What question did this study set out to answer?

To explore the relationship between Banach space non-reflexivity and the Weak Limited Principle of Omniscience in constructive mathematics.

February 8, 2026Open Access

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

Key Points

To explore the relationship between Banach space non-reflexivity and the Weak Limited Principle of Omniscience in constructive mathematics.
Calibrated logical strength of non-reflexivity in Banach spaces over Bishop-style constructive mathematics.
Used a classical metatheory to explore implications of the bidual gap and WLPO.
Formalized findings in Lean 4 with no custom axioms or undefined terms.
Documented a multi-agent AI workflow that aided in the exploration and discovery process.
Demonstrated that WLPO is equivalent to the existence of a bidual gap in certain real Banach spaces.
Extracted an Ishihara kernel from the bidual gap and provided insights into intuitionistic implications.
Clarified that the bidual gap can be located precisely at the WLPO in constructive settings.

Abstract

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.

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