The Projective Blow-Up of the Probability Simplex: Turing Jump Degrees, a Stationary Fixed-Point Theorem, and Π⁰₂-Completeness
Thompson H.I. Spencer · Zenodo (CERN European Organization for Nuclear Research) · 2026
We study the iterated projective blow-up of the probability simplex Δ^(N−1), replacing its boundary seam at each stage with a copy of ℝP^(N−2) recording the direction of approach. The resulting tower {E_n}_(n≥1) of exceptional divisors yields three results. Via Shoenfield's Limit Lemma, E_n corresponds exactly to the n-th Turing jump ∅^(n) for all N ≥ 2. Coupling the blow-up geometry to a Feller–Markov kernel via a wired product 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, with Δ⁰₂ direction marginal. The SCC is Π⁰₂-complete and inexpressible in any single chart of ℝP^(N−2); an oracle for ∅^(2) is necessary and sufficient to decide chart consistency. An arithmetic appendix identifies the reals, hyperreals, and surreals as tower levels E₁, E₂, and ⋃_α E_α respectively. Submitted May 2026 to The Notre Dame Journal of Formal Logic. Companion papers: Spencer 2026h (Three Structural Results on the Riemann Hypothesis, submitted Algebra & Number Theory, April 2026) and Spencer 2026i (The Riemann Hypothesis as a Necessary Consequence of Self-Consistency, submitted Journal of the Mathematical Society of Japan, March 2026).