{
  "date": "2026-11-06",
  "version": 2,
  "status": "Typeset v2 with final exposition review complete and table-source wording aligned with the user's simplified headings; compilation and affected-page visual review passed.",
  "pdf": "output/pdf/shifted-even-beta.pdf",
  "pdf_sha256": "acb04d7ea754bb108db03c06d40cc73e790596daf453abd5f57498e524538f36",
  "pages": 20,
  "source_files": {
    "paper/references.tex": "f0f3327d1053c45e2be3a1bbff5a23f0a519a7637769ad9bc027c760375c9cde",
    "paper/sections/arithmetic.tex": "78af9b7805ac9292d4a08065a1171a8b2f89445e29f8250e6c866e5e1f2dd2fe",
    "paper/sections/checks.tex": "a16a548224bbd1c70856c43872de81ed3d36a77145bb72c624c3c5928c5b6f7a",
    "paper/sections/construction.tex": "ffeeb15dd029eab4a674a4c6126c7380f2541064489e2bfeb4565dacfc6e02bd",
    "paper/sections/contradiction.tex": "2162e0ba6f2c85496d7fe13dc1a6d58548d2ef8f6f7deb1c272edc1ae66d1ed4",
    "paper/sections/decay.tex": "4e03981d38afcad6a30dccef404a676a8d661cdf566fdd142a40123559ae4476",
    "paper/sections/dilation.tex": "5a15be562b7692647645ad3c4a32c196f217c80f74f93c88f6db42f241f1f367",
    "paper/sections/endpoints.tex": "8f514a229f28097d89f150cc3da716b5faec507a5bc983d2aa8b3f60a3c2a70c",
    "paper/sections/gram.tex": "3aa5ce431b1200624ba493bda81b84e13d427a524cd0e21a514a8949394de9fe",
    "paper/sections/introduction.tex": "00a43de22b3daa8dcbb6ba31c2f533372d8292129f3101b3070e43a0d3f8d82c",
    "paper/sections/optimization.tex": "f421b3a85669f3bc096344008ebee2a946cb679837160c6640bff486c02dab45",
    "paper/shifted-even-beta.tex": "107f25106696b0b7a76685f0ef99940d82b0b564f8f450f52e74f085c5de75f7"
  },
  "independent_audit_record": "review/independent-audit-2026-11-06.json",
  "audited_plain_text_sha256": "603211b50609dc4a4f6646815e7192b61bb8d264c93868d39921f1e0843454e7",
  "independent_audit_scope": "Underlying general argument and recorded certificates. The subsequent typeset exposition and worked example were reviewed separately; the independent report is not an audit of this PDF.",
  "mathematical_review": "review/mathematical-review.txt",
  "exposition_review": "review/exposition-review.txt",
  "proof_correspondence": [
    {
      "manuscript": "Section 1",
      "audited_proof": "Theorems A and B; Sections 6 and 7",
      "content": "The 19m-1 result and current general family are unchanged mathematically, with endpoint notation C_B=a_B+1. The historical v1 theorem is reproduced separately for comparison, and specific endpoints and literature comparisons are retained."
    },
    {
      "manuscript": "Section 2",
      "audited_proof": "Section 1",
      "content": "Odd grids, normalization, differentiation, forms and endpoint criterion."
    },
    {
      "manuscript": "Section 3",
      "audited_proof": "Section 2",
      "content": "Differentiated kernels, independent integer polynomials, homogeneous multipliers and positive Gram matrix. The worked example was added after the audit."
    },
    {
      "manuscript": "Section 4",
      "audited_proof": "Section 3",
      "content": "Progression divisibility, pole coefficients and grouped integrality."
    },
    {
      "manuscript": "Section 5",
      "audited_proof": "Section 4",
      "content": "Uniform Stirling profile, Cauchy estimate, infinite tail and rational decay criterion."
    },
    {
      "manuscript": "Section 6",
      "audited_proof": "Section 5",
      "content": "Prime dilation, coefficient extraction and the surviving character factor. A single-grid example motivates the dilation, and the coefficient scaling is spelled out."
