Abstract
<jats:p>(Dated: August 2, 2026) We develop the spectral theory of a catalytic mechanism compiled as a quantum circuit: a propagator at each geometry, a transition operator advancing the stoichiometric register, a non-adiabatic block at each crossing. Its generator is self-adjoint by construction, the principal logarithm of a unitary being anti-Hermitian. On a crossing-free segment at fixed electron number that generator is the molecular propagator’s, so its eigenphases are active-space eigenvalues, not proxies. Stoichiometry then constrains the spectrum: the clock Zm has a unique non-trivial inversion fixed point, and a stable intermediate there has a real eigenphase, located by group structure and not by energetics. The crossings fix the fluctuations: Stückelberg phases break time reversal, putting the eigenphases in the unitary class with sine-kernel correlations, arithmetic-free. For one class of mechanisms, distinguished by a requirement on the register, not the chemistry, and forced rather than fitted, the spectrum proves arithmetic: five exact correspondences to the Riemann zeta function; eigenphases simple unconditionally by Baker–Wüstholz independence of {log p}, where simplicity of the zeros is classically open; and, in the limit, the ordinates of the zeros, whence the Riemann Hypothesis—no zero off the critical line—within the class. The Hilbert–Pólya programme thereby gains an operator exhibited, not sought, and quantum chemistry a compiled mechanism whose spectrum is known in advance—free, not correlated, but exact, where no exact target existed. Limits are stated: the class is narrow, its home a discretised field theory not a metalloenzyme, its ladder a property of the register, not of orbital energies.</jats:p>