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      "id": "work-clay-problem-status",
      "title": "Yang-Mills & the Mass Gap",
      "authors": [
        "Clay Mathematics Institute"
      ],
      "primary_url": "https://www.claymath.org/millennium/yang-mills-the-maths-gap/",
      "doi": null,
      "arxiv_id": null,
      "role": "official-target",
      "notes": "A current status page is not a complete literature census. Recheck it before any later problem-status change."
    },
    {
      "id": "work-jaffe-witten-official",
      "title": "Quantum Yang–Mills Theory",
      "authors": [
        "Arthur Jaffe",
        "Edward Witten"
      ],
      "primary_url": "https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf",
      "doi": null,
      "arxiv_id": null,
      "role": "official-target",
      "notes": "Historical discussion of research progress is not a 2026 survey. The editorial proof map is an explanatory decomposition, not an additional official list of prize conditions. Do not infer the PDF publication year from the URL upload path."
    },
    {
      "id": "work-clay-prize-rules",
      "title": "Rules for the Millennium Prize Problems",
      "authors": [
        "Clay Mathematics Institute"
      ],
      "primary_url": "https://www.claymath.org/millennium-problems/rules/",
      "doi": null,
      "arxiv_id": null,
      "role": "official-target",
      "notes": "Read the linked complete rules when the process becomes relevant. Scientific acceptance and official prize recognition are separate statuses."
    },
    {
      "id": "work-osterwalder-schrader-1973",
      "title": "Axioms for Euclidean Green's functions",
      "authors": [
        "Konrad Osterwalder",
        "Robert Schrader"
      ],
      "primary_url": "https://link.springer.com/article/10.1007/BF01645738",
      "doi": "10.1007/BF01645738",
      "arxiv_id": null,
      "role": "foundational-source",
      "notes": "The detailed theorem was not reread in this planning pass. Cite together with the 1975 extension and correction; do not reduce full reconstruction hypotheses to positivity alone."
    },
    {
      "id": "work-osterwalder-schrader-1975",
      "title": "Axioms for Euclidean Green's functions II",
      "authors": [
        "Konrad Osterwalder",
        "Robert Schrader"
      ],
      "primary_url": "https://link.springer.com/article/10.1007/BF01608978",
      "doi": "10.1007/BF01608978",
      "arxiv_id": null,
      "role": "foundational-source",
      "notes": "Exact regularity and growth assumptions require reading the full theorem before presenting a technical reconstruction checklist."
    },
    {
      "id": "work-gross-wilczek-1973",
      "title": "Ultraviolet Behavior of Non-Abelian Gauge Theories",
      "authors": [
        "David J. Gross",
        "Frank Wilczek"
      ],
      "primary_url": "https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1343",
      "doi": null,
      "arxiv_id": null,
      "role": "foundational-source",
      "notes": "Perturbative ultraviolet information does not construct the complete interacting theory or establish its infrared gap. This planning pass checked the abstract and metadata only."
    },
    {
      "id": "work-politzer-1973",
      "title": "Reliable Perturbative Results for Strong Interactions?",
      "authors": [
        "H. David Politzer"
      ],
      "primary_url": "https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1346",
      "doi": null,
      "arxiv_id": null,
      "role": "foundational-source",
      "notes": "This is not a nonperturbative existence or spectral-gap result. Check conventions and coefficients in the paper before using it for a worked calculation."
    },
    {
      "id": "work-balaban-1989-large-field-ii",
      "title": "Large field renormalization. II. Localization, exponentiation, and bounds for the R operation",
      "authors": [
        "Tadeusz Bałaban"
      ],
      "primary_url": "https://link.springer.com/article/10.1007/BF01238433",
      "doi": "10.1007/BF01238433",
      "arxiv_id": null,
      "role": "research-source",
      "notes": "Follow-up selected-section reading recovers the conditional statement and model scope. Earlier abstract-only/access-blocked assessments remain in history; no complete proof or later-history audit is claimed."
    },
    {
      "id": "work-chatterjee-2021-confinement",
      "title": "A probabilistic mechanism for quark confinement",
      "authors": [
        "Sourav Chatterjee"
      ],
      "primary_url": "https://arxiv.org/abs/2006.16229v5",
      "doi": "10.1007/s00220-021-04086-y",
      "arxiv_id": "2006.16229",
      "role": "research-source",
      "notes": "These hypotheses must not be identified with decay of one vacuum correlator. This does not establish the required weak-coupling continuum control for the Millennium problem."
    },
    {
      "id": "work-cao-nissim-sheffield-2025-dynamical",
      "title": "Dynamical approach to area law for lattice Yang-Mills",
      "authors": [
        "Sky Cao",
        "Ron Nissim",
        "Scott Sheffield"
      ],
      "primary_url": "https://arxiv.org/abs/2509.04688v2",
      "doi": null,
      "arxiv_id": "2509.04688",
      "role": "research-source",
      "notes": "Retain the exact beta thresholds in Definition 1.4, normalization discussion in Remark 1.5, and group restrictions in Theorem 1.6 when implementing a technical record. Auxiliary stochastic estimates and a fixed-regulator area law do not themselves prove a four-dimensional physical continuum gap."
    },
    {
      "id": "work-chevyrev-shen-2026-renormalisation",
      "title": "Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs",
      "authors": [
        "Ilya Chevyrev",
        "Hao Shen"
      ],
      "primary_url": "https://arxiv.org/abs/2503.03060v2",
      "doi": "10.1007/s00205-025-02163-3",
      "arxiv_id": "2503.03060",
      "role": "research-source",
      "notes": "The counterterm is not a physical mass-gap result. Read the full main theorem to confirm geometry, group assumptions, state spaces, and approximation class before extending the dossier."
    },
    {
      "id": "work-guo-li-yang-zhu-2025-bootstrap",
      "title": "Bootstrapping SU(3) Lattice Yang-Mills Theory",
      "authors": [
        "Yuanhong Guo",
        "Zeyu Li",
        "Gang Yang",
        "Guorui Zhu"
      ],
      "primary_url": "https://arxiv.org/abs/2502.14421v2",
      "doi": "10.1007/JHEP12(2025)033",
      "arxiv_id": "2502.14421",
      "role": "research-source",
      "notes": "Exact constraints, numerical solver certification, and convergence of an infinite hierarchy must be distinct claims. Short-loop bounds do not constitute a continuum spectrum or mass-gap theorem."
    },
    {
      "id": "work-liu-yang-2026-loop-equations",
      "title": "Direct and Indirect Loop Equations in Lattice Yang-Mills Theory",
      "authors": [
        "Xizhe Liu",
        "Gang Yang"
      ],
      "primary_url": "https://arxiv.org/abs/2601.04316v2",
      "doi": null,
      "arxiv_id": "2601.04316",
      "role": "research-source",
      "notes": "This record does not assert convergence of truncated hierarchies, a continuum construction, or a spectral-gap result. Detailed completeness and independence claims require checking the text."
