We present Relational Mathematical Realism (RMR), a discrete framework in which physicalreality emerges from relational updates on a 137-element integer registry. Five structural primitives tetrahedral vacuum geometry, registry partition, graph resonance dynamics, sector mapping, and overflow-fission mechanics — are stated axiomatically and used to derive eight quantitative predictions with no tunable continuous parameters: three charged lepton mass ratios, one neutrino mass-squared splitting ratio, one baryon mass ratio, and three strange baryon mass predictions. All eight agree with experiment to better than 1.1%. The central finding is that the integer set {137, 136, 17, 3} — the registry total, its substrate factor, the substrate prime from 136 = 8 × 17,and the K3 graph order — appears without modification across every sector. We employ an honestthree-tier classification: five predictions [T1] follow strictly from the primitives; three [T2] require two structural postulates whose numerical content is fixed by the same integers but whose derivation from the base primitives remains open. The Gell-Mann—Okubo baryon octet relation emerges as a structural identity. We catalog eleven open questions by tractability and identify JUNO and DUNE as the decisive near-term falsification tests via the integer prediction R ≡ Δm2 32/Δm2 21 = 33.