    },
    {
      "manuscript": "Section 7",
      "audited_proof": "Section 5",
      "content": "Vandermonde cancellation and infinitely many nonzero cleared integers."
    },
    {
      "manuscript": "Section 8",
      "audited_proof": "Sections 6 and 7.1; endpoint certificates",
      "content": "General uniform family, including the 19m-1 one-value case, and 12 displayed individual certificates. The sharper one-value formula is now in Appendix B."
    },
    {
      "manuscript": "Appendix A",
      "audited_proof": "Section 8",
      "content": "Moment identity, finite-check scope and independent audit note."
    },
    {
      "manuscript": "Appendix B",
      "audited_proof": "Sections 7.1 and 7.2",
      "content": "Sharper uniform one-value formula below 18.58m, numerical limiting multiplier, and exact exclusions for other integer zero-range parameters. The finite-m profile identity and its correction are explicit."
    }
  ],
  "exact_checks": {
    "commands": [
      "python3 -B checks/checks.py",
      "python3 -B checks/endpoints.py",
      "python3 -B checks/worked_example.py"
    ],
    "result": "All passed.",
    "structural_counts": {
      "kernel_evaluations": 57,
      "gram_entries": 49,
      "positive_principal_minors": 19,
      "seed_targets": 9,
      "reconstructions": 54,
      "reflection_coefficients": 1308,
      "differentiated_grid_zeros": 332,
      "grouped_integrality": 4516,
      "selected_pole_congruences": 404,
      "other_pole_bounds": 1940,
      "constant_congruences": 19,
      "beta_coefficient_bounds": 202
    },
    "structural_total": 8909,
    "endpoint_certificates": 21,
    "uniform_parameter_records": 6,
    "integer_r_exclusions": 7,
    "all_12_typeset_parameter_rows_match_results": true,
    "worked_example": "checks/worked_example.json",
    "audited_checker_sources_and_outputs_byte_identical": {
      "checks.py": true,
      "checks.json": true,
      "endpoints.py": true,
      "endpoints.json": true,
      "optimization.py": true,
      "optimization.json": true
    },
    "file_sha256": {
      "checks/checks.json": "521a335da27bd06946e032b06536129c09e0595d054e140c96e29b93386d4f97",
      "checks/checks.py": "98095881df253cc815ad1758ea372ce49c7e5153c16b29c869891ab66f45205f",
      "checks/endpoints.json": "977b324f352ea43e680f9aafce95d99f56850e81639ca1a1b7e1fc2936d39d4d",
      "checks/endpoints.py": "1a4a6395f887c1aca0b2887aed4c7a5619a99e24c2bd5521f27fa77b73e15a9f",
      "checks/optimization.json": "cff0f17479f70663911407e769bcad019c4da46aea7be18c46cb183231771cc0",
      "checks/optimization.py": "1377723dd4c9be3e40ad2076dd07871d7ac38968c51e44646a055000185744a4",
      "checks/worked_example.json": "01b514b0cd21690388172c1702dd183f2e10155697b4f263582123e1ddac5604",
      "checks/worked_example.py": "42007aa4b8bd7fc8c5184539303141e54d4060c9a8be0801be5b38949dba322d"
    },
    "replay_scope": "Replayed during the initial v2 assembly. Checker sources and results remained byte-identical through the subsequent exposition and headline revisions; full replays were not repeated. During publication preparation, the worked-example checker was rerun after removing a trailing blank line; it passed with identical output. Added explanatory identities and later formula, table and source comparisons are recorded in the v2 revision records."
  },
  "numerical_exploration": {
    "command": "python3 -B checks/optimization.py",
    "result": "Reproduced the limiting multiplier 18.57225289992264 in double precision.",
    "scope": "Numerical exploration only; exact theorem bounds are checked separately."
