Finite-Speed Arithmetic Operators and Structural Rigidity in the Birch–Swinnerton–Dyer Framework

We develop a deterministic arithmetic framework based on a logarithmic band decompositionof the natural numbers, termed the Standing–Sitting Band Framework(SSBF). Within this framework we construct a canonically normalized self-adjoint operatoron ℓ2(N) whose adjacency rules encode finite-speed arithmetic propagation.The associated quadratic form is shown to be nonnegative, with exact cancellationoccurring only on a finite atomic subspace. Uniform coercivity beyond this scaleenforces diagonal rigidity: every ℓ2 kernel element is supported on finitely many logarithmicbands and is uniquely determined by finite data. As a consequence, all local–global obstruction spaces defined intrinsically within the SSBF framework are finitedimensional.When specialized to elliptic curves over Q, discrete second differences of the N´eron–Tate canonical height define bounded kernel elements of the associated SSBF operator.Diagonal exhaustion yields arithmetic closure: every bounded kernel element arisesuniquely from a rational point, and the dimension of the global kernel equals thearithmetic rank.Finite-speed propagation implies compact spectral support, while canonical positivityenforces positive type. Together these properties force a Paley–Wiener extensionand functional equation by classical harmonic analysis. Under this analytic closure, theSSBF trace coincides with the Weil explicit formula, and the arithmetic rank equalsthe analytic order of vanishing at the central point.All arguments are carried out within Zermelo–Fraenkel set theory with the axiomof choice.