    },
    {
      "id": "work-athenodorou-teper-2021-spectrum",
      "title": "SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology",
      "authors": [
        "Andreas Athenodorou",
        "Michael Teper"
      ],
      "primary_url": "https://arxiv.org/abs/2106.00364v3",
      "doi": null,
      "arxiv_id": "2106.00364",
      "role": "research-source",
      "notes": "The data support numerical spectroscopy, not a rigorous lower bound on the entire spectrum. Verify exact sampled values of N, tables, scale setting, and statistical/systematic errors before transcribing benchmark values."
    },
    {
      "id": "work-unsal-yaffe-2008-center-stabilized",
      "title": "Center-stabilized Yang-Mills theory: confinement and large N volume independence",
      "authors": [
        "Mithat Ünsal",
        "Laurence G. Yaffe"
      ],
      "primary_url": "https://arxiv.org/abs/0803.0344v2",
      "doi": "10.1103/PhysRevD.78.065035",
      "arxiv_id": "0803.0344",
      "role": "foundational-source",
      "notes": "Deformation, geometry, hierarchy of scales, and large-N assumptions must remain visible. This does not itself establish the undeformed finite-N R4 target."
    },
    {
      "id": "work-hayashi-tanizaki-unsal-2026-center-vortex",
      "title": "Center-vortex semiclassics with non-minimal 't Hooft fluxes on R2 x T2 and center stabilization at large N",
      "authors": [
        "Yui Hayashi",
        "Yuya Tanizaki",
        "Mithat Ünsal"
      ],
      "primary_url": "https://arxiv.org/abs/2505.07467v2",
      "doi": "10.1007/JHEP02(2026)126",
      "arxiv_id": "2505.07467",
      "role": "research-source",
      "notes": "The dilute-gas regime NL Lambda much less than 1 and twist assumptions are essential. No continuation to ordinary undeformed R4 is established by this register entry. First submission, v2 revision, and journal year are distinct dates."
    },
    {
      "id": "work-luscher-flow",
      "title": "Properties and uses of the Wilson flow in lattice QCD",
      "authors": [
        "Martin Lüscher"
      ],
      "primary_url": "https://arxiv.org/abs/1006.4518v3",
      "doi": "10.1007/JHEP08(2010)071",
      "arxiv_id": "1006.4518",
      "role": "teaching-source",
      "notes": "Selected source passages checked by the notation-crosswalk editor on 2026-10-24; no full-proof audit."
    },
    {
      "id": "work-balaban-1987-rg-i",
      "title": "Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions",
      "authors": [
        "Tadeusz Balaban"
      ],
      "primary_url": "https://scispace.com/pdf/renormalization-group-approach-to-lattice-gauge-field-3ygai0vxt1.pdf",
      "doi": null,
      "arxiv_id": null,
      "role": "research-source",
      "notes": "General small-field conditional construction uses compact semisimple matrix groups. The separately stated d4 running-coupling Theorem 2 is formulated for SU(2), with proof deferred to another paper. Do not use the latter to remove the former's assumption."
    },
    {
      "id": "work-balaban-1988-convergent-expansions",
      "title": "Convergent renormalization expansions for lattice gauge theories",
      "authors": [
        "Tadeusz Balaban"
      ],
      "primary_url": "https://doi.org/10.1007/BF01217741",
      "doi": "10.1007/BF01217741",
      "arxiv_id": null,
      "role": "research-source",
      "notes": "Conditional inductive package for effective densities; local analytic/gauge-invariant regular, R and boundary contributions. Corollary 3 controls an integrated density with constants independent of eta and torus but depending on the effective coupling; the preceding theorems retain the coupling/constant assumptions."
    }
  ],
  "versions": [
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      "work_id": "work-clay-problem-status",
      "label": "Undated official webpage, retrieved 2026-10-24",
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      "url": "https://www.claymath.org/millennium-problems/rules/",
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      "url": "https://link.springer.com/article/10.1007/BF01645738",
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      "url": "https://link.springer.com/article/10.1007/BF01608978",
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      "url": "https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1343",
      "publication_state": "published",
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      "url": "https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1346",
      "publication_state": "published",
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    {
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      "url": "https://link.springer.com/article/10.1007/BF01238433",
      "publication_state": "published",
      "source_date": {
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    {
      "id": "version-chatterjee-2021-confinement-v5",
      "work_id": "work-chatterjee-2021-confinement",
      "label": "arXiv:2006.16229v5, revised 2021-03-26; DOI 10.1007/s00220-021-04086-y",
      "url": "https://arxiv.org/abs/2006.16229v5",
      "publication_state": "published",
      "source_date": {
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        "precision": "day"
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      "checked_at": "2026-10-24",
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      "change_note": null,
      "first_submitted": {
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        "precision": "day"
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    {
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      "url": "https://arxiv.org/abs/2509.04688v2",
      "publication_state": "preprint",
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      "first_submitted": {
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        "precision": "day"
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    },
    {
      "id": "version-chevyrev-shen-2026-renormalisation-v2",
      "work_id": "work-chevyrev-shen-2026-renormalisation",
      "label": "arXiv:2503.03060v2, revised 2026-01-21; ARMA 250 (2026), article 11; DOI 10.1007/s00205-025-02163-3",
      "url": "https://arxiv.org/abs/2503.03060v2",
      "publication_state": "published",
      "source_date": {
        "value": "2026-01-21",
        "precision": "day"
      },
      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": null,
      "first_submitted": {
        "value": "2025-03-04",
        "precision": "day"
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    {
      "id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
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      "label": "arXiv:2502.14421v2, revised 2025-12-05; JHEP 12 (2025) 033; DOI 10.1007/JHEP12(2025)033",
      "url": "https://arxiv.org/abs/2502.14421v2",
      "publication_state": "published",
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        "precision": "day"
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      "change_note": null,
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      "url": "https://arxiv.org/abs/2601.04316v2",
      "publication_state": "preprint",
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      "first_submitted": {
        "value": "2026-01-07",
        "precision": "day"
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    },
    {
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      "work_id": "work-athenodorou-teper-2021-spectrum",
      "label": "arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082",
      "url": "https://arxiv.org/abs/2106.00364v3",
      "publication_state": "published",
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        "precision": "day"
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      "checked_at": "2026-10-24",
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      "change_note": null,
      "first_submitted": {
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        "precision": "day"
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    {
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      "label": "arXiv:0803.0344v2, revised 2008-03-22; PRD 78 (2008), 065035; DOI 10.1103/PhysRevD.78.065035",
      "url": "https://arxiv.org/abs/0803.0344v2",
      "publication_state": "published",
      "source_date": {
        "value": "2008-03-22",
        "precision": "day"
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      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": null,
      "first_submitted": {
        "value": "2008-03-03",
        "precision": "day"
      }
    },
    {
      "id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
      "work_id": "work-hayashi-tanizaki-unsal-2026-center-vortex",
      "label": "arXiv:2505.07467v2, revised 2025-08-05; JHEP 02 (2026) 126; DOI 10.1007/JHEP02(2026)126",
      "url": "https://arxiv.org/abs/2505.07467v2",
      "publication_state": "published",
      "source_date": {
        "value": "2025-08-05",
        "precision": "day"
      },
      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": null,
      "first_submitted": {
        "value": "2025-05-12",
        "precision": "day"
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    {
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      "url": "https://arxiv.org/abs/2509.04688v1",
      "publication_state": "preprint",
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        "value": "2025-09-04",
        "precision": "day"
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      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": "Historical source metadata retained; the earlier full text was not technically compared.",
      "title_at_version": "Dynamical approach to area law for lattice Yang-Mills"
    },
    {
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      "label": "arXiv:1006.4518v3, 27 January 2014; JHEP 08 (2010) 071; DOI 10.1007/JHEP08(2010)071",
      "url": "https://arxiv.org/abs/1006.4518v3",
      "publication_state": "published",
      "source_date": {
        "value": "2014-01-27",
        "precision": "day"
      },
      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": "Selected equations (1.1)–(1.4), section 3.4 equation (3.3), Appendix A equations (A.1)–(A.2) and normalization paragraph checked. The arXiv v3 record reports corrected figures 2/3 and t0/a² values in Table 1 following an analysis-program bug. No table values from this work are transcribed here.",
      "journal_published": {
        "value": "2010-08",
        "precision": "month"
      }
    },
    {
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      "work_id": "work-balaban-1987-rg-i",
      "label": "CMP 109 (1987), 249-301; published-article scan",
      "url": "https://scispace.com/pdf/renormalization-group-approach-to-lattice-gauge-field-3ygai0vxt1.pdf",
      "publication_state": "published",
      "source_date": {
        "value": "1987",
        "precision": "year"
      },
      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": "Selected supporting passages examined; no complete dependency-proof audit."