  },
  "compilation": {
    "command": "python3 -B tools/build_paper.py",
    "passes": 3,
    "unresolved_references": false,
    "overfull_boxes": false,
    "underfull_warnings": [],
    "record": "paper/build-record.json"
  },
  "visual_review": {
    "renderer": "Poppler pdftoppm, 100 dpi",
    "pages": 20,
    "all_pages_reviewed": true,
    "method": "Page 3 was visually inspected after the final table-source wording change. The other 19 pages are pixel-identical to the user-edited 20-page PDF checked during the final exposition review.",
    "final_revision_changed_pages": [
      3
    ],
    "result": "No clipped text, broken formulas, table collisions or page-layout defects found."
  },
  "identity_review": {
    "scope": "Changed and new public files, extracted canonical PDF text and PDF metadata.",
    "pdf_author": "twinheterodox",
    "private_identity_or_identifying_home_paths_found": false,
    "private_audited_snapshot_published": false,
    "third_party_pdfs_redistributed": false
  },
  "earlier_pdfs_unchanged": {
    "even-beta-independence/output/pdf/even-beta-independence.pdf": "83fff93aebeff3b79fb2f316df94518321a7b7dab815c7ab37d8ebf0b5cc5fe8",
    "output/pdf/beta4-determinant-obstructions.pdf": "cdb528bc1836aed5a7c4ad6105ac3165ce57fb56d06763ea69a4b3676c3fe40a",
    "output/pdf/beta4-investigations.pdf": "828f7a5cbe20343ad67c230c1eca0d94d39c88120fbda1ca5ef7092d07b8b3f0",
    "finite-even-beta/output/pdf/finite-even-beta.pdf": "8fe8cef9a6e1a5644892881dd9b784cf9eedac7ffc4a506e37e741ab478c0312",
    "even-beta-counts/output/pdf/even-beta-counts.pdf": "f6848fa9304693f1b96a12ca9c2268511f350eb7c3e607054ddb0d4d3fe0c2a5"
  },
  "historical_review_scope": "Dated records other than independent-audit-2026-11-06 and the v2-prefixed revision records describe v1. See review/README.txt. Prior versions of tracked review files remain in Git history.",
  "publication_status": "Version 2 is preserved in the Git revision containing this record, tagged shifted-even-beta-v2. Version 1 is preserved at shifted-even-beta-v1.",
  "formal_verification_claimed": false,
  "bibliography_update": {
    "citation_key": "Tcount",
    "published_version": "Hexagon 2611.00139v1",
    "url": "https://hexagonmath.org/2611.00139v1",
    "verification": "Official citation metadata confirms the title, author and publication. The downloaded 20-page PDF has text identical to the previously read count paper apart from the added Hexagon identifier stamp.",
    "source_record": "sources/reading-record.json"
  },
  "latest_exposition_revision": "review/v2-theorem-comparison-2026-11-06.json",
  "preceding_exposition_revision": "review/v2-v1-comparison-revision-2026-11-06.json",
  "latest_wording_revision": {
    "date": "2026-11-06",
    "section": "1.1",
    "change": "Restored the v1 paragraph stating the four earlier I(s) bounds and explaining why discarding the first 1, 2, 9 and 49 even beta values leaves an irrational member in each interval.",
    "previous_pdf_sha256": "cb5c3caa1b02d4c27d26790bf4b2b88512335d49d08cad618a1bcc3ddf7b95a8",
    "mathematical_content_unchanged": true,
    "validation": "The restored paragraph matches v1 verbatim; the current endpoints are unchanged. No mathematical tests rerun for this prose-only change."
