What the Product Formula Allows: A Rigidity Theorem and an Effective-Weight Test for the ATS Height Subchain

The abc Gap Series: Part 6 — is the bridge allowed to bend? 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 the mathematical core of the repair programme run against the center of Kirti Joshi's Arithmetic Teichmüller Space (ATS) chain: the chain uses height contributions that depend place-selectively on the deformation parameter y, while a product formula continues to hold. The main result is an unconditional rigidity theorem: if a family of real exponents (λv) satisfies the full standard product formula ∏v |x|vλv = 1 for all x ∈ L×, then λv is constant — for every number field L, independently of class number. A finite-fibre factorization lemma extends this to effective weights, and an exhaustion proposition shows that under the source's own normalization identity ((5.3.3) of ATS II(½)) the y-dependence of the normalization weights and of the underlying absolute values cancel exactly: the weighted degree and the weighted projective height of the source's Definition 8.2.1 reduce, by direct substitution, to their classical y-independent values. The consequences are stated conditionally, in three typed cases: for quantities factoring through effective weights, the product formula leaves at most a y-dependent global scalar, which carries no place-selectivity; for quantities not of this class — in particular log-volumes and measures — the rigidity theorem yields no conclusion, and their status is a separately named measure-pinning obligation. No error claim is made against any statement of the audited sources: the finding is that the audited passages, in the exact versions specified, do not by themselves supply a place-selective metric carrier for the height subchain; structures outside the audited passages were not examined. ─────── ✦ ───────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.