Recoverability after Knowledge Invalidation: Contingent Safety Budgets and the Limits of Observable Evidence

Knowledge invalidation creates a contingent recovery problem: a staged successor may fail verification while a potentially stale old version remains servable and requires safe isolation. This working preprint characterizes an exact restricted finite frontier for two old identities and two staged successors under a known full-support joint witness-validity law, exact paid verification, individual paid quarantine, and zero unsafe terminal risk. Under a hard worst-path budget K, safe recovery is infeasible below K = 2; V₂ = 0, V₃ = max(u, v) + j, and Vₖ = u + v for K ≥ 4. The result is related to classical first-failure task sequencing but does not claim a new general planning algorithm. Separate synthetic and isolated software diagnostics examine the limits of observable evidence. Independent semantic warrants, real-world all-route safety, and complete lifecycle costs remain unestablished. Research status: Working preprint, not peer reviewed. The mathematical proof has not received independent formal certification. Supporting materials: https://github.com/k3ijo-miyamoto/knowledge-recoverability The public repository provides selected source code, fixed fixtures, and bounded results under their respective licenses. It is not a complete independently reproduced experimental package.