The Bridge Certificate: A Line-Level Audit Instrument for Inter-Theoretic Transfer Arguments, with Four Applications to the abc Literature (DRAFT)

Abstract

Part 3 of the abc Gap Series — a connected sequence of papers that locate, formalize, verify, and repair the central gap in the Mochizuki/Joshi approaches to the abc conjecture (Part 1: Forensic Diagnosis, DOI 10.5281/zenodo.21899747; tool supplement: DOI 10.5281/zenodo.21924253). Parts appear individually as they are completed. This paper proposes an audit instrument for a recurring dispute shape in arithmetic geometry: passages that move a numerical readout out of one construction into a container shared with another, and then read an inequality off the result. The parties to such disputes agree on almost every formula and disagree about whether a particular passage is an argument. The instrument — a bridge certificate — fixes seven obligations that any admissible passage from a qualitative linkage to a numerical membership statement must discharge, arranges them as a nine-row ledger to be filled line by line against the wording of a primary text, and constrains how findings may be recorded: verdicts are about certificate form, never about mathematical truth; every finding carries a verbatim quotation with a line anchor; gaps are typed rather than graded; every appeal to standard literature is entered in an import ledger; and every claim of the form „this is not in the text“ is reported together with the search terms used. Four applications are given, and they do four different things. Against the proof of one corollary of inter-universal Teichmüller theory the ledger locates an unmet obligation: the decisive membership is stated to follow „formally“ from a pair of declared properties rather than being derived. Against the follow-up paper named as its quantification site, the ledger closes off that site: the computation there is explicit and self-contained at every node it names, but consumes the disputed inequality as an input rather than producing it. Against an independent alternative programme in a different theory the ledger serves as a comparison control and returns a differently shaped gap at a different address. Against two downstream applications it records inheritance: the disputed step is imported in a single sentence and never discussed. Claim discipline: No error in the audited sources is asserted, and no claim on the abc conjecture is made. All verdicts concern certificate form; the instrument is proposed, not validated beyond these four applications. Files: English version (29 pp.), German version (30 pp.), and a combined bilingual PDF.