{
  "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 public Hexagon beta search on 5 November did not expose a record for this title; the confirmed public GitHub manuscript location is cited.",
      "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."
    },
    {
      "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"
    }
  ],
  "updated_on": "2026-11-06"
}