The Silence Principle and the Provability of NP ≠ P in ZF

We prove NP ≠ P in ZF. We define a Δ₀ relation Silent (i, p, n): the clocked Turing machine Mᵢ with oracle p runs for n steps without querying any index j ≤ nₚ. We prove in ZF the Silence Principle: ∀e∈ω ∀p∈ℙ ∃n∈ω n > nₚ ∧ (∀i<e) Silent (i, p, n). By iterating this principle we construct an explicit Δ₁ language p∞ ∈ NP such that every clocked polynomial-time machine disagrees with p∞ on some input. Hence ZF ⊢ p∞ ∈ NP ∖ P, and therefore ZF ⊢ NP ≠ P. The construction is finitary, effective, and provably non-relativizing, thereby circumventing the Baker–Gill–Solovay barrier. The separation is Σ₂⁰ and hence absolute for the standard model of arithmetic. *ATTENTION: This version supersedes v1. * Version 1 contained a fatal logical contradiction: it simultaneously claimed ZF ⊢ NP≠P and that P=NP is Pi₂⁰-complete, which would imply ZF is inconsistent. That claim is retracted. This v2 proves only ZF ⊢ NP≠P. All proofs are self-contained in ZF. The core construction is unchanged. *Do not cite v1. Cite this v2. *