Quantum Mechanics of Integers

Foundations of the Quantum Mechanics of Integers: An Axiomatic Reformulation of Number Theory via Self-Adjoint Operators and Planck-Scale Regularization. Abstract This paper formally establishes the theoretical and axiomatic foundations for a new independent scientific discipline: the Quantum Mechanics of Integers. For centuries, analytic number theory has operated under the continuous idealization of the real number line (ℝ), generating systemic logical vulnerabilities, unphysical singularities, and undecidable loops due to the assumption of infinite divisibility. We break with this platonic tradition by establishing that the set of integers (ℤ) is an emergent non-local topological manifold M_d ⊂ ℤ³ rigidly governed by the laws of quantum and continuum mechanics. We construct the Arithmetic Fock Space over the discrete Hilbert space ℓ²(M_d), where the infinite sequence of prime numbers functions not as a stochastic sequence of scalars, but as the unique, irreducible orthogonal basis vectors and quantum eigenstates (⟨p_i│p_j⟩ = δ_ij). Within this granular matrix, arithmetic operations are rigorously redefined under a native nomenclature as physical state transitions driven by explicit transfer, injection, and dissipative operators. Multiplicative composition is formalized as multi-particle quantum entanglement via tensor products ⨂|pᵢ^(αᵢ)⟩, where exponents serve as discrete occupation numbers. To prevent non-physical divergences, we introduce the granular Planck Dissipation Limit (ℓ_P) as a structural arithmetic axiom, providing a strict ultraviolet cutoff that guarantees global smoothness (C∞) and complete deterministic control over the grid. Furthermore, by evaluating the algebraic breakdown of multi-dimensional spaces through the lens of geometric frustration, we prove that the infinite-dimensional Fock space undergoes an unconditional dimensional collapse, autoconfining its active rank to exactly three independent spatial variables to preserve metric conservation. This treatise delivers the independent language, syntax, and axiomatic architecture necessary to transition number theory from an abstract symbolic game into a rigorous branch of physical mechanics, establishing that the fundamental properties of integers are the hydrodynamics of the vacuum in its primordial state.