Phase Before Probability: Why Stable Route Combination Forces a Compositional Invariant

Phase is forced by cancellation, before probability or physics enters the story. This paper continues a constructive sequence deriving only what is logically forced once incompatible closure criteria induce non-commuting stabilization operations. When distinct stabilization routes matter, a state-only description becomes lossy, but explicit route-tracking produces unbounded descriptive growth (“route explosion”). To summarize non-equivalent routes without erasing forced distinctions, the framework is compelled to introduce a contribution carrier and a combination rule. Regrouping invariance forces associativity; null contributions force an identity element; cancellation adequacy forces inverses, pushing the contribution space toward an abelian group (on the generated representational classes). Purely nonnegative scalar accumulation cannot support cancellation, so a phase-like orientation invariant appears prior to probability assignments, amplitudes, Hilbert space axioms, or physical interpretation. Finally, requiring descriptive boundedness under iterative route concatenation pressures the minimal orientation structure toward a compact (periodic) group. The forward pressure is then explicit: extract a nonnegative invariant magnitude from phase-bearing summaries in a way that is phase-invariant, additive over exclusive outcomes, and stable under composition—setting up the next paper’s derivation of a quadratic (Born-like) invariant.