An All-Multiplicity R12 Discrete-Value Theorem for Half-Collinear Single-Minus Yang-Mills Amplitudes

We prove an all-multiplicity discrete-value theorem for the fixed-frame adjacent-two-negative-frequency R12 sector of half-collinear single-minus Yang-Mills amplitudes. Away from the walls of the canonical ordered-partition recursion, the stripped tree amplitude satisfies A ∈ {-1,0,+1} for every multiplicity n ≥ 4. The proof proceeds through an exact triangular suffix reduction, a diagonal sign gauge, nested endpoint-selection rules, and exact alternation of active contributions. The final active set also provides a necessary-and-sufficient recursive criterion distinguishing the three values. The n=3 case is treated separately and follows directly from the imported recursion. This version substantially revises and strengthens the preprint released on 11 August 2026. It adds a more complete fixed-frame formulation, an explicit local derivation of positive-block vanishing, the separate n=3 treatment, the recursive value characterization, an exact off-R12 counterexample with A=+5, expanded literature positioning, and the final disclosure and authorship statements.