Can the Existential Quantifier Be Compressed by a Mathematical State Machine?

This is the formal entry into the classical P vs. NP question — but without picking a side right away. This round establishes the core method of the entire series — the dual-hypothesis rehearsal method: build both H_=: P=NP and H_≠: P≠NP to their own strongest versions, and have them attack each other under the same computational model, the same resource-accounting rules, and the same correctness standard. It isolates the existential quantifier ∃w inside a polynomial-time verifier, defines EX_V(x)=⋁_w V(x,w), and asks whether it can be collapsed by a unified, finite, exact, polynomially resource-bounded mathematical state machine. It introduces, for the first time, the full resource vector (description length, construction time, state memory, precision, transition time, decoding time), the variable-by-variable elimination thought experiment, and the three barrier-reviewers of relativization, natural proofs, and algebrization.