Can the Standard Model of particle physics and general relativity be constructed without infinity? This book demonstrates that they can. Starting from the finite sets and exact arithmetic of Bounded Foundations (Volume 2 in the series), Unified Finite Physics constructs every algebraic structure needed for the Standard Model and general relativity on a single finite simplicial complex. The gauge group SU(3) × SU(2) × U(1) is carved from finite matrix spaces. The Lie algebras, Clifford algebras, and spin groups are built with integer structure constants and explicit multiplication tables. The five Standard Model multiplets (fifteen Weyl spinors per generation) are constructed as finite-dimensional representations. Gauge fields are connections assigning group elements to edges. Gravity is encoded in edge lengths with curvature measured by deficit angles. The Yang-Mills action and the Regge action are both derived from algebraic uniqueness conditions, not postulated by analogy with continuum theories. The four sectors - gauge fields, fermions, the Higgs mechanism, and gravity - are assembled into one action on one finite simplicial complex, with one finite partition function. All six anomaly cancellation conditions are verified by exact rational arithmetic. Every observable is a computable ratio of finite sums. No divergences arise. No renormalisation is needed. No infinite-dimensional path integral is defined. The path integral is a finite sum. Part I presents the constructions in plain language accessible to anyone with an interest in the foundations of physics. Part II presents the formal mathematics with full definitions, theorems, proofs, and dependency lines. Appendix A contains the complete dependency map (137 items, 273 edges, zero cycles). Appendix B contains a complete Isabelle/HOL theory - importing only Main, using no library beyond the bounded computational substrate - that verifies every construction with 78 tests collected into a single master check. The theory type-checks in Isabelle 2024/2025. The master check evaluates to true.