    },
    {
      "id": "version-balaban-1988-convergent-expansions-1988",
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      "label": "CMP 119 (1988), 243-285; published-article scan",
      "url": "https://doi.org/10.1007/BF01217741",
      "publication_state": "published",
      "source_date": {
        "value": "1988",
        "precision": "year"
      },
      "checked_at": "2026-10-24",
      "prior_version_id": null,
      "change_note": "Selected supporting passages examined; no complete dependency-proof audit."
    }
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      "workflow_visibility": "public",
      "target_relation": "direct-ingredient",
      "statement": "Conditional on that running-coupling hypothesis, effective densities have the form and satisfy the conditions and bounds specified in [III] section 2. Part II then states ultraviolet-stability bounds (0.1), with constants independent of renormalization step, torus and background field as specified immediately after the displayed bound.",
      "plain_language": "The effective lattice densities satisfy the cited ultraviolet-stability package provided all effective couplings stay sufficiently small. The source retains that hypothesis; this is not a continuum existence or mass-gap theorem.",
      "domain": {
        "dimension": "Four-dimensional Euclidean pure gauge lattice setting",
        "gauge_group": "Compact semisimple matrix Lie group G contained in U(N), as defined in [I] pp251-252; do not replace with only SU(2)",
        "matter": "None",
        "geometry": "Finite periodic torus with regularly refined lattices, from [I] introduction; no infinite-volume QFT is reconstructed here",
        "regulator": "Lattice spacing epsilon in the discrete refining sequence of [I], successive block/coarse-lattice transformations; [III] procedure stops at the unit lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": "Periodic identifications on the finite torus, as in [I].",
        "normalization": "Wilson plaquette action with normalized trace tr(1)=1, [I] Eq.(0.2); [III] initial density has inverse bare-coupling-squared Wilson weight and vacuum-energy normalization E",
        "unchecked_details": [
          "Temperature and theta are not assigned beyond the cited Wilson-action setup."
        ]
      },
      "assumptions": [
        "The entire sequence of effective coupling constants remains in (0,gamma] with gamma sufficiently small (Part II Theorem 1).",
        "The effective-density representation and conditions are the full package of [III] section 2, not merely a generic small-coupling slogan.",
        "The associated geometric, regularity, analyticity and constant restrictions inherited from [I] and [III] remain part of the statement."
      ],
      "conclusion": "Conditional on that running-coupling hypothesis, effective densities have the form and satisfy the conditions and bounds specified in [III] section 2. Part II then states ultraviolet-stability bounds (0.1), with constants independent of renormalization step, torus and background field as specified immediately after the displayed bound.",
      "limitations": [
        "Part II page 355 explicitly retains the coupling assumption: the proposed removal relies on the second-order perturbative result called Theorem 2 in [I], whose proof the article says had not yet been published. This review does not assert that the proof remains unpublished today; that later-history question was not resolved.",
        "No claim of full-proof review, reconstruction of a nontrivial continuum QFT, infinite-volume control of all physical observables, or physical Hamiltonian mass gap.",
        "The full [I] and [III] proofs, [II] cluster-expansion proof and [IV] first R-operation paper were not audited. Background-field/propagator/cluster-expansion dependencies in earlier references remain unread here.",
        "No assertion is made that all effective couplings stay small for a prescribed continuum tuning; the cited theorem assumes this.",
        "Equations have poor OCR. Exact stability-bound formulas were not retypeset from OCR. Identified objects, assumptions and locators are supported by readable text; no claimed pixel verification for these scans.",
        "Canonical source identity is the journal article/DOI; the readable scan is third-party hosted and is not redistributed."
