{
  "date": "2026-11-05",
  "scope": "Cited sources reused from the local library. Earlier search and reading scopes are retained; the 6 November proof-method attribution pass adds precise reading scopes for Lai-Yu, Kumar-Tosi and Lai-Sprang. No third-party PDF is redistributed.",
  "sources": [
    {
      "citation_key": "H12",
      "title": "Riemann P-scheme, monodromy and diophantine approximations",
      "authors": "Marc Huttner",
      "version": "Publisher version, Indagationes Mathematicae 23 (2012), 522-546",
      "source_url": "https://doi.org/10.1016/j.indag.2012.03.002",
      "local_pdf_filename": "Riemann-P-scheme-monodromy-and-Diophantine-approximations_Huttner-2012.pdf",
      "pdf_sha256": "6c8fd5bca3a3fc3e4cd3db4f393633c1fd428552c35a1913a462e897e0f39377",
      "retrieved_at_utc": "2026-11-04T00:29:36.842387+00:00",
      "reread_on": "2026-11-05",
      "actual_scope_this_pass": "Reread end of Section 7 and Remark 5 (printed pp.545-546). The remark states the same shifted-block conclusion as Sorokin. Earlier complete relevant statement readings retained; full technical proofs and dependencies not independently audited.",
      "historical_scope": "Introduction pp.522-525; Theorem 1 and opening proof pp.529-530; Section 5.1 construction and Theorem 2 statement pp.537-539; Section 6.0.1 and all Section 7 pp.542-546. Printed p.545 rendered and visually checked. Theorem 3 excludes z=1 and z=-1. Remark 5 states a Sorokin-type shifted-block result with an unspecified starting threshold. Full technical proofs in Sections 3-6 and imported dependencies not audited.",
      "unread_or_unaudited_dependencies": "Direct statement and hypothesis comparison, not independent proof verification. Section 7 explicitly gives arithmetic consequences without proofs. Earlier Huttner (2006), Riemann/Chudnovsky inputs and other technical dependencies not fully audited here. No exhaustive optimization of this construction.",
      "reused_existing_pdf": true,
      "role": "Method attribution and statement comparison; not an imported proof proposition.",
      "section_1_1_revision_read_scope": "Remark 5 in the local source text."
    },
    {
      "citation_key": "K26",
      "title": "One of beta 4 beta 6 through beta 36 is irrational",
      "authors": "Shawn Krisman",
      "version": "2610.00285v2",
      "source_url": "https://hexagonmath.org/pdf/2610.00285v2",
      "local_pdf_filename": "One-of-beta-4-beta-6-through-beta-36-is-irrational_krisman-even-beta-2026.pdf",
      "pdf_sha256": "4ac2ff67f77637f70fe9db664773d1d45795acd7fbd605714ab751b64b95d65b",
      "retrieved_at_utc": "2026-10-30T17:18:01.250398+00:00",
      "reread_on": "2026-11-05",
      "actual_scope_this_pass": "Reread introduction and method, partial fractions and derivative expansion (Section 3), and grouped denominator argument (Section 4.3, Lemma 4.6). Kernel/Gram extension developed independently here. Prior complete 34-page reading retained in historical record; Lean/certificate build not rerun.",
      "historical_scope": "All 34 pages read, including denominator proof, real-axis estimate, saddle-point nonvanishing, and formalisation scope. Three large finite certificates are described in the paper but their full data live in the cited repository; not independently checked or built here.",
      "unread_or_unaudited_dependencies": "Finite certificate data and claimed Lean build not independently reproduced here.",
      "reused_existing_pdf": true,
      "role": "Method attribution and statement comparison; not an imported proof proposition.",
      "section_1_1_revision_read_scope": "Theorem 1.1 in the local source text."
