Functional Decomposition of the Four Fundamental Interactions: A Minimal Physical Distinction-Operation Closure for Finite Distinction Systems

The Standard Model and general relativity describe four known fundamental interactions: strong, electromagnetic, weak, and gravitational. X3 does not claim to derive the Standard Model gauge group, coupling constants, particle masses, scattering amplitudes, electroweak symmetry breaking, or quantum gravity. Instead, it proposes a functional decomposition of the four known interactions as a minimal physical distinction-operation closure for finite distinction systems. A finite physical system capable of maintaining persistent distinctions must implement four non-equivalent operation classes: hadronic/baryonic encapsulation of stable material tokens, connection and remote detectability among tokens, identity transformation or flavor-changing selective update, and global causal-geometric / stress-energy accounting. In the observed physical world these primary roles are realized respectively by the strong interaction, electromagnetism, the weak interaction, and gravity. The mapping is primary-role, not exclusive-role: real interactions can contribute to multiple physical processes, but each known interaction carries one operation class that the others do not generically replace. The paper separates the functional closure thesis from lower-status gauge-group minimality claims, distinguishes new particles or portals from genuinely new operation primitives, and gives falsification conditions for both. Deterministic normal-form demonstrations illustrate operation coverage, remove-one-operation failure, redundant fifth-mediator comparison, fifth-force classification, and the relation of X3 to nearby FDS physical-bridge papers.