VR-Audit: A Programme of Operational Audits of Classical Theorems

This preprint presents VR-Audit, an open-ended programme of applying the two-register apparatus of VR-Forms to classical theorems of mathematics. Each audit establishes an operational corollary of a classical theorem through the transit pattern: classical machinery is used as a black box in the formal register, while operationality of the result is established separately through wrapping predicates and explicit witness construction.

VR-Audit is the fifth work in the VR Cycle, following VR (A Formal System), VR-Numbers, VR-Sets, and VR-Forms. The first four works are foundational; VR-Audit is applied, demonstrating that the apparatus works on classical mathematics with realistic complexity, using mathlib and the predecessor Lean cycles as black-box dependencies.

This document is intended as an accumulating record of the VR-Audit programme. Subsequent audits will be added as new parts within future versions of this preprint, each version superseding the previous as the complete record to date. This avoids fragmentation across many small preprints while keeping each individual audit citable through its part-numbered section and corresponding Lean tag.

Version 1.0.0 contains the first audit: Hahn–Banach for operational Hilbert spaces via Riesz representation.

The theorem states: given an operational Hilbert space E, an operational located subspace M, and an operational normable linear functional f on M, there exists a continuous linear functional g on E that restricts to f on M, has the same operator norm, and is itself operational (its values on E's dense sequence are computable, and its operator norm is computable).

The proof is a demonstration of the VR-Forms transit pattern in pure form. Mathlib's Riesz representation is invoked as a black box in the formal register, yielding a vector ξ ∈ M with classical existential status. The extension g(x) = ⟨x, ξ⟩ is then defined explicitly. Operationality of g is extracted through the orthogonal projection identity: ⟨x, ξ⟩ = ⟨P_M(x), ξ⟩ = f(P_M(x)), where the right-hand side is computable by locatedness of M (making projection operational) and operationality of f. Norm equality is inherited from Riesz isometry.

The Specker boundary — a constructive obstacle where bounded sequences may have non-computable limits — is avoided by structure: Riesz produces an explicit vector, not a supremum; orthogonal projection produces a specific element, not a sequence limit. At no step does the proof invoke constructions vulnerable to the obstacle.

The accompanying Lean 4 formalisation contains 17 public objects across 6 files (~1427 lines), with main theorem HahnBanachOperational_Hilbert. All objects carry axiom profile [propext, Classical.choice, Quot.sound] — the standard mathlib ceiling. No sorry, no admit. The first non-trivial instance is EuclideanSpace ℝ (Fin n) (n-dimensional Euclidean space), demonstrating that the OperationalHilbertSpace typeclass is non-vacuous on a non-trivial infinite family of structures.

Cost comparison: VR-Audit's operational proof is approximately 150 lines (with ~600 lines of cycle infrastructure), contrasting with an estimated 3000–5000 lines for a Bishop-style constructive rewrite, and 2000–4000 lines for the classical proof of Hahn–Banach in mathlib together with its immediate dependencies. The efficiency multiplier is in the 5–25× range. The multiplier does not measure new mathematical content; it measures the reorganisation enabled by the two-register apparatus.

The preprint includes ten methodological observations accumulated during the work on Audit One, candidates for subsequent audits (Banach–Steinhaus, open mapping, closed graph, spectral theorem for self-adjoint compact operators, Stone–Weierstrass), and discussion of the programme's position relative to neighbouring frameworks (Bishop-style constructive analysis, reverse mathematics, computable analysis, Weihrauch reducibility).

Accompanying Lean 4 formalisation: DOI 10.5281/zenodo.20363739.