Arithmetic Positivity and Absolute Hodge Classes: Why HR Positivity Does Not Strengthen Deligne's Absoluteness

DRAFT version. This is an advanced draft and remains subject to revision. The paper establishes a boundary result for arithmetic positivity and absolute Hodge classes; it does not claim a proof of the Hodge conjecture or of Deligne's question. We introduce arithmetic positivity conditions (AP1–AP3) inspired by the Hodge–Riemann bilinear relations and prove that APp(X) = AbsHodgep(X): the AP conditions are automatically satisfied by all absolute Hodge classes. This is a negative result — the Hodge–Riemann framework does not strengthen Deligne's absoluteness criterion. Version 1.3 keeps that boundary theorem explicit and updates the bibliography and internal Zenodo references after a source audit; AP', AP'', and AP''' remain framed as special-family provenance filters rather than as new global Hodge proof stages. CHANGELOG Changes in Version 1.3 (May 2026) Bibliography maintenance: RFEP/FST and RH self-citations now use stable Zenodo Concept DOIs and current live Zenodo titles. Bibliography correction: Moriwaki, Bakker--Brunebarbe--Tsimerman, and Bhatt--Morrow--Scholze citation years/keys synchronized with DOI/Crossref/Numdam metadata. German edition cleanup: Legacy transliterations in the German source were converted to real umlauts and the German build now uses T1/UTF-8 encoding. No mathematical claim change: Version 1.3 is a source-audit maintenance release; the AP = AbsHodge boundary result and guardrails remain unchanged. Changes in Version 1.2 (May 2026) Major: Strict-review corrections integrated: finite pullback is restricted to the pullback ample cone or an ample-cone comparison hypothesis; finite pushforward is treated as conjectural rather than proved. Major: AP', AP'', and AP''' added as special-family filters: provenance-defect vector, conditional orbit-rigidity scheme, and anti-advice/uniformity filter. Major: Result classification sharpened: AP = AbsHodge remains the global theorem; AP''/AP''' can only support algebraicity inside separately verified controlled families. Minor: English and German versions synchronized after the strict review; combined bilingual PDF regenerated from the current language PDFs. DE/EN: EN, DE, and combined PDFs reflect the 2026-05-06 local Strict-Review-Build. Changes in Version 1.1 (March 2026) No substantive changes. AP1 sign ((-1)p) already correct. Recompiled with latest cross-references.