    },
    {
      "citation_key": "S07",
      "title": "On the Zudilin-Rivoal theorem - Russian original",
      "authors": "V. N. Sorokin",
      "version": null,
      "source_url": "https://www.mathnet.ru/php/getFT.phtml?jrnid=mzm&paperid=3742&what=fullt",
      "local_pdf_filename": "On-the-Zudilin-Rivoal-theorem-Russian-original_sorokin-beta-2007.pdf",
      "pdf_sha256": "a1d4353f0e5e03b0f7dfb550b3fe65fc229ea3e0a3a1719480dbf51bd2987914",
      "retrieved_at_utc": "2026-10-30T17:20:37.490390+00:00",
      "reread_on": "2026-11-05",
      "actual_scope_this_pass": "Reread introduction and Theorem S (printed pp.912-913), and the shifted construction, Assertions 1-5 and proof of Theorem S (printed pp.920-922). This review compares the stated shifted conclusion and checks its parameter dependence. Historical all-12-page reading retained; imported normality/Gelfond inputs not independently audited.",
      "historical_scope": "All 12 PDF pages of Russian original, including Mellin/Rodrigues construction, symmetry, denominators and asymptotic argument. Imported generalized Nikishin/Angelesco normality and Gelfond two-log theorem not audited.",
      "unread_or_unaudited_dependencies": "Imported generalized Nikishin/Angelesco normality and Gelfond input not independently audited. Numerical specializations take the published construction and nonvanishing as input.",
      "reused_existing_pdf": true,
      "role": "Method attribution and statement comparison; not an imported proof proposition.",
      "section_1_1_revision_read_scope": "Theorem S in the local source text."
    },
    {
      "citation_key": "Tcount",
      "title": "Explicit lower bounds for the number of irrational even beta values",
      "authors": "twinheterodox",
      "version": "Manuscript of 4 November 2026, current 20-page exposition",
      "source_url": "https://github.com/twinheterodox/beta4/tree/main/even-beta-counts",
      "repository_commit": "06bd732c81d48125d619ea2e9bf514878d774a86",
      "local_pdf_location": "../even-beta-counts/output/pdf/even-beta-counts.pdf",
      "pdf_sha256": "f6848fa9304693f1b96a12ca9c2268511f350eb7c3e607054ddb0d4d3fe0c2a5",
      "actual_read_scope": "Main theorem, construction, Gram, arithmetic, decay, dilation, contradiction and endpoint sections read during the shifted-interval investigation; full adaptation independently audited in its plain-text form. Cited for construction provenance and comparison; the present proof is self-contained apart from standard classical inputs.",
      "bibliographic_link_note": "The v2 bibliography uses the confirmed Hexagon 2611.00139v1 publication. The earlier GitHub location and source hash remain recorded above.",
      "section_1_1_revision_read_scope": "Introduction, finite count table, general endpoint theorem and Section 8 endpoints.",
      "method_attribution_read_scope": "Introduction method paragraph and Sections 2, 3, 4, 6 and 7 checked against the shifted construction: rational functions, homogeneous polynomial multipliers, Gram representation, grouped denominator cancellation, prime dilation and final elimination. Section numbers verified against the source assembly order.",
      "published_version": {
        "version": "Hexagon 2611.00139v1",
        "source_url": "https://hexagonmath.org/2611.00139v1",
        "pdf_url": "https://hexagonmath.org/pdf/2611.00139v1",
        "publication_date": "2026-11-06",
        "retrieved_at_utc": "2026-11-06T18:27:12.675733+00:00",
        "local_pdf_filename": "Explicit-lower-bounds-for-the-number-of-irrational-even-beta-values_hexagon-2611.00139v1.pdf",
        "pdf_sha256": "126ac901b8d8fee1af67fa37b5d99a611cf90550f7a0325c55b97c6d379a3862",
        "comparison": "Extracted PDF text agrees with the previously read canonical 20-page manuscript after removing the added Hexagon identifier stamp and normalizing whitespace.",
        "reading_scope": "Reused the argument and comparison reading recorded above. Checked publication metadata and full text identity apart from the identifier stamp; no fresh proof audit."
