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 Eₙ of exceptional divisors unifies three results. Turing Tower Theorem. Via Shoenfield's Limit Lemma, the n-th exceptional divisor Eₙ corresponds exactly to the n-th Turing jump ∅^ (n): a direction in Eₙ 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 E₁, E₂, and ∪_α E_α respectively.
Thompson H.I. Spencer (Mon,) studied this question.