The Bridge Certificate Applied to the Joshi ATS Series: A Ledger Audit (DRAFT)

Abstract

The abc Gap Series: Part 4 — inspect the blueprint DRAFT / work in progress. This is a versioned working paper of an ongoing research programme; newer versions supersede older ones. No claim about the truth of the abc conjecture, the correctness of IUT, or the correctness of the ATS series is made. We audit Kirti Joshi's Arithmetic Teichmüller Space (ATS) series — a detailed, publicly available repair programme for the disputed transport step of Mochizuki's Corollary 3.12 — against the bridge-certificate standard of this series (Parts 2–3). The audit is a pre-specified four-row form test (ledger rows R2, R4, R6, R8, fixed in the audit mandate before the test), not a complete certification and not a proof audit. On the historical selected-form questions, the source makes the theta-link scaling law, the assigned j²-compensation site, and the asymmetric Frobenius mode visible. These are narrower address and visibility findings, not current C5 budget, C7 hypothesis-coverage, or complete E8 mode findings. The corrected mapping assigns F-R2a to C4 (dependent on C6), F-R2b and F-R6a to C6, and F-R6b to C7; F-R8 is a triggered E8 diagnostic outside the seven-component crosswalk. A documented normalization convention is partially delivering (I), not a derived cross-container compatibility. The indexed log-volume, hull, measure, and weight obligations remain open. Dated external status notices are recorded separately from the form verdicts and now precede the synthesis. A1 concerns ATS IV's normalization and measure bridge; A2 concerns ATS III's actual tensor-box inclusion and applicable hypotheses. The published Parts 5 and 6 and the corrective normalization supplement v1.2 are cited with explicit limits: the effective-weight classification is conditional and does not imply unconditional F-R6a, global branch closure, or an E2 measure theorem. Corrective scope: our own generic volume paraphrase. The cited ATS III lemma is specific to a tensor-product box with integral constituents and stated weights. With normalized additive Haar measure on Zp, the translated ball U = 1 + pZp contains 1 but has measure 1/p < |1|p = 1. This refutes our unrestricted paraphrase, not Joshi's special lemma, theorem, IUT, or abc. No complete proof audit or new mathematical certification is claimed. ─────── ✦ ───────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. Changes in Version 0.3 (2026-10-02) Corrects the C4/C6/C5/C7 crosswalk, documents normalization as partial I, and distinguishes triggered E8 from the universal core. Separates historical address/visibility predicates from the current quantified budget, hypothesis-anchor and mode-coverage obligations; no retrospective full-certificate score is claimed. Retracts this paper's generic volume paraphrase and replaces it with the cited tensor-product-box hypotheses; a translated-ball counterexample refutes only the unrestricted paraphrase. Separates ATS IV normalization/measure obligations from ATS III actual box containment, and limits effective-weight imports to their stated hypotheses. Updates published series citations, places corrected notices before the synthesis, and synchronizes the English and German manuscripts. Provides two separately checked 13-page PDFs; preserves the previous record and the user-designed series directory. No new mathematical proof certification is asserted.