      }
    },
    {
      "citation_key": "LZ22",
      "title": "At least two of zeta(5), zeta(7), ..., zeta(35) are irrational",
      "authors": "Li Lai; Li Zhou",
      "version": "2103.00904v2",
      "source_url": "https://arxiv.org/abs/2103.00904",
      "local_pdf_filename": "At-least-two-of-zeta-5-zeta-7-zeta-35-are-irrational_2103.00904v2.pdf",
      "pdf_sha256": "33c0d91dd1209fbf767be9c94c09c305124be3a1d6773a1341f1a01d22ad0369",
      "retrieved_at_utc": "2026-10-27T21:54:42.713348+00:00",
      "actual_read_scope": "Section 6 and references, pp.12-14. Other zeta arguments not reviewed.",
      "priority_read_scope": "Opening introduction and finite-beta result screened again; prior reading of Section 6 and its proof/references, pp.12-14, retained. Used as the beta(2) through beta(10) baseline.",
      "unread_or_unaudited_dependencies": "Other zeta arguments and all imported arithmetic inputs not independently audited.",
      "reread_on": "2026-11-06",
      "actual_scope_this_pass": "Theorem 6.1 and opening proof reread on 6 November 2026. Earlier reading of Section 6 and its references retained. Used only for the initial beta(2) through beta(10) comparison.",
      "reused_existing_pdf": true,
      "role": "Historical comparison; not an imported proposition in this proof."
    },
    {
      "citation_key": "OAI26",
      "title": "Catalan's constant is irrational",
      "authors": "OpenAI",
      "version": "24 September 2026; PDF unchanged at Git commit 4eaee87548250b2682affda7d05d947c05554bae, checked 3 November 2026",
      "source_url": "https://raw.githubusercontent.com/openai/math/4eaee87548250b2682affda7d05d947c05554bae/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf",
      "local_pdf_filename": "Catalans-constant-is-irrational-OpenAI_openai-catalan-2026.pdf",
      "pdf_sha256": "4023d0344abceb79dc46a16380dac3a662193f9d4f23e6c340b2162068b04c0e",
      "retrieved_at_utc": "2026-10-30T17:11:44.759270+00:00",
      "actual_read_scope": "All 44 pages previously read, including finite-place elimination, Frobenius nonvanishing, Hardy-space interpolation, uniform energy bound and rational certificates. For this introduction update, reread the definition of G=beta(2), Theorem 1.1 and the closing proof in Section 8; visually inspected p.1. Reused the archived PDF and verified its hash.",
      "formalization_scope_read": "Official lean/docs/005.md states that the formalization proves irrationality of the defining alternating series. ComparatorChallenges/Catalan.lean states that same real-number assertion; it is a comparator interface, not the proof module. These statements were read for attribution; no Lean build was performed here.",
      "formalization_scope_url": "https://raw.githubusercontent.com/openai/math/main/lean/docs/005.md",
      "unread_or_unaudited_dependencies": "The rational certificate recipes and imported facts were read but not independently audited or replayed in this update. No new proof audit or formal-verification claim is made.",
      "reread_on": "2026-11-06",
      "actual_scope_this_pass": "Definition G=beta(2), Theorem 1.1 and historical comparison reread on 6 November 2026. Earlier 44-page reading and pinned-version check retained. No fresh complete audit or Lean build.",
      "reused_existing_pdf": true,
      "role": "Historical comparison; not an imported proposition in this proof."
