Paper 42 in the Constructive Reverse Mathematics (CRM) of Mathematical Physics series. This paper applies the axiom calibration framework to the cosmological constant problem -- the alleged 10¹20 discrepancy between quantum field theory's "prediction" of vacuum energy and the observed value. The problem decomposes into three logically distinct claims with different constructive status. Claim I (Dissolved): The 10¹20 ultraviolet discrepancy is a regulator-dependent artifact. The unregularized vacuum energy does not converge to a real number at any level of the constructive hierarchy. Different regularization schemes (cutoff, dimensional, zeta-function) produce different finite values. A regulator-dependent quantity has no empirical content within the framework. The CRM analysis provides a formal, machine-checked expression of the post-naturalness position articulated by Martin (2012), Bianchi-Rovelli (2010), and Hossenfelder (2019): the 10¹20 is a property of a calculational choice, not of quantum field theory. Claim II (Reclassified): The "naturalness" argument is a Bayesian prior, not a mathematical derivation. The Hollands-Wald axioms prove that the cosmological constant is a free parameter of the renormalized theory. Naturalness resides outside the BISH/LPO deductive hierarchy. Claim III (Identified, LPO): The genuine constraint is the 55-decimal-place cancellation between the bare cosmological constant and the electroweak/QCD vacuum condensates. Computing the exact interacting condensates requires the thermodynamic limit (Fekete's subadditive lemma, Paper 29), showing that LPO suffices for the fine-tuning equation. The residual fine-tuning is an LPO equality -- logically mundane, at the same level as every thermodynamic limit in physics. The Casimir effect proves the scaffolding diagnostic works: energy differences are BISH-computable and experimentally verified; absolute vacuum energies are regulator-dependent and physically meaningless. Main formalized results: - vacuumₑnergydivergent: Unregularized vacuum energy is unbounded (BISH) - predictionᵣegulatordependent: Two valid schemes yield different values (BISH) - lambdafreeₚarameter: Wald ambiguity -- Lambda is a free parameter (BISH) - casimirbish: Casimir energy difference has computable Cauchy modulus (BISH) - condensateₗpo: Exact interacting condensate exists given LPO (LPO) - rgᵣunningbish: Perturbative RG running via Picard-Lindelof (BISH) - fineₜuningₗpo: Fine-tuning equation holds given LPO (LPO) - cccalibration: 7-part assembly theorem (BISH + LPO) - ccₘaster: Master theorem with full axiom audit Axiom profile (#print axioms ccₘaster): 11 physics bridge axioms + 1 CRM axiom (bmcfromₛubadditive = Fekete equiv LPO, Paper 29) + 3 Lean infrastructure (propext, Classical. choice, Quot. sound). Zero sorry. Zero warnings. Technical details: - Lean 4: leanprover/lean4: v4. 28. 0-rc1- Mathlib4 (pinned via lake-manifest. json) - Build: cd P42CosmologicalConstant && lake build- Status: 0 errors, 0 warnings, 0 sorry- Size: 10 modules, ~830 lines
Paul Chun-Kit Lee (2026) studied this question.