ITU Tier 1+ #43: Meta-Mathematics (K_meta)

Tier 1+ Pass-1.5 paper 43 of 45. ITU-derived meta-mathematics on axiom + proof + model + consistency + provability. Mathlib growth + AlphaProof IMO 2024.7.25 + ARC-AGI o3 87.5%. Defines K_meta = -log ρ_meta as the operator-algebraic modular Hamiltonian on H_axiom ⊗ H_proof ⊗ H_model ⊗ H_consistency ⊗ H_provability. K_meta inherits from K_QG via the CLPW 2023 type II crossed-product specialised to this scale. Numerical results. Lean Mathlib 2018-2024: LoC 20K -> 1.1M (55x), theorems 4K -> 220K, exponential doubling 1.11 yr; projected 2030: 2-3M lines + 500K+ theorems. AI math benchmarks 2018-2025: IMO 0->3->7->14->28 (+14 jump 2024 AlphaProof silver), ARC-AGI 0->21->55->87.5% (o3 2024.12.20 crossed human ~85% Chollet AGI threshold), FrontierMath 0->2->15->25-30%. Topics covered. Hilbert 1900 23 problems + Program + Gödel 1931 Incompleteness + Tarski 1933 truth undefinability + Church-Turing 1936 Halting + Gentzen 1936 PA consistency, ZFC + Cohen 1963 forcing CH independent Fields 1966 + Solovay 1970 + Woodin determinacy + Shelah PCF, Friedman 1975 + Simpson 1999 reverse math Big Five + Bourbaki Éléments + Quine NF, Mochizuki 2012 IUT + ABC conjecture contested + Scholze-Stix 2018 + RIMS 2021 + Atiyah RH 2018 rejected + Wiles Fermat 1994/95, Viazovska 2016 sphere packing dim 8 Fields 2022 + Cohn-Kumar-Miller-Radchenko 2017 dim 24 + Hales Kepler Flyspeck Coq+Isabelle + Polymath Tao-Gowers + Polymath 8 Zhang twin primes 2013-14, Eilenberg-Mac Lane 1945 categories + Lawvere ETCS + Grothendieck SGA topos + Lurie 2009 inf-cat + Voevodsky 2009 HoTT univalence + Cubical Coquand-Cohen-Mörtberg 2015, Coq 1989 + Isabelle/HOL 1986 + Agda 2007 + Lean 4 2021 de Moura + Mathlib 1M+ lines 2024.10 + Liquid Tensor Experiment 2020-22 Scholze+Buzzard + PFR Tao 2023.11 in 3 weeks, AlphaProof + AlphaGeometry 2 IMO 2024.7.25 silver + trained on Lean autoformalized + AlphaGeometry 1 2024.1 Nature gold-level + OpenAI Math Olympiad 2024-25, Chollet 2019 ARC-AGI + o3 2024.12.20 87.5% AGI threshold + o1 2024.9.12 + ARC Prize 2024 $1M + Tao+o1 2024.10 'mediocre grad student', FunSearch 2023.12 Nature DeepMind cap-set + AlphaTensor 2022.10 Nature matrix mult 4x5 47 vs Strassen 49 + AlphaCode 2 + FrontierMath Tao 2024.10 25%, ITU Pass-1.5 cross-cutting: 43 papers Lean K_X formalization ($150-300K each, ~$10M+ total Pass-2) + ITU axiom Lean-formalizable via Tomita-Takesaki + CLPW 2023. 45-vertex polytope #43 top couplings: #17 K_QG (0.95), #23 Math (0.95), #22 MathPhys (0.92), #2 AI (0.92), #25 Holo (0.92), #33 Lang (0.85). Ten falsifiable predictions: P_avg=0.56, S/M/W=1/6/3: Mathlib 2M lines 2027 (0.75), AlphaProof IMO gold 2026 (0.65), Logical AI + GPT-5 Mathlib formal 2026 (0.65), ITU axiom Lean-formalized 2027 (0.55), RH LLM-assisted attempt 2030 (0.30), Mochizuki ABC resolution 2030 (0.40). Pass-2 roadmap: ~$2.5M: Mathlib analytics ($1M) + ITU axiom Lean formalization Tomita-Takesaki + CLPW ($500K) + AI math AlphaProof+ARC-AGI+FrontierMath partnerships ($1M).