    },
    {
      "citation_key": "LY20",
      "title": "A note on the number of irrational odd zeta values",
      "authors": "Li Lai; Pin Yu",
      "version": "1911.08458v2",
      "source_url": "https://arxiv.org/abs/1911.08458v2",
      "local_pdf_filename": "A-note-on-the-number-of-irrational-odd-zeta-values_Lai-Yu_1911.08458v2.pdf",
      "pdf_sha256": "4df3e0fed2d792d72f55007bb5d5e04495d672de79167765577c07513d224878",
      "retrieved_at_utc": "2026-11-02T21:11:39.835237+00:00",
      "priority_read_scope": "Introduction refreshed for construction/attribution. Earlier repair reading of arithmetic Propositions 3.1-3.2 and Lemma 3.3 with their supplied proofs, plus rational-function definition, retained from parent record.",
      "unread_or_unaudited_dependencies": "Full counting theorem and all cited dependencies not independently audited.",
      "earlier_selected_proof_reading": "Arithmetic Propositions 3.1-3.2 and Lemma 3.3 with the supplied proofs; relevant rational-function definition and statements. Used for the Kumar--Tosi repair attempt, whose finite arithmetic proof is reproduced explicitly. Full theorem and all cited dependencies not independently audited.",
      "historical_scope": "Abstract and opening introduction excerpts (pp.1-2) screened. Totient-zero construction, arithmetic lemmas and full theorem proof remain unread in this update.",
      "reread_on": "2026-11-06",
      "actual_scope_this_pass": "Introduction and construction attribution; Section 2, including the denominator/zero sets and Definition 2.3; Section 5 coefficient-elimination setup through equations (5.1)-(5.3). This confirms the rational-grid and generalized-Vandermonde elimination methods cited here. Lai-Yu explicitly credit Fischler-Sprang-Zudilin for the earlier elimination framework.",
      "role": "Attribution for rational-grid construction and coefficient elimination. No theorem from this source is imported into the proof.",
      "reused_existing_pdf": true,
      "local_pdf_hash_rechecked_on": "2026-11-06"
    },
    {
      "citation_key": "KT26",
      "title": "Irrationality of values of $L$-functions for real Dirichlet characters",
      "authors": "Veekesh Kumar; Riccardo Tosi",
      "version": "2610.41458v1",
      "source_url": "https://arxiv.org/abs/2610.41458v1",
      "local_pdf_filename": "Irrationality-of-L-values-for-real-Dirichlet-characters_Kumar-Tosi_2610.41458v1.pdf",
      "pdf_sha256": "ee3d7849086a2cdd9f9cb7515565564a60ca7c3edf809242702f2896e2ec096b",
      "retrieved_at_utc": "2026-11-02T21:09:06.442351+00:00",
      "priority_read_scope": "Primary arXiv history checked: v1 remains latest. Main theorem/asymptotic scope and construction origin refreshed. Prior repair investigation read rational-function, denominator, relevant prime-valuation and analytic arguments and final optimization; see parent record.",
      "unread_or_unaudited_dependencies": "Full asymptotic theorem and every dependency not independently audited or repaired. Existing arithmetic counterexamples remain separate from this priority conclusion.",
      "earlier_selected_proof_reading": "Repair attempt: read the rational-function definition, denominator arguments, relevant prime-valuation/nonvanishing calculations, the analytic-estimate section and final optimization. Reconstructed the corrected B=1, T=4 finite family directly. No comprehensive independent audit of every paper dependency.",
      "historical_scope": "PDF/text pp.1-11 (introduction, criterion, notation, linear-form construction, denominator Lemmas 13-14), final proof pp.29-31, and full LaTeX Section 3.6 (4_Denominators.tex lines 603-819); 2_Notation.tex inspected for precise hypotheses. PDF p.2 rendered visually. Remaining prime-valuation Sections 3.3-3.5 and analytic Section 4 not fully audited. Several apparent v1 inconsistencies recorded in review-2026-11-02.txt.",
      "reread_on": "2026-11-06",
      "actual_scope_this_pass": "Introduction methodological discussion; complete Section 2 (Linear forms), read in 3_Linear_forms.tex; beginning of Section 5 (Proof of the main result), through the final irrationality lemma and parameter setup. Verified that the linear-form section is numbered 2, despite the source filename. The citation concerns the adaptation of the rational-grid construction and coefficient elimination to real Dirichlet characters.",