      ],
      "source_locators": [
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Printed page 355/PDF page 1: Theorem 1 and following qualification",
          "url": "https://scispace.com/pdf/large-field-renormalization-ii-localization-exponentiation-h8b7ih93xo.pdf"
        },
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Printed pages 355-356: inequality (0.1) and stated independence of constants",
          "url": "https://scispace.com/pdf/large-field-renormalization-ii-localization-exponentiation-h8b7ih93xo.pdf"
        },
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Printed page 392/PDF page 38: concluding representation (1.104) discussion and references",
          "url": "https://scispace.com/pdf/large-field-renormalization-ii-localization-exponentiation-h8b7ih93xo.pdf"
        },
        {
          "version_id": "version-balaban-1987-rg-i-1987",
          "label": "Introduction, printed pp251-252: torus, lattice scales, compact semisimple matrix group and Wilson action (0.2)",
          "url": "https://scispace.com/pdf/renormalization-group-approach-to-lattice-gauge-field-3ygai0vxt1.pdf"
        },
        {
          "version_id": "version-balaban-1987-rg-i-1987",
          "label": "Printed p259: Theorems 1 and 2; p260 opening qualification",
          "url": "https://scispace.com/pdf/renormalization-group-approach-to-lattice-gauge-field-3ygai0vxt1.pdf"
        },
        {
          "version_id": "version-balaban-1988-convergent-expansions-1988",
          "label": "Introduction pp243-245: Wilson initial density, E normalization, RT iteration to unit lattice",
          "url": "https://scispace.com/pdf/convergent-renormalization-expansions-for-lattice-gauge-29jyexd38e.pdf"
        },
        {
          "version_id": "version-balaban-1988-convergent-expansions-1988",
          "label": "Section 2 pp254-264, especially density (2.18), effective action (2.23), localized terms (2.25)-(2.42), Theorems 1-2 and Corollary 3 (2.50)",
          "url": "https://scispace.com/pdf/convergent-renormalization-expansions-for-lattice-gauge-29jyexd38e.pdf"
        }
      ],
      "limits": [
        {
          "parameter": "lattice refinement and renormalization steps",
          "operation": "Successive T followed by R transformations of the effective densities, through the stated multiscale construction",
          "fixed_quantities": [
            "Wilson-action convention and geometric setup of [I]",
            "all effective couplings assumed in (0,gamma] for sufficiently small gamma",
            "complete density conditions and bounds of [III] section 2"
          ],
          "uniformity": "Part II inequality (0.1) states constants independent of renormalization step, torus and background field under its assumptions; this is not a physical infinite-volume mass-gap bound.",
          "order_notes": "[III] stops at unit lattice. A reconstructed continuum theory and its physical spectrum are not supplied by this statement."
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "UV",
          "relation": "conditional",
          "explanation": "Controls the specified multiscale effective-density estimates only under the all-scale small-coupling hypothesis. Regulator removal with physical reconstruction and an infrared spectral gap are further questions."
        }
      ],
      "assessment_ids": [
        "assessment-uv-stability-seed-20261023",
        "assessment-uv-stability-20261024",
        "assessment-balaban-selected-sections-followup-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Constructive RG",
        "Foundations",
        "Ultraviolet"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [
        "A deeper proof course would need [II], [IV], earlier technical dependencies and later history of the deferred coupling calculation. No conclusion here depends on declaring that calculation unpublished today."
      ],
      "obligation_note": null
    },
    {
      "id": "claim-center-divergent-potential",
      "title": "Unbroken slab center symmetry implies a divergent potential",
      "dossier_url": "/research/results/confinement-criterion/",
      "claim_type": "conditional-theorem",
      "workflow_visibility": "public",
      "target_relation": "conditional-bridge",
      "statement": "Theorem 2.2 obtains a rectangular-loop bound with a static potential diverging at large separation under the defined slab-center condition.",
      "plain_language": "The stated slab-center symmetry condition yields a static potential that diverges with separation, for center-charged probe representations.",
      "domain": {
        "dimension": "Lattice d ≥ 2",
        "gauge_group": "Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation",
        "matter": "Pure gauge",
        "geometry": "Lattice slabs and boxes",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Definition 2.1 unbroken slab center symmetry; irreducible unitary representation nontrivial on the center."
      ],
      "conclusion": "Theorem 2.2 obtains a rectangular-loop bound with a static potential diverging at large separation under the defined slab-center condition.",
      "limitations": [
        "The conclusion is a divergent static potential, not the stronger rectangular area law. Definition 2.3 uniform exponential decay is an additional premise of the separate Theorem 2.4 record, not a premise asserted here.",
        "The slab-center hypothesis must itself be established in the intended regime. This result does not remove the lattice cutoff or provide a physical continuum mass-gap theorem."
      ],
      "source_locators": [
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Section 2.1 (model)",
          "url": "https://arxiv.org/html/2006.16229v5"
        },
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Definition 2.1 and Theorem 2.2 (slab center symmetry and divergent static potential)",
          "url": "https://arxiv.org/html/2006.16229v5"
        }
      ],
      "limits": [],
      "obligation_links": [
        {
          "obligation_id": "volume",
          "relation": "conditional",
          "explanation": "A conditional lattice confinement implication under the stated slab-center condition; it neither establishes the stronger area-law hypothesis nor completes the continuum target."
        }
      ],
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        "assessment-center-divergent-potential-seed-20261023",
        "assessment-center-divergent-potential-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Confinement",
        "Center symmetry",
        "Correlation inequalities"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": null
    },
    {
      "id": "claim-uniform-decay-area-law",
      "title": "Uniform boundary-condition decay implies an area law",
      "dossier_url": "/research/results/confinement-criterion/",
      "claim_type": "conditional-theorem",
      "workflow_visibility": "public",
      "target_relation": "conditional-bridge",
      "statement": "Theorem 2.4 obtains an area law using the exponential-correlation hypothesis of Definition 2.3, uniform under arbitrary boundary conditions.",
      "plain_language": "Theorems relate a precise unbroken-center condition to confinement, and a stronger correlation-decay hypothesis to a rectangular Wilson-loop area law.",
      "domain": {
        "dimension": "Lattice d ≥ 2",
        "gauge_group": "Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation",
        "matter": "Pure gauge",
        "geometry": "Lattice slabs and boxes",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": "Every boundary condition admitted in Definition 2.3, not only a selected vacuum.",
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Definition 2.3, including its quantifiers over boxes, boundary conditions and local observables; center-charged representation."
      ],
      "conclusion": "Theorem 2.4 obtains an area law using the exponential-correlation hypothesis of Definition 2.3, uniform under arbitrary boundary conditions.",
      "limitations": [
        "The boundary-condition-uniform hypothesis is stronger than decay in one vacuum correlator. Its required weak-coupling continuum control is not established here.",
        "Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied."
      ],
      "source_locators": [
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Section 2.1 (model)",
          "url": "https://arxiv.org/html/2006.16229v5"
        },
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Definition 2.3 and Theorem 2.4 (uniform correlation decay and area law)",
          "url": "https://arxiv.org/html/2006.16229v5"
        }
      ],
      "limits": [],
      "obligation_links": [
        {
          "obligation_id": "volume",
          "relation": "conditional",
          "explanation": "Uniform boundary-condition decay implies an area law: The boundary-condition-uniform hypothesis is stronger than decay in one vacuum correlator. Its required weak-coupling continuum control is not established here."
