The Joshi ATS Series under the Bridge-Certificate Standard: A Targeted Substance Audit and an All-Place Rigidity Classification of Effective Normalization Weights (DRAFT)

Abstract

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. This paper reports a targeted substance audit of selected load-bearing internal steps of Kirti Joshi's Arithmetic Teichmüller Space (ATS) series, following the form audit of Part 4 of this series. The method is a proof-spine cartography of the published chain, followed by a line-anchored verification of selected heavily load-bearing own results and a full-text analysis of the normalization mechanism; IUT imports, the existence base, and steps outside the selected sites are not audited. The line-anchored audit verifies the formal bookkeeping at several nodes, isolates reconstructible citation and restriction obligations, and leaves the full-box containment required by the measure lower bound open. Separately, the rigidity theorem of Part 6 yields a classification of exponent realizations under named all-place hypotheses: under the pointwise normalization identity ((5.3.3) of ATS II(½)) the weighted degree and projective height are exactly classical; under the full weighted product formula alone, effective weights are constant across places and leave at most a y-dependent global scalar; for quantities not established to lie in the effective-weight class — including log-volumes and measures — the rigidity theorem gives no conclusion. Because ATS III also uses a notation in which VL may denote only non-archimedean places, the application of the all-place theorem to that passage is conditional on an explicit reconciliation of the place range, which the audited text does not supply. The result is a source-bound non-derivability finding for the height subchain under stated hypotheses — not an error verdict on ATS, and not a conclusion about the measure and log-volume subchain. A response-verification decision tree records the routes by which a future revision or reply could change the findings. Part 5 of the abc Gap Series. The abc Gap Series — series directory (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: Admissibility Tests for Transport Arguments and Globally-Summed Positivity Claims — (planned) 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 — this record 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) 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) Companion (meta) paper: The abc Landscape: Obstructions, Failed Routes, and HCT Diagnostics — DOI 10.5281/zenodo.21916900 (latest version). The landscape paper is the complete memory of all routes; the series parts and capsule papers present selected results as independently citable units. The manuscript is provided in English and German.