  },
  "version_comparison_note": {
    "date": "2026-11-06",
    "location": "Page 2, after the opening one-value theorem",
    "change": "The comparison now identifies and cites the earlier version before reproducing its general theorem. Matching endpoint formulas and numerical multipliers are retained; the headings are Earlier bound and Theorem 1.2.",
    "previous_pdf_sha256": "f16481fac6ea412474a18cdabbe2d5e2792b7ab9a3a732eeb94ac4f4572d54b9",
    "mathematical_claims_unchanged": true,
    "validation_record": "review/v2-theorem-comparison-2026-11-06.json"
  },
  "context_revision": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "877bf973e7e73ea20358a96233155850f723c95f3c8533899ebde4eeaddde43e",
    "change": "Introduces the earlier result as a prior version of this paper, states why it is recalled, cites bibliography entry [10], and uses the headings Earlier bound and Theorem 1.2.",
    "validation": {
      "mathematical_displays_unchanged": true,
      "abstract_unchanged": true,
      "historical_statement_unchanged": true,
      "current_statement_unchanged": true,
      "checker_sources_and_results_unchanged": true,
      "citation_key": "TshiftV1",
      "bibliography_number": 10,
      "bibliographic_metadata": "Uses the supplied Hexagon 2611.00160v1 citation metadata; the public landing page returned HTTP 200 with the matching title.",
      "full_mathematical_checkers_rerun": false
    },
    "independent_audit_scope_unchanged": true
  },
  "opening_comparison_revision": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "d95d89e47b3b39f64dc375641c98b58b2161e20ecf7f2e785db15b0ade07f3d4",
    "change": "The transition explicitly states that the earlier paper [10] proved Theorem 1.1 with 67 in place of 19, before introducing the comparison of the general bounds.",
    "mathematical_content_unchanged": true,
    "validation": "Only the introductory transition changed in the manuscript. Recompiled in three passes with no unresolved references or box warnings; inspected both changed pages. Mathematical checkers were not rerun for this prose-only change.",
    "independent_audit_scope_unchanged": true,
    "wording_clarification": "Replaced 'More generally, it' with 'More generally, that paper' to make the reference explicit. Only page 2 changed; compilation and visual review passed."
  },
  "cited_heading_revision": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "9f30a55825e45f1f52c3327f8999eabfaff63913865cf5b83d9beb992a5cce19",
    "change": "Replaced the heading Earlier bound with a citation to TshiftV1, Theorem 1.1, rendered as [10, Theorem 1.1].",
    "validation": "Checked the theorem number and statement in the submitted v1 PDF and its repository source. Only the unnumbered theorem heading changed in the manuscript; compilation and visual review passed. Mathematical checkers were not rerun.",
    "mathematical_content_unchanged": true,
    "independent_audit_scope_unchanged": true
  },
  "four_exposition_refinements": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "c72c9fe90466e7593a7d6c1826e43b3f6e84cfa241446825796fc4e83fa22c50",
    "changes": [
      "Renamed the normalization C_n, including its prime and seed specializations, to kappa_n. Endpoint constants remain C_B.",
      "Explained in the proof overview that B candidate polynomial multipliers produce B seed vectors, which form the columns of the matrix.",
      "Defined s as the even upper index and stated that q_b(n) and rho_nu(n) are rational before explaining the irrationality contradiction.",
      "Stated that the odd summation grid has spacing 2/b and that the tail estimate uses the denser grid of spacing 1/b."
    ],
    "validation": {
      "displayed_formulas_unchanged_after_notation_and_grid_label_normalization": true,
      "abstract_and_main_theorems_unchanged": true,
      "sole_remaining_C1_is_endpoint_constant_19": true,
      "pages": 20,
      "unresolved_references": false,
      "overfull_boxes": false,
      "underfull_warnings": [],
      "changed_pages_visually_reviewed": [
        3,
        4,
        5,
        9,
        11,
        12,
        14,
        15,
        18
      ],
      "full_mathematical_checkers_rerun": false,
      "checker_sources_and_results_unchanged": true
    },
    "independent_audit_scope_unchanged": true,
    "mathematical_claims_unchanged": true
  },
  "endpoint_table_citation": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "9db2889aa6798c0dc126958349fd601ab5437c9f68ff609532d61ae51a5cada7",
    "change": "Replaced Earlier endpoint with Endpoint from [9], using the Tcount bibliography citation, consistently with the preceding comparison table.",
    "validation": "Only the table heading changed in the manuscript. Compilation passed without reference or box warnings; page 3 was visually checked. No mathematical checker replay was needed."
