arXiv:2606.13954v2 Announce Type: replace Abstract: Modern distributed systems, ranging from distributed tracing platforms to blockchain oracle networks, frequently rely on processes holding partial observations to collectively certify a global execution state. Under Byzantine failures, a single malicious process withholding or fabricating partial states renders sound verification impossible, as genuine observations remain indistinguishable from fabricated assertions. To overcome this impossibility, we study input overlap x: the minimum number of processes that initially witness each element of a target execution I. We formalize this challenge as the x-reconstruction problem in asynchronous message-passing systems with up to t failures, where the goal is for all correct processes to recover I completely given an input overlap x. We establish tight matching upper and lower bounds for solvability: under crash failures, x-reconstruction is solvable if and only if x >= t + 1; under Byzantine failures, x-reconstruction is solvable if and only if x >= 2t + 1. Finally, we connect these structural bounds to semantic verification. For globally dependent safety properties (such as consensus, linearizability, k-set agreement, and leader election), sound verification under Byzantine faults requires a certificate of size c = t + 1 to guarantee at least one honest report. Consequently, sound distributed verification under Byzantine faults is achievable if and only if the input overlap satisfies x >= 2t + 1.