Sharp coincidence bounds under a shared anchored clock

Finite event labels derived from a single recording share one clock, so their attainable labels must be chosen jointly. This technical note studies fixed-window coincidence counts with fixed finite nonnegative pair weights over continuous strictly increasing clock maps agreeing with finitely many anchors. For n>=1 events at distinct source coordinates strictly between two consecutive anchors with label duration D, and a fixed closed coincidence window Delta>0, the sharp lower count equals the minimum internal edge weight of a contiguous partition into at most K=min(n,ceil(D/Delta)) nonempty blocks. The sharp upper count is the total candidate-pair weight. Both endpoints are attained by explicit piecewise-linear shared clocks. Standard contiguous-partition dynamic programming supplies an O(K n^2) computation after interval-cost preprocessing. For general finite rational inputs, enumeration of joint coincidence patterns reduces exact extrema to strict and weak difference constraints. An inherited six-event model has sharp bounds [4,9] although each pair separately can be non-coincident. The results concern the declared coordinate model; physical timing and recording-support calibration remain separate inputs.