        }
      ],
      "assessment_ids": [
        "assessment-uniform-decay-area-law-seed-20261023",
        "assessment-uniform-decay-area-law-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Confinement",
        "Center symmetry",
        "Correlation inequalities"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": null
    },
    {
      "id": "claim-dynamical-area-law",
      "title": "Volume-uniform lattice area law at strong coupling",
      "dossier_url": "/research/results/lattice-area-law/",
      "claim_type": "theorem-as-stated",
      "workflow_visibility": "public",
      "target_relation": "restricted-model",
      "statement": "Theorem 1.6 gives rectangular Wilson-loop area-law bounds in the stated small-beta lattice regime with constants independent of lattice volume.",
      "plain_language": "Stochastic estimates for conditional slab models yield rectangular Wilson-loop area-law bounds uniform in the lattice volume.",
      "domain": {
        "dimension": "Lattice d ≥ 2",
        "gauge_group": "Theorem 1.6 lists U(N), SU(N), SO(2(N−1)), N ≥ 2. A nonempty positive-beta SO(M) regime requires even M ≥ 4.",
        "matter": "Pure gauge",
        "geometry": "Finite periodic lattice boxes; rectangular loops subject to size bounds",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": "Periodic box; rectangular side lengths at most L/2.",
        "normalization": "Equation (1.2) uses exp(N beta sum_p ReTr Q_p), where N is matrix size. For SO(M), that coefficient is M beta. beta* = 1/[8(d−1)] for U/SU, and 1/[16(d−1)] − 1/[8M(d−1)] for SO(M), with beta < beta*.",
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Definition 1.4 coupling restriction and Theorem 1.6 group/loop/box conditions; normalization in Remark 1.5."
      ],
      "conclusion": "Theorem 1.6 gives rectangular Wilson-loop area-law bounds in the stated small-beta lattice regime with constants independent of lattice volume.",
      "limitations": [
        "The lattice regulator remains. The auxiliary dynamical estimates are not by themselves a continuum physical Hamiltonian gap.",
        "Definition 1.4 gives beta*=1/[8(d-1)] for U/SU and beta*=1/[16(d-1)]-1/[8M(d-1)] for SO(M); beta<beta* and the stated group-parameter restrictions matter. Odd SO is excluded. Finite-regulator result; stochastic estimates are not a physical continuum gap."
      ],
      "source_locators": [
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 1, action (1.2), Definition 1.4, Remark 1.5, Theorem 1.6 and Remark 1.7",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 2, slab criterion and dependency discussion",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Printed page 2 visually inspected in local Poppler rendering",
          "url": "https://arxiv.org/abs/2509.04688v2"
        }
      ],
      "limits": [
        {
          "parameter": "lattice box size L",
          "operation": "Increase volume in the fixed-regulator regime",
          "fixed_quantities": [
            "beta",
            "d",
            "N"
          ],
          "uniformity": "Constants C,c in C exp(−c area) depend on beta,d,N but not L.",
          "order_notes": "No cutoff-removal limit is established."
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "volume",
          "relation": "ingredient",
          "explanation": "Volume-uniform lattice area law at strong coupling: The lattice regulator remains. The auxiliary dynamical estimates are not by themselves a continuum physical Hamiltonian gap."
        }
      ],
      "assessment_ids": [
        "assessment-dynamical-area-law-seed-20261023",
        "assessment-dynamical-area-law-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Confinement",
        "Strong coupling",
        "Stochastic methods"
      ],
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      "obligation_note": null
    },
    {
      "id": "claim-stochastic-counterterm-uniqueness",
      "title": "Gauge covariance selects the stochastic mass counterterm",
      "dossier_url": "/research/results/stochastic-renormalization/",
      "claim_type": "theorem-as-stated",
      "workflow_visibility": "public",
      "target_relation": "related-method",
      "statement": "The source establishes uniqueness of the gauge-covariant Yang–Mills mass renormalization for local stochastic three-dimensional Yang–Mills–Higgs dynamics.",
      "plain_language": "Gauge covariance selects a unique Yang–Mills mass renormalization in the local stochastic construction, strengthening control over approximations.",
      "domain": {
        "dimension": "3D",
        "gauge_group": "Compact Lie group in the source setup",
        "matter": "Yang–Mills–Higgs",
        "geometry": "Three-dimensional torus; auxiliary stochastic time",
        "regulator": "Regularized stochastic dynamics",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Compact Lie group and the specified representation on T3; non-anticipative mollifier class; finite counterterm shifts in the stated equivariant classes; local possibly explosive dynamics of Theorem 1.6."
      ],
      "conclusion": "The source establishes uniqueness of the gauge-covariant Yang–Mills mass renormalization for local stochastic three-dimensional Yang–Mills–Higgs dynamics.",
      "limitations": [
        "A stochastic counterterm is not a physical spectral gap. The result concerns three-dimensional local dynamics, not a four-dimensional continuum vacuum theory.",
        "Local possibly explosive dynamics and counterterm uniqueness. No invariant 3D Yang-Mills measure or physical Hamiltonian mass-gap theorem is established. Complete dependency proofs and all function-space estimates not audited. HTML conversion truncated; PDF used."
      ],
      "source_locators": [
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Sections 1.1-1.2",
          "url": "https://arxiv.org/abs/2503.03060v2"
        },
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Theorem 1.6 and Remark 1.7, printed pages 7-8, in PDF",
          "url": "https://arxiv.org/abs/2503.03060v2"
        }
      ],
      "limits": [
        {
          "parameter": "regularization scale",
          "operation": "Remove the mollifier within the specified local stochastic construction",
          "fixed_quantities": [
            "three-dimensional torus",
            "admissible initial data and local stopping regime"
          ],
          "uniformity": "Only the source’s local-dynamics theorem is invoked; global time and invariant measure are separate.",
          "order_notes": null
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "continuum",
          "relation": "ingredient",
          "explanation": "Gauge covariance selects the stochastic mass counterterm: A stochastic counterterm is not a physical spectral gap. The result concerns three-dimensional local dynamics, not a four-dimensional continuum vacuum theory."
        }
      ],
      "assessment_ids": [
        "assessment-stochastic-counterterm-uniqueness-seed-20261023",
        "assessment-stochastic-counterterm-uniqueness-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Stochastic quantization",
        "Lower dimensions",
        "Renormalization"
      ],
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      "unresolved_checks": [],
      "obligation_note": null
    },
    {
      "id": "claim-bootstrap-loop-constraints",
      "title": "Multiple-trace loop equations and positivity constraints",
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      "claim_type": "method",
      "workflow_visibility": "public",
      "target_relation": "related-method",
      "statement": "The method combines multiple-trace loop equations with positive-semidefinite matrices of Wilson-loop observables to restrict admissible expectation values.",
      "plain_language": "Multi-trace loop equations and positivity constrain plaquette expectations. The added twist-reflection condition is proved only in two dimensions.",
      "domain": {
        "dimension": "2D, 3D, 4D lattice",
        "gauge_group": "SU(3), with SU(2) comparisons",
        "matter": "Pure gauge",
        "geometry": "Euclidean lattice",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Specified observable basis, trace normalization and exact identities; each reflection condition retains its dimensional domain."
      ],
      "conclusion": "The method combines multiple-trace loop equations with positive-semidefinite matrices of Wilson-loop observables to restrict admissible expectation values.",
      "limitations": [
        "Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.",
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked."
