Seven Problems, One Language: Executable Subfields as the Unit of Work

Hard problems — most visibly the Clay Millennium Prize Problems (P vs NP; Navier–Stokes existence and smoothness; Yang–Mills existence and mass gap; the Riemann Hypothesis; the Hodge Conjecture; Birch–Swinnerton-Dyer; with Poincaré as the solved control) — rarely stall for lack of conjectures; they stall because validity does not survive transport. Proofs, arguments, numerical findings, and even definitions drift when moved across regimes: assumptions, representations, toolchains, institutions, and time. In the Millennium setting, this drift is not a peripheral inconvenience; it is a central obstruction to cumulative progress, because each problem lives inside a dense ecology of “equivalent” formulations where results must travel across dialects without silently changing meaning, cost, or admissibility. Our work advances a general thesis: progress becomes systematic when hard problems are decomposed into executable subfields — regime-stamped fragments whose admissibility conditions are explicit, whose failure modes are minimized into obstructions, and whose verification surfaces are replayable under declared budgets. We unify a field-stack vocabulary (subfield, field, metafield, overfield, metaoverfield) with a transport-first discipline grounded in drift-bounded re-description and auditable acceptance. The result is a computable program for long-horizon work: claims are promoted by portable closure under admissible movement and contestable receipts, not by narrative completeness or authority.