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

Abstract

Summary. We prove an all-multiplicity discrete-value theorem in the adjacent-two-incoming R12 sector, using an alternating-sign collapse of the canonical ordered-partition recursion. We are not aware of a public statement of this result. We consider half-collinear single-minus Yang-Mills tree amplitudes in a fixed half-collinear frame with two adjacent negative-frequency legs and all remaining legs of positive frequency. Starting from the canonical ordered-partition recursion of Guevara, Lupsasca, Skinner, Strominger and Weil (arXiv:2602.12176), we derive an exact suffix reduction and a sign-gauged triangular recursion. Main result. A nested endpoint-selection rule and a successor sign law imply an all-multiplicity alternating-sign structure for the surviving suffix contributions. Consequently, every suffix preamplitude satisfies Bs ∈ {−1, 0, +1} and the full stripped amplitude satisfies AR12 ∈ {−1, 0, +1} for every n ≥ 4. The theorem holds on open R12 chambers away from the relevant bracket and composite-vertex walls. Verification. The analytic proof has undergone an internal line-by-line audit. Independent exact-rational, Z3 and OR-Tools CP-SAT computations at fixed multiplicity provide post-hoc corroboration; these computations are not used as premises of the all-multiplicity proof. Relation to the previous version. This release substantially extends the R12 suffix-partition note published on 22 February 2026 (DOI: 10.5281/zenodo.18730760), which documented the suffix-selection structure and finite-multiplicity witnesses but did not contain the all-multiplicity theorem or the analytic alternating-sign proof. Scope and status. No claim of absolute priority or of a physical observable is made. The result concerns a restricted half-collinear kinematic sector and remains subject to external peer review.