The Diagnostic in Action — Axiom Calibration of the AdS/CFT Correspondence (Lean 4 Formalization) Paper 41

DESCRIPTION: The holographic dictionary is an axiom-preserving map. We deploy the axiom calibration framework of constructive reverse mathematics (CRM) on the AdS/CFT correspondence — from the Ryu-Takayanagi formula through the FLM quantum correction and the quantum extremal surface (QES) prescription to the island formula for the information paradox. For every computation examined, bulk and boundary carry identical axiom cost. This is the first test of a physical duality for logical consistency at the level of individual computational steps — a structural constraint on AdS/CFT not previously articulated. The Fan Theorem (FT) builds the Platonic extremal surface in the unobservable bulk; the boundary computes the observable entropy without it. Holography projects away the FT cost of geometric existence. No observable prediction exceeds BISH+LPO, extending the ceiling established in Papers 1–40 to the most active area of contemporary theoretical physics. The Lean 4 formalization (955 lines, 8 modules, Mathlib) compiles with 0 errors, 0 warnings, 0 sorry. It contains 12 bridge axioms encoding physics input, 35 theorems (7 genuine machine-checked proofs), a 12-row calibration table with compile-time consistency verification, and a master theorem whose axiom audit is printed by Lean. Key results: - Axiom preservation: the holographic dictionary preserves axiom cost across all 12 calibration entries- Phase transitions are cheap: min (x, y) = (x+y-|x-y|) /2 converts LLPO comparison to BISH arithmetic- Infimum vs. minimizer: the scaffolding separation distinguishes LPO (observable entropy) from FT (bulk surface existence) - Perturbative QES is BISH (Picard-Lindelöf) ; LPO cost appears only at the semiclassical breakdown Archive contents: paper41. pdf (22 pages), paper41. tex (LaTeX source), P41AdSCFT/ (complete Lean 4 project with lake-manifest. json for reproducibility). Build: cd P41AdSCFT && lake exe cache get && lake build Part of the Foundations of Mathematical Physics series (Papers 1–42).