-transfer theorem identifies the exact arithmetic identity that a linearized Extrinsic Constitutive Law must satisfy, and a finite-rank obstruction excludes any single finite protected response sector from representing the full moment tower. An equivalent prime-side resolvent identity is derived in the absolutely convergent half-plane, avoiding any use of the Euler product inside the critical strip. The required counting and heat-trace asymptotics exclude the full spectrum of ordinary fixed-order elliptic operators on smooth compact manifolds. We then examine an explicit exponential-corridor Schr"odinger operator: it is positive and self-adjoint, has a trace-class inverse, and reproduces both leading terms of the zeta-zero counting law. Nevertheless, a large-order Bessel calculation proves that its normalized determinant differs from by a ratio asymptotic to a positive constant times on the real -axis. Allowing both the Morse potential scale and the Dirichlet endpoint to vary does not repair the determinant: the second counting coefficient fixes their spectral combination to , the power prefactor forces , and a nonzero inverse-order coefficient still excludes equality. An explicit complex-order estimate establishes the Whittaker genus-zero determinant normalization. The leading counting term also fixes the kinetic--rate invariant of the full two-exponential Morse family. Constant spectral shifts introduce an unmatched logarithmic inverse-order term. Fixed Robin data change the power match to and can cancel the first inverse-order discrepancy, but a nonzero second-order coefficient excludes the remaining candidate without a numerical normalization premise. Finite numerical moment tests and a counterexample family clarify why symmetry, kernel positivity, and truncated positivity cannot close the argument. Compatible cochains and a shared boundary law motivate a positive Schur-complement response. We distinguish its static determinant from the full spectral pencil, prove finite-to-infinite moment transfer and a trace-tightness estimate, and derive explicit prime-tail and resolvent-trace error bounds for candidate and bounded-family rejection. The Riemann hypothesis is not proved; the remaining step is precisely the construction of a source-derived positive representation satisfying the full arithmetic Gram or prime-trace identity.