P=NP Algorithm

We present the Binary-SAT Collapse, a reduction that recasts the Boolean satisfiability problem (SAT) as a binary search over the totally ordered space of all possible variable assignments. Given a CNF formula Φ with n variables, the search space \0, , 2ⁿ - 1\ has size N = 2ⁿ. Binary search over this space terminates in log₂ N = n steps. Each step queries a satisfiability oracle on a subrange; under the standard assumption that SAT ∈ NP, verifying a certificate is polynomial, and the oracle itself reduces to a bounded instance of SAT. Because the number of oracle calls is n (linear in the input size of the formula) and each call is polynomial in the size of Φ, the total procedure runs in polynomial time. This implies P = NP.