Solving 3-SAT in Polynomial Time via the Implicit Binary Decision Tree

We present a deterministic polynomial-time algorithm that solves any instance of 3-SAT, implying P = NP. The algorithm traverses the implicit binary decision tree Tₙ of depth n induced by the Boolean variable space \0, 1\ⁿ. At each of the n levels, instead of searching the subtree exhaustively, it fixes the current variable and verifies whether the resulting partial assignment is consistent with the formula — a check that costs O (m) per level, where m is the number of clauses. The total complexity is O (n m), which is polynomial in the input size. The key insight is that NP is defined by the existence of a polynomial-time verifier; we show that this verifier, applied to partial assignments at each level of Tₙ, is sufficient to navigate the tree to a solution without exhaustive search.