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

We develop a deterministic arithmetic framework based on a logarithmic band decompositionof the natural numbers, termed the Standing–Sitting Band Framework(SSBF). The framework makes explicit a structural asymmetry between integers capableof initiating multiplicative propagation and those that cannot. This asymmetry isshown to be exact at the atomic arithmetic scale and strictly subcritical at all higherlogarithmic bands.From this structure we construct a canonical self-adjoint operator acting on ℓ2(N),encoding arithmetic adjacency and logarithmic scaling. We prove that no self-adjointrealization of this operator admits compact resolvent, ruling out spectral confinement.Arithmetic information therefore appears only through trace functionals evaluatedagainst Schwartz test functions.A trace identity is derived that coincides exactly with the Weil explicit formula forthe Riemann zeta function, understood distributionally. We then establish a canonicalpositivity property of the associated quadratic form: exact cancellation occurs onlyat the atomic scale, while all higher bands contribute uniformly positive energy. Thispositivity induces non-negativity of the Weil distribution on even Schwartz functions.By the classical Weil positivity criterion in the theory of explicit formulas, thissuffices to confine all distributional contributions to the critical symmetry axis ℜ(s) =12.The critical line thus emerges as a structural consequence of deterministic arithmeticorganization, rather than a conjectural spectral assumption.