Quantum Prime Spectral Theory (QPST)

Since Hilbert’s 1910 proposal and Pólya’s later formulation, the Hilbert–Pólya conjecture has remained one of the central open problems in mathematics. It asserts that the non-trivial zeros of the Riemann zeta function should arise as the spectrum of a self–adjoint operator, yet no construction had previously succeeded in providing an explicit Hamiltonian together with a complete and rigid spectral mechanism compatible with the Riemann–Weil explicit formula.

This record collects the first five parts of the Quantum Prime Spectral Theory (QPST) program, which together constitute the first complete, explicit, and structurally closed realization of the Hilbert–Pólya conjecture in the operator–theoretic sense.

The program provides a canonical self–adjoint Hamiltonian whose trace formula reproduces the Riemann–Weil explicit formula and whose intrinsic spectral architecture rigidly identifies the zero–type spectrum associated with the Riemann zeta function. All results are obtained without invoking probabilistic models, random matrix heuristics, or conjectural statistical assumptions, and without assuming the Riemann Hypothesis.

Across these five works, QPST advances from the explicit construction of the canonical Hamiltonian to the structural closure of the Hilbert–Pólya paradigm, culminating in a forced spectral identification derived purely from geometric and operator–theoretic principles.

Contents of this collection

QPST I — A Canonical Spectral Framework for the Hilbert–Pólya Paradigm
Introduces an explicit self–adjoint Hamiltonian combining an arithmetic block encoding prime powers with a canonical Archimedean block encoding the gamma factor. A trace decomposition is established that reproduces the Riemann–Weil explicit formula at a structural level.

QPST II — Archimedean Rigidity and the Weyl Law for the Zero–Type Spectrum
Proves that the zero–type spectrum of the QPST Hamiltonian satisfies a Weyl law with the universal coefficient 1/(2π)1/(2\pi)1/(2π), inherited uniquely from the Archimedean sector. This fixes the mean spectral density associated with the critical line without assuming the Riemann Hypothesis.

QPST III — Structural Origin of the Error Term and Non–Poissonianity
Shows that purely arithmetic contributions to the error term are subdominant and that all fine spectral structure arises from an intrinsic mixed arithmetic–Archimedean channel, structurally excluding Poissonian behavior.

QPST IV — Rigidity of the Zero–Type Spectrum via the Mixed Spectral Channel
Establishes a rigidity theorem: once the mixed arithmetic–Archimedean spectral channel is fixed, the zero–type spectral functional—and, under minimal assumptions, the spectrum itself—is uniquely determined.

QPST V — Canonical Geometric Selection of the Mixed Spectral Channel
Eliminates the last remaining structural freedom by proving that compatibility with the geometry of the critical line and with Archimedean rigidity canonically selects a unique internal delocalization, given by a rotation by π/2\pi/2π/2 up to gauge equivalence. The resulting mixed spectral channel is identified with the classical mixed term of the explicit formula, leading—via rigidity—to spectral identification without additional hypotheses.

Conceptual significance

Taken together, these works establish that:

• the critical line emerges as an intrinsic consequence of unitary spectral geometry and Archimedean normalization;

• the Weyl law and its coefficients are universal and rigid;

• fine spectral correlations are structurally enforced by arithmetic–Archimedean interference;

• once the canonical mixed spectral channel is identified, no residual spectral freedom remains.

In this sense, the QPST program provides the first complete realization of the Hilbert–Pólya conjecture, in the operator–theoretic sense: a canonical self–adjoint Hamiltonian, the correct Weyl law, a structural mechanism for spectral correlations, and a rigid identification of the zero–type spectrum, obtained entirely from internal mathematical principles.

Keywords

Hilbert–Pólya conjecture; Riemann zeta function; spectral theory; explicit formula; Weyl law; operator theory; arithmetic–Archimedean interference; quantum chaos.