arXiv:2610.29757v1 Announce Type: new Abstract: Conservatism in distributed stability certification is often treated as a limitation of the certificate itself, motivating richer Lyapunov functions, tighter relaxations, or stronger optimization. But a more fundamental limitation may arise from the reporting interface: once information is compressed, physically different systems can become indistinguishable. If stable and unstable systems produce the same report, no sound method can separate them. This paper asks where this information limit lies and how much additional information is needed to move it. We model information as an information structure that partitions admissible systems into information fibers. The strongest sound certificate for a fiber is determined by its smallest stability margin, yielding a method-independent benchmark. This separates total conservatism into information-induced and method-induced parts and characterizes information order, task sufficiency, and the stability value of additional information. Three network-node studies show that network-spectrum, scalar/eigenvalue-only passivity, and independent-local reports respectively lose network--node alignment, local directional alignment, and cross-node relation. They show that critical missing information is often relational rather than dimensional. We then move from diagnosis to information design. An information-conservatism frontier gives the smallest irreducible conservatism achievable under a prescribed budget, while its inverse gives the minimum information required for a target conservatism. A witness-pair characterization turns this into a weighted covering problem. Besides, a local sensitivity law provides a computable approximation and identifies valuable unresolved quantities. The framework indicates whether conservatism should be reduced by improving the certificate or enriching the information.