The Bidual Gap and WLPO: A Constructive Calibration of Banach Space Non-Reflexivity (Lean 4 formalization). (Axiom Calibration Series Paper 2)

We calibrate the logical strength of Banach space non-reflexivity over Bishop-style constructive mathematics (BISH). Writing “bidual gap” for the failure of surjectivity of the canonical embedding JX : X → X∗∗, we show—in a classical metatheory—that over BISH the Weak Limited Principle of Omniscience (WLPO) is equivalent to the existence of a bidual gap for some real Banach space (our BidualGapStrong). The forward implication proceeds by classically extracting an Ishihara kernel from a gap; the kernel-to-WLPO step is intuitionistic. Both directions are formalized in Lean 4 with 0 sorry and 0 custom axioms. We document the multi-AI-agent workflow used to build the formalization—notably, the equivalence was discovered during AI–human mathematical exploration—and discuss observations on agentic AI for constructive reverse mathematics. We also give finite-dimensional surrogates via Ces`aro means (paper-level) and include, in Appendix A, a Stone window sketch (not formalized). These results sharpen the folklore that the bidual gap is “non-constructive” by locating it precisely at WLPO over BISH.