PAPER 1 in The UAP Gödel Obstruction Series which builds on the prior preprints from the UAP series: Galois-Theoretic Invariants of Paraconsistent Determinization AND Homotopical Apophasis: Bilattice Symmetry, the Progenitor as Eilenberg–MacLane Space, and the Shadow of Classical HoTT This paper isolates the structural barrier between finite presented arithmetic data and theory-attached regime formation. We formalize presentation streams ᴘ: ℕ → Sent (ℒₐ), where the associated theory Th (ᴘ) is the deductive closure of the stream's image. The core of the paper investigates finite-local extractors E: streams → Ob (𝐁𝐑𝐞𝐠). An extractor is finite-local if its output on a stream ᴘ is determined by a finite prefix ᴘ|ɴ. We further require presentation-invariance: if Th (ᴘ) = Th (𝐪), then E (ᴘ) ≅ E (𝐪). Main Results: The Finite Perturbation Barrier: Let 𝕌 be a recursively axiomatized theory and Δ⁺, Δ⁻ be finite sets of sentences. For any finite-local presentation-invariant extractor E, we prove that E𝕌 + Δ⁺ ≅ E𝕌 + Δ⁻. Indistinguishability of Complementary Extensions: As a corollary, for any sentence θ where both 𝕌 + θ and 𝕌 + ¬θ are recursively axiomatized, E𝕌 + θ ≅ E𝕌 + ¬θ. Obstruction Geometry: Since the first obstruction class β₁ is invariant under isomorphism, it follows that β₁ (E𝕌 + Δ⁺) = β₁ (E𝕌 + Δ⁻). Conclusion: The results demonstrate that finite-local extraction is fundamentally too weak to recover theory-sensitive independence geometry. To capture the distinctions necessary for Gӧdel-style obstruction theorems, one must leave the level of presentation prefixes and utilize arithmetically definable extractors sensitive to the Π₁ threshold of consistency. This identifies the transition point in the series from bounded inspection to genuinely theory-attached arithmetic formation.
David Betzer (Sat,) studied this question.