One Meta-Theory, Three Clay-Problem Closures

Six Birds Theory has produced three landed Clay-class conditional closures: Navier–Stokes regularity, the Riemann Hypothesis, and P vs NP, each under the SB closure assumption of Foundations I. This paper does not re-prove those substrate results. It extracts their common meta-theoretic form for a reader who treats Six Birds as a typed framework for organizing attempted closures of open conjectures. The account has two complementary halves. The negative half, Foreclosure, studies a typed primary carrier, meaning an attempted proof construction whose type signature targets a designated conjecture, and audits admissible carriers through a finite state space whose coverage steps are recorded as explicit guarded obligations: non-target-equivalent, target-equivalent tautological, bridge-failed, or matrix-element-terminal. The positive half, Recognition-Mode Landing, states a seven-stage route for trace-state-only substrates: instead of deriving the load-bearing source from framework primitives, it imports that source as named recognition under closure formation. The v5 refinement states the joint thesis: derivation routes remain subject to foreclosure, while recognition-mode landings are structurally distinct peer routes under the same closure assumption. A third part, the Calibration Family, provides the typed bookkeeping that makes this comparison uniform across substrates: calibrations, bridges-as-theorems, directed bridge types, coverage geometry, framework-type mismatch, and named functorial gates. Layer-instantiation examples appear throughout: the three substrate validation rows (NS, RH, and PvNP), a thirteen-entry calibration seed catalog of typed detectors, two named coverage holes, and three named functorial gates from the PvNP track. The formal catalog contains seventeen rows: three partial structural derivations, eight primary theorem-grade rows, and six primary obligation rows. The body gives the mathematical proofs or obligation analyses, while Lean serves as the mechanization companion and audit trail. The nine honest obligation axioms are recorded with opacity guards, so their assumptions are explicit rather than silently instantiated; the full validator chain cross-checks the inventory, manifest, and trust base. The scope is deliberately limited: BSD and Hodge are out of scope except as future work or prediction, and no substrate closure is strengthened here to a standard-ZFC unconditional claim.