  },
  "v1_endpoint_table_comparison": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "e815623ef1425bab23b38461da621234d8548f19b0d2d4d35f08b6969b64f7df",
    "source_citation": "TshiftV1, Hexagon 2611.00160v1",
    "source_git_commit": "a3f56ac5865736555fb0379b94a712038efcf99f",
    "source_pdf_sha256": "9f2157cc8fd9c910fb47f65d5283f83fcc1a7ce4b012823c4ed3688122dbd1f9",
    "shifted_interval_entries": [
      {
        "first_index": 4,
        "m": 3,
        "endpoint": 198,
        "source": "Section 8.2, specific choice"
      },
      {
        "first_index": 6,
        "m": 5,
        "endpoint": 330,
        "source": "Section 8.2, specific choice"
      },
      {
        "first_index": 20,
        "m": 19,
        "endpoint": 1272,
        "source": "Theorem 1.1, B=1"
      },
      {
        "first_index": 100,
        "m": 99,
        "endpoint": 6632,
        "source": "Theorem 1.1, B=1"
      }
    ],
    "finite_count_entries": [
      {
        "B": 2,
        "D_v1": 2,
        "inner_illustrative": "97.1916231251975389237754454656",
        "ceiling_lower": 98,
        "ceiling_upper": 98,
        "endpoint": 196
      },
      {
        "B": 3,
        "D_v1": 4,
        "inner_illustrative": "230.875105980129872225572071323",
        "ceiling_lower": 231,
        "ceiling_upper": 231,
        "endpoint": 462
      },
      {
        "B": 4,
        "D_v1": 6,
        "inner_illustrative": "385.149738751185233542652672794",
        "ceiling_lower": 386,
        "ceiling_upper": 386,
        "endpoint": 772
      },
      {
        "B": 5,
        "D_v1": 10,
        "inner_illustrative": "692.123530297672584285170849976",
        "ceiling_lower": 693,
        "ceiling_upper": 693,
        "endpoint": 1386
      },
      {
        "B": 10,
        "D_v1": 32,
        "inner_illustrative": "2713.86126695571264664679132803",
        "ceiling_lower": 2714,
        "ceiling_upper": 2714,
        "endpoint": 5428
      }
    ],
    "finite_count_formula": "At m=1, s_B=2 ceil(D_B^(1)*(33+(45/2)*log B)), D_B^(1)=sum_{d=1}^B phi(d).",
    "ceiling_validation": "For z=(B-1)/(B+1), sum 80 terms of 2*sum z^(2j+1)/(2j+1); bound the positive remainder by 2*z^161/(161*(1-z^2)). Exact rational arithmetic gives the same ceiling at both ends.",
    "explanation": "The shifted table uses v1 Section 8.2 at first indices 4 and 6, and its 67m-1 bound at 20 and 100. The finite-count table specializes v1 Theorem 1.1 to m=1 and explicitly notes that these general bounds are weaker than the specific estimates from the count paper.",
    "validation": {
      "existing_endpoint_columns_unchanged": true,
      "all_nine_v1_entries_checked": true,
      "first_uniform_comparison_table_unchanged": true,
      "theorem_statements_and_proof_unchanged": true,
      "pages": 20,
      "unresolved_references": false,
      "overfull_boxes": false,
      "underfull_warnings": [],
      "changed_pages_visually_reviewed": [
        3
      ],
      "checker_sources_and_results_unchanged": true,
      "full_mathematical_checkers_rerun": false
    },
    "independent_audit_scope_unchanged": true
  },
  "final_table_source_wording": {
    "date": "2026-11-06",
    "previous_pdf_sha256": "e9263346ef5d4e736cb95b2948e3bd644bb9a8aa30974c4069e002840cfba120",
    "change": "Both comparison notes now begin The column from [10]. References to Section 8.2 and Theorem 1.1 explicitly refer to that earlier paper. The user's removal of (v1) from both table headings is preserved.",
    "validation": "Only explanatory prose changed. All table entries, formulas and theorem statements are unchanged. Compilation is clean and the PDF remains 20 pages.",
    "final_exposition_assessment": "The final full-manuscript review found no further substantive exposition changes needed. Its remaining table-label consistency recommendation is now resolved.",
    "mathematical_checkers_rerun": false,
    "independent_audit_scope_unchanged": true
  }
}