The Strong-CP Boundary Endpoint

This paper treats the strong-CP smallness problem as a boundary-admissibility and CP-odd topological-standing problem. The Admissibility And Standing Constraint framework is used as a mathematical constraint formalism, not as an additional QCD model, an axion substitute, a naturalness prior, or an interpretive overlay. Standard QCD supplies the comparison arena: the topological θ-term, anomalous chiral transport, quark-mass determinant phase, topological-sector weighting, and the physical quotient survivor ¯θ = [θ + arg detMq]2π. AASC determines which promotions, boundary loads, selectors, branch claims, and endpoints retain standing in that arena. The dependency direction is kernel-first: any determinate sametarget strong-CP endpoint evaluation already requires reference, standing, admissibility, and irreversibility before any candidate value can claim endpoint role. The first result is the strong-CP quotient ledger: chiral rotations, anomaly/Jacobian bookkeeping, mass-basis changes, CKM rephasings, determinant branches, vacuum relabelings, and scale/scheme transport preserve ¯θ or change target. The second result is the QCD-to-boundary bridge: because GeG is locally a total derivative but globally weights topological sectors, the quotient survivor ¯θ, not the displayed bare angle alone, is the CP-odd topological boundary class subject to standing audit. Generic nonzero classes are exchanged with CP-conjugate sector-weighting classes, while every nonzero class, including the sign-fixed π class, is non-identity topological load. The third result is the selector-burden theorem: nonzero QCD topological boundary load can have primitive fixed-scope standing only if a certified selector package, declared boundary input, positive empirical CP-odd fixing witness, or richer-scope descent supplies the missing standing-bearing source. A displayed θ-term or representative θ-coordinate, a chosen chiral basis, a determinant branch, an anomaly equation, a regulator convention, a θ-vacuum label, or an EDM upper bound does not satisfy that burden. The endpoint is AASC-internal in the precise necessity sense: any non-degenerate regime of claim-bearing endpoint use for the fixed strong-CP target already instantiates the kernel roles that the AASC consequence layer audits. In DSCP,0, no same-scope selector package survives exhaustion. Writing Cκ := {¯θ = κ} for the family of candidate quotient classes, every non-identity class is deleted from the certified primitive survivor set, while C0 is the identity class carrying no nonzero topological boundary load. The selected fixed-scope endpoint is SelDSCP,0 (C0) = 1, C0 = {¯θ = 0}. A nonzero parameterized theory QCD¯θ=ϵ is formally writable, but ϵ ̸= 0 is a declared boundary insertion, witness-driven update, or richer-scope descent rather than an internally selected primitive endpoint. Peccei-Quinn/axion, UV CP/parity, Nelson-Barr, flavor/modular, massless quark, and future nonzero EDM routes are not objections to the theorem; they are exactly the selector, input, or target-update packages the theorem says would be required. In the usual physics idiom, the result says: without new physics, the fixed strong-CP target admits no primitive nonzero ¯θ endpoint. More exactly, absent a certified selector, declared nonzero boundary input, positive empirical fixing witness, or richer-scope source-continuity package, the only candidate class that carries primitive endpoint standing is C0 = {¯θ = 0}.