Subfields as the Unit of Progress: A Modular Path Through P vs NP

Hard problems fail in two predictable ways: we overfit to tractable corners, or we accept persuasive narratives that cannot be replayed. Our work proposes a receipt-first, replay-first program for P vs NP in which the unit of progress is a subfield: a regime-bounded slice of the problem with locked scope, admissible transforms, invariants, falsifiers, and a solver proofpack. Progress becomes promotable only when accompanied by verifiable witnesses and append-only receipts that support clean-room replay. We formalize a universal budgeted hypothesis (UCCB(k)) that, if satisfied under a locked witness menu, yields a polynomial-time decision route for SAT. We then show why expander-coupled families stress this hypothesis by collapsing separator- and treewidth-style witnesses and by forcing frontier growth that breaks locally verified elimination-style compilation traces. Finally, we map the frontier cleanly: proof-carrying decisions (W6′) re-express the core question as universal polynomial certificate generation, while DRIFT-style proofpacks can provide honest intermediate deliverables (e.g., SAT ∈ BPP) without overclaiming P = NP. The result is a Clay-compatible research loop: it produces publishable negative tracks and transparent conditional frontiers, and it prevents illusory “near-proofs” via explicit stop rules.