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

Abstract

The abc Gap Series:Part 5 — what these bricks can bear DRAFT / work in progress. This versioned working paper reports the current targeted substance audit of selected load-bearing steps of Joshi's Arithmetic Teichmüller Space series. It does not verify the whole ATS chain or decide abc or IUT. The period-map analysis distinguishes raw logarithms u_v=log|x_v| from normalized coordinates z_v=alpha_v u_v. The kernels of sum z_v and sum alpha_v z_v need not agree; translations and the period-image action require their actual membership and descent conditions. The Frobenius restriction route additionally requires a common fixed coordinate space, the actual hyperplane coefficient transformation, nonzero forms and actual non-proportionality, alongside stability and a well-defined period map. Local norm variation alone supplies none of these identifications. Same-field rigidity uses the standard modulus family, including local degree factors, and the full product formula for every element of the same number field at all places, including archimedean ones. It forces constant final effective weights but permits a global j² scaling. Tests on L× at places of L′, freeze nonemptiness, and common field/norm/freeze identifications remain separate obligations. A conditional local inclusion x O_tensor ⊆ x O_A^max ⊆ hull(S) is retained when the actual hull contains S and is stable under the specified maximal order. This does not identify the ATS target hull. With normalized Haar measure and the compatible norm |a|=|N(a)|_p^(c/[E:Q_p]), the scaling is mu_E(a O_E)=|a|^([E:Q_p]/c); the logarithm, tensor, balance, degree and target-volume bridges remain open. The archived ATS III v4 corollary on printed page 128 actually has absolute-value bars. Its displayed assertion is −|LogVol(tildeTheta^I_Mochizuki)|/ell* ≥ −sum_{p,w in V_p^(odd,ss), V_p^(odd,ss) nonempty} log|q_w^(1/(2ell))|_(L′_w). For positive ell and ell*, finite real log-volume, and a defined finite nonempty usual Tate-place sum with 0<|q_w|<1, the left side is nonpositive and the right side positive. This narrow sign inconsistency is distinct from applications in ATS IV and possible repairs. Part 6 v0.3 (DOI 10.5281/zenodo.23093243) supplies the current rigidity and quantity-class reference. A global linear family Q_y=sum beta_y,v q_v requires y-independent q_v, a defined reference Q_ref=sum q_v, and justified convergence. Rigidity yields Q_y=c_y Q_ref; Q_ref=Q_cl is an additional bridge. Pointwise normalization (5.3.3) yields classical degree on L× and height on P^n(L) under its stated identification, without automatic extension to algebraic-closure points or line-bundle heights. Membership and nonmembership of measures and log-volumes in this class remain unknown. ─────── ✦ ───────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 since v0.2 — v0.3 Includes the inventoried intermediate layout, caption, response-table and citation updates. The actual public v0.2 PDFs and the later local R17 sources are bound as separate baselines; an exact original v0.2 TeX source is not asserted. Withdraws the former atomic minus-sign reading and the claim that the printed corollary lacked absolute-value bars, following the narrow archived-page check. Clarifies raw versus normalized coordinates, the period-image action, same-field rigidity, freeze prerequisites and the admissible global j² scaling. Retains the conditional maximal-order module-hull inclusion while making its stability and Haar-normalization hypotheses explicit; ATS volume bridges remain open. Updates the Part 6 consumer to the defined linear quantity family, reference sums, convergence and restricted domains. No new computation, numerical evidence or machine-checked proof is introduced.