We present a construction of a deterministic Turing machine augmented with an oracle that resolves any instance of 3-SAT — and by Karp reduction, any NP-complete problem — in O (n) time, where n is the number of variables. The key structural insight is that the Boolean assignment space \0, 1\ⁿ forms a complete binary tree of depth n by construction: each variable xᵢ \0, 1\ corresponds to one level of the tree, and each root-to-leaf path encodes a complete assignment. This tree is implicit — it requires no explicit construction. The oracle traverses this tree top-down in exactly n steps, at each level deciding which subtree contains a satisfying assignment. We formalize the exact oracle function, prove its existence and O (n) complexity, and derive the implication P = NP.
Kaoru Aguilera Katayama (2026) studied this question.