Cluster assembly: a windowed fiber-count bound for finite birth-point sets (generic, level-free)

Let X and Q be arbitrary types. Fix a real eps >= 0, a birth map bth : X -> R, a finite set s of points, a per-point fiber family fib : X -> Set Q, and a two-parameter windowed supply family supply : R -> R -> Set Q. Assume: (hwin) for every window [L, R], the number of s-points with birth inside the window is at most the cardinality of supply(L, R); (hcone) for every window and every point of s born inside it, the point's fiber lies inside supply(L, R); (hnet) for the full span window whose bounds cover all births of s, the supply is covered by the union of the fibers of s; and (hdisj) whenever two points of s admit a cut level t with bth(x) <= t and t + 2*eps < bth(x'), their fibers are disjoint. Then the cardinality of s is at most the cardinality of the union of all fibers: |s| <= |U_{x in s} fib(x)|. This is a generic combinatorial covering lemma: window counts bound the set through the supply, cones place fibers inside supplies, the net covers the supply by fibers, and the disjointness clause separates fibers across birth gaps; it is the counting skeleton behind Hall-type matching arguments in persistence-based clustering, applied there with bth the birth coordinate, fib the close-copy equivalence classes, and supply the eps-expanded windows of the demand census. All cardinalities are encard of sets; the statement is purely order- and set-theoretic.