Arithmetic Band Structure and a Canonical Trace Formula for the Riemann Zeta Function

This paper presents a fully deterministic arithmetic framework based solely on divisibility and logarithmic scale. The natural numbers are partitioned into finite logarithmic bands, and a canonical dichotomy is introduced between integers that act as primitive generators under divisibility and those that arise as composite consequences of earlier generators. The resulting arithmetic structure exhibits an irreversible multiplicative imbalance: exact balance occurs only at the minimal arithmetic scale, while all higher scales are uniformly subcritical, with no mechanism for recovery. This imbalance is encoded via a canonically defined arithmetic adjacency operator acting on ℓ2(N)\ell^2(\mathbb{N})ℓ2(N). The operator is symmetric, locally finite, self-adjointly realizable, and admits no compact resolvent, forcing arithmetic information to appear only through trace regularization against Schwartz test functions. A trace formula is derived directly from arithmetic adjacency and shown to coincide exactly with the classical Euler–Weil explicit formula, understood as an identity of tempered distributions. A canonical positivity property of the associated quadratic form is established: cancellation occurs only at the minimal arithmetic scale, while all higher scales contribute uniformly positive energy. This verifies Weil positivity as a consequence of arithmetic coercivity rather than an assumption. By Weil’s classical equivalence, this positivity rigidly confines all nontrivial zeros of compatible Dirichlet series to the critical symmetry axis ℜ(s)=12\Re(s)=\tfrac12ℜ(s)=21. Applied to the Riemann zeta function, this yields confinement of all nontrivial zeros to the critical line. The paper is fully self-contained, relies only on elementary arithmetic, functional analysis, and operator theory, and is carried out within standard Zermelo–Fraenkel set theory with the axiom of choice. It supersedes earlier exploratory manuscripts by the author and does not rely on prior versions.