      ],
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 2 setup and strategy",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2, especially equation (43)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "limits": [],
      "obligation_links": [],
      "assessment_ids": [
        "assessment-bootstrap-loop-constraints-seed-20261023",
        "assessment-bootstrap-loop-constraints-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Bootstrap",
        "Positivity",
        "Wilson loops"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": "This method or constraint does not establish a target proof obligation; its value is in organizing restricted lattice information."
    },
    {
      "id": "claim-twist-reflection-two-dimensions",
      "title": "The additional twist-reflection positivity argument is two-dimensional",
      "dossier_url": "/research/results/positivity-bootstrap/",
      "claim_type": "theorem-as-stated",
      "workflow_visibility": "public",
      "target_relation": "restricted-model",
      "statement": "Section 3.2.2 justifies the additional twist-reflection positivity condition in two dimensions.",
      "plain_language": "Multi-trace loop equations and positivity constrain plaquette expectations. The added twist-reflection condition is proved only in two dimensions.",
      "domain": {
        "dimension": "2D lattice",
        "gauge_group": "SU(3) in the paper’s setting",
        "matter": "Pure gauge",
        "geometry": "Two-dimensional Euclidean lattice",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Two-dimensional geometry and reflection operation of section 3.2.2; no transfer to 3D or 4D is inferred."
      ],
      "conclusion": "Section 3.2.2 justifies the additional twist-reflection positivity condition in two dimensions.",
      "limitations": [
        "Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.",
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked."
      ],
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 2 setup and strategy",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2, especially equation (43)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "limits": [],
      "obligation_links": [],
      "assessment_ids": [
        "assessment-twist-reflection-two-dimensions-seed-20261023",
        "assessment-twist-reflection-two-dimensions-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Bootstrap",
        "Positivity",
        "Wilson loops"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": "This method or constraint does not establish a target proof obligation; its value is in organizing restricted lattice information."
    },
    {
      "id": "claim-bootstrap-numerical-bounds",
      "title": "Finite bootstrap computations constrain plaquette values",
      "dossier_url": "/research/results/positivity-bootstrap/",
      "claim_type": "numerical-evidence",
      "workflow_visibility": "public",
      "target_relation": "numerical-support",
      "statement": "The paper reports numerical plaquette bounds from finite loop/positivity systems in two, three and four dimensions.",
      "plain_language": "Multi-trace loop equations and positivity constrain plaquette expectations. The added twist-reflection condition is proved only in two dimensions.",
      "domain": {
        "dimension": "2D, 3D, 4D lattice",
        "gauge_group": "SU(3), with SU(2) comparisons",
        "matter": "Pure gauge",
        "geometry": "Euclidean lattice",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Finite truncation and solver implementation; a numerical optimum is distinct from an independently checked rigorous certificate."
      ],
      "conclusion": "The paper reports numerical plaquette bounds from finite loop/positivity systems in two, three and four dimensions.",
      "limitations": [
        "Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.",
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked."
      ],
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 6.2 (2D SU(3) numerical results)",
          "url": "https://arxiv.org/html/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 7.2, Figure 14 (3D SU(3))",
          "url": "https://arxiv.org/html/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 7.4, Figure 16 (4D SU(3))",
          "url": "https://arxiv.org/html/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 8, Figure 17 (summary)",
          "url": "https://arxiv.org/html/2502.14421v2"
        }
      ],
      "limits": [],
      "obligation_links": [],
      "assessment_ids": [
        "assessment-bootstrap-numerical-bounds-seed-20261023",
        "assessment-bootstrap-numerical-bounds-20261024",
        "assessment-bootstrap-numerical-bounds-cross-review-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Bootstrap",
        "Positivity",
        "Wilson loops"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": "This method or constraint does not establish a target proof obligation; its value is in organizing restricted lattice information."
    },
    {
      "id": "claim-direct-indirect-equations",
      "title": "Elimination beyond a loop cutoff supplies indirect relations",
      "dossier_url": "/research/results/loop-equations/",
      "claim_type": "method",
      "workflow_visibility": "public",
      "target_relation": "related-method",
      "statement": "The framework generates direct loop equations and relations obtained by eliminating longer intermediate loops, with SU(2) applications in two, three and four lattice dimensions.",
      "plain_language": "A systematic construction of direct loop equations and relations induced by eliminating longer loops improves the organization of lattice constraints.",
      "domain": {
        "dimension": "2D, 3D, 4D lattice",
        "gauge_group": "SU(2) applications",
        "matter": "Pure gauge",
        "geometry": "Euclidean lattice",
        "regulator": "Lattice",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Finite retained loop basis, exact input identities and elimination rules; no convergence of the full hierarchy is claimed."
      ],
      "conclusion": "The framework generates direct loop equations and relations obtained by eliminating longer intermediate loops, with SU(2) applications in two, three and four lattice dimensions.",
      "limitations": [
        "An equation-generation method is not a proof that finite truncations converge, nor a spectral or continuum construction result.",
        "Completeness in reported finite comparisons is not convergence of the infinite hierarchy. Auxiliary loop-space enlargement, truncation and projected relations remain explicit; no continuum or spectral result."
      ],
      "source_locators": [
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Section 1",
          "url": "https://arxiv.org/abs/2601.04316v2"
        },
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Section 5.1, Table 6",
          "url": "https://arxiv.org/abs/2601.04316v2"
        },
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Section 5.2 improved filtering and Section 5.3 application examples",
          "url": "https://arxiv.org/abs/2601.04316v2"
        }
      ],
      "limits": [],
      "obligation_links": [],
      "assessment_ids": [
        "assessment-direct-indirect-equations-seed-20261023",
        "assessment-direct-indirect-equations-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Bootstrap",
        "Loop equations",
        "2026"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": "This method or constraint does not establish a target proof obligation; its value is in organizing restricted lattice information."
    },
    {
      "id": "claim-continuum-glueball-spectrum",
      "title": "Continuum-extrapolated pure SU(N) glueball spectra",
      "dossier_url": "/research/results/glueball-spectrum/",
      "claim_type": "numerical-evidence",
      "workflow_visibility": "public",
      "target_relation": "numerical-support",
      "statement": "Monte Carlo calculations estimate low-lying glueball masses at sampled values of N, with continuum extrapolations at fixed N and a subsequent large-N analysis.",
      "plain_language": "A systematic numerical benchmark for glueball masses, string tensions, and related observables, with continuum extrapolations at fixed N.",
      "domain": {
        "dimension": "3+1D",
        "gauge_group": "SU(N), N = 2, 3, 4, 5, 6, 8, 10, 12",
        "matter": "Pure gauge",
        "geometry": "Finite periodic lattice volumes with volume studies",
        "regulator": "Lattice with numerical continuum extrapolation",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Wilson plaquette action; finite operator bases, statistical sampling and extrapolation assumptions; separate finite-volume and cutoff analyses."
