P ≠ NP as Binary Limit: Certainty, Probability, and the Unbridgeable Gap

We present a structural argument that P ≠ NP by establishing that the two complexity classes occupy categorically distinct epistemic domains. P operates at the binary limit — a verified solution is either 0 or 1, with zero residual uncertainty. NP operates in the continuous interval (0,1) — any search through an incompletely defined solution space requires at least one assumption about the variable set, and probability is not certainty by definition. Three progressive analogies — the blank Sudoku grid, the unwalked lunar surface, and Escher's infinite staircase — establish why the NP search space cannot be reduced to P certainty. This framework develops from the state-transition approach of Brown (2026) and presents the Binary Limit as the most compressed and fundamental expression of the separation. Standalone structural framework. Supersedes the state-transition framing of 10.5281/zenodo.19323766 as the primary structural account of P ≠ NP separation.