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

We introduce arithmetic positivity conditions (AP1–AP3) inspired by the Hodge–Riemann bilinear relations and prove that AP^p(X) = AbsHodge^p(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 1.5 preserves that boundary theorem, keeps the v1.4 reviewer guardrails and citation maintenance, adds the disclosure-placement maintenance update, and keeps AP', AP'', and AP''' framed as special-family provenance filters rather than as new global Hodge proof stages.

CHANGELOG

Changes in Version 1.5 (July 2026)

• Disclosure placement: the AI/KI disclosure was moved from the post-abstract frontmatter to the final substantive paragraph after the bibliography in both language versions.
• PDF synchronization: English and German PDFs were rebuilt from the current TeX sources, and the bilingual combined PDF was regenerated from the final language PDFs.
• Live-description flow preserved: this local file contains only the v1.5 changes block; the full Description is to be fetched from the latest live Zenodo record and merged by paper_publisher.py.
• Status boundary: this is a layout and publication-hygiene maintenance candidate. It does not add a new Hodge theorem, does not upgrade the AP/absolute-Hodge boundary result, and does not close the open source-bound cycle and matched negative-control gates.

Changes in Version 1.4 (June 2026)

• Proof/paper synchronization: AP3 language for general real ample classes now uses Hodge-type positivity rather than literal cycle-intersection language; the Kunneth passage is restricted to primitive product-polarization sign tests.
• Reviewer guardrails: constructive Hodge strategies are no longer dismissed globally; the Picard/Kunneth example is limited by an explicit Neron-Severi comparison hypothesis; JD-Hodge now requires a cycle-realization gate; CDK is stated as a theorem on Hodge loci, not algebraicity loci.
• Status guardrails: finite pushforward evidence is now explicitly a non-proof remark; conditional orbit-rigidity is labeled as a template; test-family controls are marked as planned candidates until a real ledger row exists.
• Scale and erosion controls: D_scale, Reef_E, negative controls and unknown-status handling are recorded as follow-up ledger gates rather than as completed closure claims.
• Bibliography maintenance: RFEP/FST and RH self-citations now use stable Zenodo Concept DOIs with the latest live titles and visible versions checked on 2026-06-02.
• German/searchability cleanup: the German source and PDF build use Latin Modern plus ToUnicode mappings, improving extraction of umlauts, ß and ligatures from the PDF text layer.
• PDF design cleanup: English and German frontmatter are separated from the first normal section; the combined bilingual PDF was rebuilt from the final language PDFs with robust metadata.

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.