      ],
      "conclusion": "Monte Carlo calculations estimate low-lying glueball masses at sampled values of N, with continuum extrapolations at fixed N and a subsequent large-N analysis.",
      "limitations": [
        "Numerical estimates depend on statistical and systematic control. They do not prove a lower bound on the complete physical spectrum or construct the continuum theory.",
        "Quoted numerical uncertainty is not a rigorous spectral lower bound or a guarantee that every systematic effect is covered. Spin assignment, poor continuum fits and multiparticle/ditorelon contamination require the paper's caveats. No simulation reproduced."
      ],
      "source_locators": [
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Sections 5.3-5.7; continuum extrapolation equation (27)",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Tables 23-30 (sampled groups)",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Table 24, printed page 74, A1++ ground state",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Table 34, printed page 83, 0++ gs SU(3)",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Table 34 PDF page 84 visually inspected; 0++ gs SU(3) cell 3.405(21), no star",
          "url": "https://arxiv.org/abs/2106.00364v3"
        }
      ],
      "limits": [
        {
          "parameter": "a² sigma",
          "operation": "Numerical continuum extrapolation via equation (27)",
          "fixed_quantities": [
            "N",
            "dimensionless observable definition"
          ],
          "uniformity": "Finite-volume and fit-systematic checks are numerical controls, not a rigorous uniform spectral bound.",
          "order_notes": "Fixed-N continuum extrapolations precede the separate large-N fits."
        },
        {
          "parameter": "1/N²",
          "operation": "Separate large-N extrapolation where reported",
          "fixed_quantities": [
            "continuum-extrapolated dimensionless normalization"
          ],
          "uniformity": "Sampled finite ranks and fit assumptions; no theorem for every compact simple group.",
          "order_notes": null
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "gap",
          "relation": "numerical-support",
          "explanation": "Continuum-extrapolated pure SU(N) glueball spectra: Numerical estimates depend on statistical and systematic control. They do not prove a lower bound on the complete physical spectrum or construct the continuum theory."
        }
      ],
      "assessment_ids": [
        "assessment-continuum-glueball-spectrum-seed-20261023",
        "assessment-continuum-glueball-spectrum-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Glueballs",
        "Lattice spectroscopy",
        "Numerical evidence"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": null
    },
    {
      "id": "claim-deformed-small-circle",
      "title": "Confinement mechanisms in center-stabilized small-circle Yang–Mills",
      "dossier_url": "/research/results/semiclassical-confinement/",
      "claim_type": "controlled-approximation",
      "workflow_visibility": "public",
      "target_relation": "restricted-model",
      "statement": "A double-trace deformation stabilizes center symmetry and permits controlled small-circle semiclassical dynamics; large-N volume independence has separate symmetry hypotheses.",
      "plain_language": "Center-stabilized and twisted compactifications make nonperturbative mechanisms analytically accessible, including monopole and center-vortex descriptions.",
      "domain": {
        "dimension": "4D with one compact direction",
        "gauge_group": "SU(N)",
        "matter": "Pure gauge with double-trace deformation",
        "geometry": "R3 × S1",
        "regulator": "Continuum compactified effective description",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Double-trace deformation; compactification and weak-coupling hierarchy; the small-circle regime cannot be extrapolated without error control."
      ],
      "conclusion": "A double-trace deformation stabilizes center symmetry and permits controlled small-circle semiclassical dynamics; large-N volume independence has separate symmetry hypotheses.",
      "limitations": [
        "The deformation, twist, geometry, and regime matter. Continuing to undeformed pure Yang–Mills on R⁴ requires additional control.",
        "At fixed circle length, N to infinity exits the Abelian semiclassical regime. Section 3.3 requires correlated scaling to retain control; no extension to undeformed finite-N R4 is established."
      ],
      "source_locators": [
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Section 1",
          "url": "https://arxiv.org/abs/0803.0344v2"
        },
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Sections 3.1-3.2, equations (3.31)-(3.37) and (3.44)-(3.47)",
          "url": "https://arxiv.org/abs/0803.0344v2"
        },
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Section 3.3, equation (3.48) and large-N limits",
          "url": "https://arxiv.org/abs/0803.0344v2"
        }
      ],
      "limits": [
        {
          "parameter": "NL Lambda",
          "operation": "Remain much smaller than one in the semiclassical regime",
          "fixed_quantities": [
            "specified deformation or twist",
            "geometry and expansion assumptions"
          ],
          "uniformity": "No uniform continuation to undeformed R4 is established.",
          "order_notes": "Taking N large at fixed L leaves the stated Abelian/dilute regime; correlated scaling is a different limit."
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "gap",
          "relation": "ingredient",
          "explanation": "Confinement mechanisms in center-stabilized small-circle Yang–Mills: The deformation, twist, geometry, and regime matter. Continuing to undeformed pure Yang–Mills on R⁴ requires additional control."
        }
      ],
      "assessment_ids": [
        "assessment-deformed-small-circle-seed-20261023",
        "assessment-deformed-small-circle-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Semiclassics",
        "Compactification",
        "Confinement"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": null
    },
    {
      "id": "claim-twisted-center-vortices",
      "title": "Center-vortex semiclassics with non-minimal twist",
      "dossier_url": "/research/results/semiclassical-confinement/",
      "claim_type": "controlled-approximation",
      "workflow_visibility": "public",
      "target_relation": "restricted-model",
      "statement": "The source develops center-vortex semiclassics for SU(N) on R2 × T2 with coprime non-minimal ’t Hooft flux in the small-torus regime.",
      "plain_language": "Center-stabilized and twisted compactifications make nonperturbative mechanisms analytically accessible, including monopole and center-vortex descriptions.",
      "domain": {
        "dimension": "4D with two compact directions",
        "gauge_group": "SU(N)",
        "matter": "Pure gauge",
        "geometry": "R2 × T2 with ’t Hooft twist",
        "regulator": "Continuum compactified effective description",
        "global_form": null,
        "signature": "Euclidean",
        "temperature": null,
        "theta": null,
        "boundary_conditions": null,
        "normalization": null,
        "unchecked_details": [
          "Null domain fields are not inferred from a conventional default."
        ]
      },
      "assumptions": [
        "Coprime non-minimal twist p with gcd(N,p)=1; NLΛ ≪ 1; controlled dilute-vortex expansion."
      ],
      "conclusion": "The source develops center-vortex semiclassics for SU(N) on R2 × T2 with coprime non-minimal ’t Hooft flux in the small-torus regime.",
      "limitations": [
        "The deformation, twist, geometry, and regime matter. Continuing to undeformed pure Yang–Mills on R⁴ requires additional control.",
        "Controlled semiclassical approximation, not a rigorous full R4 construction. Generalized-anomaly statements have separate geometric hypotheses. No decompactification or undeformed-target conclusion inferred."
