The Projective Blow-Up of the Probability Simplex: Turing Jump Degrees, a Stationary Fixed-Point Theorem, and Π⁰₂-Completeness

We study the iterated projective blow-up of the probability simplex Δ^(N-1) at its boundary, replacing the seam ∂Δ^(N-1) at each stage with a copy of real projective space RP^(N-2) recording the direction of approach. The resulting tower {E_n} of exceptional divisors unifies three results. Turing Tower Theorem. Via Shoenfield's Limit Lemma, the n-th exceptional divisor E_n corresponds exactly to the n-th Turing jump ∅^(n): a direction in E_n is computable from ∅^(n) but not from ∅^(n-1). The result holds for all N ≥ 2. Stationary Fixed-Point Theorem. A Feller–Markov kernel on Δ^(N-1) and the blow-up geometry are coupled via a wired kernel on Δ^(N-1) × RP^(N-2). The Schauder–Tychonoff theorem yields a stationary measure μ* satisfying the Seam Consistency Condition (SCC) — a chart-overlap compatibility condition on RP^(N-2) — in all overlaps simultaneously; under isotropic refresh, μ* is unique. Its direction marginal is Δ⁰₂. Π⁰₂-Completeness. The SCC is Π⁰₂-complete and is not expressible as a Σ⁰₁ sentence in the language of any single chart of RP^(N-2). The projective space RP^(N-2) is the minimal structure in which the SCC is expressible. An oracle for ∅^(2) is necessary and sufficient to decide chart consistency from within a single chart. An arithmetic appendix identifies the reals, hyperreals, and surreals as instances of the blow-up tower at levels E_1, E_2, and ∪_α E_α respectively.