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

The abc Gap Series: Part 3 — certify the bricks Part 3 of the abc Gap Series — a connected sequence of preprints studying proposed form audits and transfer arguments 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 of its proposed form contract for passages from a qualitative linkage to a numerical membership statement, 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 comparative discrimination case 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 and has no independent method validation from these four partial applications. No completed positive control for obligation 7 is reported. Files: English version (35 pp.) and German version (37 pp.); no combined PDF is included in version 0.3. Corrective scope of version 0.3 The application recipe is a prospective protocol distilled from partial historical cases, rather than a claim that every case executed all hygiene steps. The Case-C synopsis now cites ATS IV v2 section 6.10, Theorem 6.10.1. The LANA equation retains its original notation η^q = η_S^anab (9-1). The LANA crosswalk proposes roles; it does not prove a typed implication to the certificate's order/half-line condition. A volume-equivalence quotient does not establish equality of concrete pilot classes. Choice of S, object and quotient representatives, normalized evaluation, orientation and the required comparison bound remain open. The seven certificate obligations are a proposed specification, with no proved universal necessity for every mathematical proof. Case D reports a directed downstream import chain; no dependency cycle, method-effectiveness result or ATS contradiction is established. Deutsche Kurzfassung: Das Rezept ist prospektiv; historische Teilanwendungen werden nicht als vollständige Durchführung ausgegeben. ATS-Abschnitt und LANA-Notation sind berichtigt. Die getypte Objekt-, Auswertungs- und Halbgeradenbrücke bleibt offen. Die Korrektur ist keine Beweis-, Theorie- oder Gesamtpaper-Abnahme. ─────── ✦ ───────seriesdirectory (each part lists the whole series; planned parts are updated with title and DOI upon publication): ✔ Part 1: The Gap in IUTchIII, Corollary 3.12 — DOI 10.5281/zenodo.19960781 (latest version)✔ Part 2: The Bridge-Contract Criterion: A Form Audit for Transport Arguments, with an Instantiation in the abc Literature — DOI 10.5281/zenodo.21960476 (latest version) ☞ Part 3: The Bridge Certificate — DOI 10.5281/zenodo.21925803 (latest version) ↳ Part 4: The Bridge Certificate Applied to the Joshi ATS Series: A Ledger Audit — DOI 10.5281/zenodo.21951155 (latest version) Part 5: The Joshi ATS Series under the Bridge-Certificate Standard: A Targeted Substance Audit and an All-Place Rigidity Classification of Effective Normalization Weights — DOI 10.5281/zenodo.21956398 (latest version) Part 6: Repairing the ATS Chain? A Rigidity Theorem for Product-Formula Weights and an Effective-Weight Test for the Height Subchain — DOI 10.5281/zenodo.21952486 (latest version) ↳ coming soon - Part 7: The bridge certificate applied to our own programme (self-application; working title) — (planned; appears in sync with the landscape paper) ✎ Tool supplement: The Global Normalization Contract Lemma — DOI 10.5281/zenodo.21924253 (latest version) ℹ Companion (meta) paper: From Landscape to Atlas: Multi-Route Cartography of an Ongoing Expedition Toward the abc Conjecture (formerly: The abc Landscape) — DOI 10.5281/zenodo.21916900 (latest version). The atlas paper is the complete memory of all routes; the series parts and capsule papers present selected results as independently citable units. ─────── ✦ ───────changelog Changes in Version 0.3 (October 2026) Distinguishes the prospective recipe from partial historical case execution; preserves open hygiene rows and records the chronology. Corrects the Case-C synopsis to ATS IV v2 section 6.10/Theorem 6.10.1 and restores the original LANA eta notation in equation (9-1). Limits the LANA crosswalk to proposed roles: no quotient-based equality or proved typed order implication; required object/evaluation/bound premises remain open. Limits the seven obligations to the proposed specification; no universal proof-necessity, C7 positive-control validation, dependency-cycle finding or mathematical refutation is claimed. Synchronizes the corrective EN/DE text and 38 labels; retains the prepared numbered tables and synopses, checks all 72 PDF pages and German Unicode text. Publishes separate English and German PDFs and a self-contained GitHub source bundle with the unchanged disclosure include. No new combined PDF, empirical result or theorem promotion. Version 0.2 changes (2026-08-16): This version aligns the paper with the finalized methodology of Part 2 of the series (The Bridge-Contract Criterion, DOI 10.5281/zenodo.21960476), which appeared after v0.1. Under Part 2's hardened obligation-6 semantics (a common numerical evaluation alone never discharges the compatibility obligation, and a stipulated invariance is not a proof of one), two published codings are explicitly revised rather than silently corrected: in Case A, obligation 6 moves from discharged to not discharged (the gap sits at obligations 6 and 7), and in Case C, obligation 6 moves from discharged to partly discharged (compatibility holds only between the sums and is place-wise disclaimed by the source). Further alignment changes: the R6 row is split into R6a/R6b, the paper's ledger is scoped and labelled BC-Ledger v0, R8 is clarified as a triggered extension, Case C is relabelled as a comparative discrimination case, and the LANA mapping is extended. A ``Changes in version 0.2'' section in the paper records each recoding with its reasoning; the central finding of v0.1 (the decisive membership obligation is not discharged at the audited step) is unchanged. From this version the record carries the English and German manuscripts as separate files, without a combined PDF.