Degree-Local BSD Flows on Rational Isogeny Graphs: Valuation Potentials and Edge-Level Compensation in Cremona Isogeny Classes

We study rational prime-isogeny graphs of elliptic curves over Q using a stratified sample of 50,000 Cremona isogeny classes, and examine how Birch–Swinnerton-Dyer invariants redistribute along prime-isogeny edges. The prime-isogeny graphs are overwhelmingly sparse: among 49,763 successfully processed classes, 99.45% have first Betti number zero, and among 21,367 prime-isogeny edges, 99.97% have degree 2 or 3. For a prime-degree isogeny, exact rational factorization of the Sha-free ratio R_I(phi) shows pure l-power support for every edge in the sample, with zero exceptions. Using the database analytic Sha orders, the Sha-inclusive ratio R_J(phi) satisfies the same support condition, and hence provides a BSD-consistency audit rather than an independent determination of Sha. The real period and regulator ratios also reconstruct numerically as pure l-powers, and their exponents satisfy the edge compensation identity on all 21,367 edges. These observations are assembled into a graph-theoretic framework: an l-isogeny carries only l-adic BSD flow, while prime-to-l components are rigid — a behavior compatible with standard isogeny theory. Since the isogeny graphs are almost always trees, BSD variation is naturally described by valuation potentials on vertices rather than by cycle cocycles; the rare cyclic classes serve as consistency checks on the data pipeline. The novelty is not the individual identities, many of which follow from classical isogeny theory, but their systematic assembly into a reproducible, edge-level, graph-theoretic audit over a large Cremona sample.