Frontier Geometry, Cut Sets, Obstruction Cover, and Relative Exhaustion

Addresses the question most often misjudged in long-horizon research: once many routes have been proved, refuted, blocked, or quotiented away, what exactly is the “genuinely still-open remainder,” and when is it legitimate to upgrade that remainder into a claim-level exhaustion result? Defines the active frontier, the quotiented frontier, weighted frontier mass, frontier components, and closure distance; generalizes graph-theoretic cut sets into typed hypergraph cut sets (route cuts, hypothesis cuts, obstruction cuts, bridge cuts, scope cuts, representation cuts, and mixed cuts); and introduces the Certified Cut and the Obstruction Cover. Only when RouteCompleteness, CutCompleteness, ObstructionCoverage, ScopeFidelity, and ParentBridge all hold at once can “every observed route is blocked” be upgraded into parent-problem-level relative exhaustion — and explicitly keeps the four layers Observed / Admissible / Relative Mathematical / Absolute Mathematical exhaustion separate, never collapsing into one another. Also establishes a six-level exhaustion ladder (EXH0–EXH5) and a frontier-reopening geometry.