      "role": "Attribution for the real-Dirichlet-character adaptation only. The claimed asymptotic theorem is not used; the earlier recorded v1 arithmetic issues remain outside the present argument.",
      "reused_existing_pdf": true,
      "local_pdf_hash_rechecked_on": "2026-11-06"
    },
    {
      "citation_key": "LS25",
      "title": "Many $p$-adic odd zeta values are irrational",
      "authors": "Li Lai; Johannes Sprang",
      "version": "2306.10393v2",
      "source_url": "https://arxiv.org/abs/2306.10393v2",
      "local_pdf_filename": "Many-p-adic-odd-zeta-values-are-irrational_Lai-Sprang_2306.10393v2.pdf",
      "pdf_sha256": "240092a619e23b666ed61a363e45bd4940ed0a4326c916526df84504c03b3941",
      "retrieved_at_utc": "2026-11-02T21:11:39.709993+00:00",
      "priority_read_scope": "Introduction refreshed for auxiliary-prime antecedent. Prior pp.1-6 and full proofs of Lemmas 2.2-2.3 read; elementary criterion algebra checked directly.",
      "unread_or_unaudited_dependencies": "Remaining p-adic construction and complete theorem proof not read in full. This is an antecedent, not a real-beta finite result.",
      "historical_scope": "PDF pp.1-6 read, including full proofs of Lemmas 2.2 and 2.3; p.6 rendered visually. Auxiliary-prime criterion checked directly as elementary algebra. Remaining p-adic construction and theorem proof not read in full.",
      "reread_on": "2026-11-06",
      "actual_scope_this_pass": "Section 2.1, statements and proofs of Lemmas 2.2 and 2.3. Verified that Lemma 2.3 is the real/Archimedean auxiliary-prime criterion: the constant coefficient has strictly smaller auxiliary-prime valuation than all other coefficients, preventing vanishing under assumed rationality.",
      "role": "Attribution for auxiliary-prime nonvanishing. The elementary contradiction is proved explicitly in the manuscript.",
      "reused_existing_pdf": true,
      "local_pdf_hash_rechecked_on": "2026-11-06"
    },
    {
      "citation_key": "TshiftV1",
      "title": "Irrational even beta values in shifted intervals",
      "authors": "twinheterodox",
      "version": "Hexagon 2611.00160v1",
      "source_url": "https://hexagonmath.org/2611.00160v1",
      "repository_commit": "a3f56ac5865736555fb0379b94a712038efcf99f",
      "repository_pdf_path": "shifted-even-beta/output/pdf/shifted-even-beta.pdf",
      "pdf_sha256": "9f2157cc8fd9c910fb47f65d5283f83fcc1a7ce4b012823c4ed3688122dbd1f9",
      "local_copy_status": "Existing submitted v1 PDF retained in repository history and local records; no new PDF download.",
      "pdf_hash_scope": "The submitted 17-page PDF, not a checksum of a subsequently stamped public download.",
      "bibliographic_metadata_checked_on": "2026-11-06",
      "landing_page_check": "HTTP 200; title and identifier match the supplied Hexagon citation metadata.",
      "actual_read_scope": "The submitted v1 source and PDF were read during the exposition comparison. The general theorem, totient sum and ceiling formula were checked for the restatement. This update verifies citation identity and adds the bibliography entry; it is not a new independent proof audit.",
      "reused_existing_pdf": true,
      "role": "Historical comparison with the earlier version of the same paper; no proof proposition is imported."
    }
  ],
  "updated_on": "2026-11-06",
  "v2_assembly": {
    "date": "2026-11-06",
    "scope": "Retained the v1 source-reading scopes. Updated the count-paper citation to confirmed Hexagon 2611.00139v1 metadata; its published PDF text matches the previously read manuscript apart from the identifier stamp. No fresh literature search or independent audit of cited papers was undertaken.",
    "proof_source": "Independently audited v2 argument identified in review/independent-audit-2026-11-06.json.",
    "method_attribution": "Odd-grid construction and parity choice; differentiation; Gram multipliers; grouped denominator clearing; prime dilation and coefficient elimination are attributed in the introduction and beside the relevant proof steps."
  }
}