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

Abstract

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 a source-bound correction of the mathematical core of the ATS repair programme: a fixed-field, full-place rigidity test and its application to identified linear quantity families. It separates local normalization inputs, coordinate variation, arithmetic domains, and unclassified measure objects. The rigidity theorem has an explicit arithmetic contract: for one fixed number field L, its full standard place family, and every x ∈ L×, real final weights (γv) satisfying the weighted standard product formula must be constant across places. A prescribed j2 on a nonempty place set admits the constant global extension; only unequal final weights are excluded. Finite-fibre redistribution gives the same restriction on effective weights. Under ATS II(½) (5.3.3), the identified degree on L× and projective height on Pn(L) reduce to their classical values. Local height terms are defined on representative tuples and their sum descends by the product formula. Every normalized summand of the source stabilization on L× is then constant, so its supremum has that same value. The consequences remain conditional and typed: membership in the effective-weight class requires a proved linear representation with parameter-independent local terms and a convergent reference sum, not merely dependence on the weights. The full weighted product formula leaves at most a parameter-dependent global scalar; the pointwise normalization identity removes that scalar. Log-volumes, hulls, and measures have neither membership nor nonmembership proved here. Their representation and measure-pinning obligation remain open. The extension to Pn(‾L), ATS III ample-line-bundle heights, and ATS IV log-volume comparisons require separate bridges. Coordinate kernels may differ while their pullbacks agree; no whole source theorem is certified by that observation. The negative conclusion is limited to verified quantity/domain pairs: the fixed-field linear families do not retain place-selective metric variation under the stated product-formula hypotheses. This does not refute the algebraic-extension, line-bundle, or measure apparatus, establish absence of another operative carrier, close the repair programme, or decide abc/IUT. The source restriction on local comparisons in ATS IV §1.5 concerns the particular lower/upper log-volume comparison across two arithmeticoids. Published Part 5 is cited with its conditional R6 positions; the full-chain and P5 obligations remain open. ─────── ✦ ───────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 R18 inventory corrections distinguish unequal final place weights from admissible global extensions and require a linear effective-weight representation with a defined reference sum. Degree, projective height, and stabilization results are pinned to L× and Pn(L); algebraic-extension, line-bundle, field/place, and log-volume bridges remain separate. Tables and conditional cases now distinguish local inputs, coordinates, coefficients, and global quantity families. Measure membership and nonmembership remain unclassified; global closure claims are withdrawn. Published Parts 2 and 5 are cited; Parts 1, 3, 4 and GNC are pinned to checked cited versions. The core rigidity and finite-fibre redistribution proofs are unchanged. English and German manuscripts are synchronized.