Quantifier Monitor Game and the Finite Certificate Hierarchy: From Limit Observation to the Σ₂⁰/Π₂⁰ Boundary

Places Round 20's Limit Separation Monitor inside the arithmetical hierarchy: P=NP is equivalent to ∃i∀x R(i,x), a Σ₂⁰-type statement; P≠NP is equivalent to ∀i∃x¬R(i,x), a Π₂⁰-type statement, and the monitor's eventual-stabilization/unbounded-advance dichotomy corresponds exactly to this quantifier swap. Further, by embedding the standard c.e. sets FIN (finite) and INF (infinite), it's proved that for a general monotone computable monitor, the stabilization problem can reach Σ₂⁰-completeness and the unboundedness problem can reach Π₂⁰-completeness — so there is no certificate system, universal across all monitors, of the form “guess a finite witness, verify it with a computable verifier.” But the most important safeguard against misreading this round is: this must never be swapped in for “P/NP has no finite mathematical proof.” P/NP is a fixed proposition, not an index problem of the form “feed in an arbitrary monitor and ask yes/no”; an inductive proof can already cover ∀n in finitely many steps by its very nature. So this round splits finite certificates into three tiers — Prefix Witness, Uniform Mechanical Certificate, and Structural Mathematical Proof — and only the third tier is the genuine opening where P/NP remains open; any argument that claims to cover an infinite obligation with a finite proof owes a Quantifier Compression Debt (QCD), and must account for its generalization mechanism.