The End of Exhaustive Search: A General Theory of Solving Hard Problems by Certified Transport

Hard problems are rarely blocked by a lack of clever ideas; they are blocked by the inability to make those ideas travel. Exhaustive search, brute enumeration, and ad hoc heuristics fail at scale not only because exploration explodes, but because the “reason” an instance is solved often collapses when the instance is transformed, encoded, reduced, or verified in a different environment. This paper proposes a general alternative: solving by certified transport. Progress is measured in transportable certificates — finite verifiers with explicit obligations — rather than in faster exploration. We present a domain-agnostic discipline for producing such certificates: admissible transformations must be accompanied by replayable evidence artifacts that remain independently checkable under bounded resources. Central to the theory is kappa margin preservation: every step must maintain a quantitative safety gap that prevents a claim from “closing to zero” under transport (as an epsilon regularity threshold in analysis, a correlation decay rate or spectral gap in physics, or a quantitative separation in discrete regimes). When the margin cannot be preserved, the system must abstain rather than overclaim. The proposed framework is compatible with, but logically independent from, the Multitime Receipt Standard (MRS): a normative receipt interface that specifies how artifacts are pinned, replayed deterministically, and contested across environments. Under this view, MRS is an admissibility layer for “what is signable,” while certified transport is the problem-solving thesis for “what is progress.” Together they support honest partial closures, rigorous failure classification, and cumulative advancement without treating exhaustive search as the default closure mode.