The Formal Foundations of Closure-Space Mathematics

CSM's first-version formal foundations. The core question: after a long-horizon mathematics research program has accumulated a large body of propositions, hypotheses, proof attempts, counterexamples, obstructions, and bridges, can it be organized into a verifiable, replayable, updatable relative-global mathematical space that gives a typed determination of which regions are closed, which remain open, and which are only locally obstructed? Absorbs, but is not equivalent to, two prior internal theory lines (LSI-PSD, UCT/UGC-CUR), elevating them into a new research object: closure space itself. Three core principles: Observed Proof Space ≠ Admissible Proof Space ≠ Mathematical Reality; Route Closure ≠ Theorem Proof; and the Globality Typing Principle (any claim of being “global” must state which axis it is global along). Defines the typed closure-space object, closure debt, reopening, relative-global closure grade, route-completeness certificate, and the first-version data model for the NS Relative-Global Closure Space. Research status: a theoretical framework, a system of definitions, formal propositions, and a plan for later proofs — not a completed proof of any open mathematical problem.