Schroeder-Bernstein matching assembly: two injective demand-copy maps yield a copy equivalence satisfying the gate clauses I and II

Fix a level alpha and a radius eps >= 0. Let DX and DY be diagram-like persistence diagrams over the extended-real plane with finite support, with off-diagonal copies (points carrying copy indices below the multiplicity) and diagonal slots. Call an off-diagonal copy DEMANDED when the diagonal cannot absorb it at radius eps (either prominent -- strictly above alpha in both coordinates and not eps-absorbable -- or the high-far QSE bundle). Assume there are injective maps on demanded copies in BOTH directions: F from the demanded X-copies into the Y-copies, and G from the demanded Y-copies into the X-copies, such that (a) every prominent copy maps to a copy eps-close to it in BOTH coordinates, and (b) every copy whose point lies in the strict QSE quadrant maps to a copy eps-close in the birth coordinate -- and if that image lands in the strict quadrant, the closeness upgrades to both coordinates. Then there exists a bijection gamma between ALL X-copies and ALL Y-copies (diagonal slots included) that satisfies the matching gate clauses I and II: quadrant membership and eps-closeness are preserved in both directions. The proof is a Schroeder-Bernstein argument: demanded copies ride the given injections, non-demanded copies are absorbed to eps-close diagonal points, injective natural-number codes separate the sectors by parity, and Mathlib's relational Schroeder-Bernstein theorem converts the two injections into the global bijection. This is the assembly engine of the persistence-clustering interleaving matching: it upgrades one-sided Hall-type copy injections on the demanded sectors to a full matching equivalence of the copy structures. All structures (copies, diagram-likeness, quadrants, demands, closeness, the gate predicate) are the platform's Definitions.