      ],
      "source_locators": [
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Section 2.3 center-vortex semiclassics",
          "url": "https://arxiv.org/abs/2505.07467v2"
        },
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Section 2.3.2, equations (47)-(51)",
          "url": "https://arxiv.org/abs/2505.07467v2"
        },
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Section 2.4 opening anomaly-matching hypotheses",
          "url": "https://arxiv.org/abs/2505.07467v2"
        }
      ],
      "limits": [
        {
          "parameter": "NL Lambda",
          "operation": "Remain much smaller than one in the semiclassical regime",
          "fixed_quantities": [
            "specified deformation or twist",
            "geometry and expansion assumptions"
          ],
          "uniformity": "No uniform continuation to undeformed R4 is established.",
          "order_notes": "Taking N large at fixed L leaves the stated Abelian/dilute regime; correlated scaling is a different limit."
        }
      ],
      "obligation_links": [
        {
          "obligation_id": "gap",
          "relation": "ingredient",
          "explanation": "Center-vortex semiclassics with non-minimal twist: The deformation, twist, geometry, and regime matter. Continuing to undeformed pure Yang–Mills on R⁴ requires additional control."
        }
      ],
      "assessment_ids": [
        "assessment-twisted-center-vortices-seed-20261023",
        "assessment-twisted-center-vortices-20261024"
      ],
      "dependencies": [],
      "tags": [
        "Semiclassics",
        "Compactification",
        "Confinement"
      ],
      "last_substantive_update": "2026-10-24",
      "unresolved_checks": [],
      "obligation_note": null
    }
  ],
  "assessments": [
    {
      "id": "assessment-uv-stability-seed-20261023",
      "claim_ids": [
        "claim-uv-stability"
      ],
      "version_ids": [
        "version-balaban-1989-large-field-ii-1989"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Publisher abstract (refers to Theorem 1)",
        "Bibliographic record"
      ],
      "finding": "Publisher abstract says that bounds for the R operation complete the ultraviolet-stability proof formulated in Theorem 1 for four-dimensional pure gauge field theories.",
      "limits": [
        "Exact group, lattice, finite-volume, and coupling assumptions must be recovered from Theorem 1 before a more detailed claim is published. No full continuum construction or physical mass-gap theorem is asserted by this record.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-center-divergent-potential-seed-20261023",
      "claim_ids": [
        "claim-center-divergent-potential"
      ],
      "version_ids": [
        "version-chatterjee-2021-confinement-v5"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2.1; Definition 2.1; Theorem 2.2"
      ],
      "finding": "Theorem 2.2 obtains confinement with a divergent static potential from the defined slab-center condition for center-charged representations. Theorem 2.4 adds exponential correlation decay uniform under arbitrary boundary conditions to establish an area law.",
      "limits": [
        "These hypotheses must not be identified with decay of one vacuum correlator. This does not establish the required weak-coupling continuum control for the Millennium problem.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-uniform-decay-area-law-seed-20261023",
      "claim_ids": [
        "claim-uniform-decay-area-law"
      ],
      "version_ids": [
        "version-chatterjee-2021-confinement-v5"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Definition 2.3; Theorem 2.4"
      ],
      "finding": "Theorem 2.2 obtains confinement with a divergent static potential from the defined slab-center condition for center-charged representations. Theorem 2.4 adds exponential correlation decay uniform under arbitrary boundary conditions to establish an area law.",
      "limits": [
        "These hypotheses must not be identified with decay of one vacuum correlator. This does not establish the required weak-coupling continuum control for the Millennium problem.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-dynamical-area-law-seed-20261023",
      "claim_ids": [
        "claim-dynamical-area-law"
      ],
      "version_ids": [
        "version-cao-nissim-sheffield-2025-dynamical-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 1, especially Theorem 1.6 and definitions preceding it",
        "Section 2, slab criterion",
        "Section 3, Proposition 3.2 and semigroup estimates"
      ],
      "finding": "Theorem 1.6 gives rectangular Wilson-loop area-law bounds with volume-independent constants in the stated small-beta 't Hooft-normalized lattice regime, for the specified unitary, special unitary, and even orthogonal groups. Section 3 develops conditional sigma-model estimates.",
      "limits": [
        "Retain the exact beta thresholds in Definition 1.4, normalization discussion in Remark 1.5, and group restrictions in Theorem 1.6 when implementing a technical record. Auxiliary stochastic estimates and a fixed-regulator area law do not themselves prove a four-dimensional physical continuum gap.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-stochastic-counterterm-uniqueness-seed-20261023",
      "claim_ids": [
        "claim-stochastic-counterterm-uniqueness"
      ],
      "version_ids": [
        "version-chevyrev-shen-2026-renormalisation-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Abstract",
        "Version history and journal reference"
      ],
      "finding": "The abstract states uniqueness of the Yang–Mills mass renormalization yielding gauge-covariant solutions in the previously constructed local 3D stochastic Yang–Mills–Higgs setting.",
      "limits": [
        "The counterterm is not a physical mass-gap result. Read the full main theorem to confirm geometry, group assumptions, state spaces, and approximation class before extending the dossier.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-bootstrap-loop-constraints-seed-20261023",
      "claim_ids": [
        "claim-bootstrap-loop-constraints"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Abstract",
        "Section 2 (setup and strategy)",
        "Section 3.2.2 (twist reflection positivity)",
        "Section 4 (loop equations)"
      ],
      "finding": "Wilson-loop equations and positivity constrain SU(3) lattice observables with 2D, 3D, and 4D numerical applications. The additional twist-reflection positivity argument in section 3.2.2 is justified in 2D.",
      "limits": [
        "Exact constraints, numerical solver certification, and convergence of an infinite hierarchy must be distinct claims. Short-loop bounds do not constitute a continuum spectrum or mass-gap theorem.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-twist-reflection-two-dimensions-seed-20261023",
      "claim_ids": [
        "claim-twist-reflection-two-dimensions"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 3.2.2 (two-dimensional twist reflection positivity)"
      ],
      "finding": "Wilson-loop equations and positivity constrain SU(3) lattice observables with 2D, 3D, and 4D numerical applications. The additional twist-reflection positivity argument in section 3.2.2 is justified in 2D.",
      "limits": [
        "Exact constraints, numerical solver certification, and convergence of an infinite hierarchy must be distinct claims. Short-loop bounds do not constitute a continuum spectrum or mass-gap theorem.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-bootstrap-numerical-bounds-seed-20261023",
      "claim_ids": [
        "claim-bootstrap-numerical-bounds"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "selected-sections",
      "examined": [
        "Abstract",
        "Section 2 (setup and strategy)",
        "Section 3.2.2 (twist reflection positivity)",
        "Section 4 (loop equations)"
      ],
      "finding": "Wilson-loop equations and positivity constrain SU(3) lattice observables with 2D, 3D, and 4D numerical applications. The additional twist-reflection positivity argument in section 3.2.2 is justified in 2D.",
      "limits": [
        "Exact constraints, numerical solver certification, and convergence of an infinite hierarchy must be distinct claims. Short-loop bounds do not constitute a continuum spectrum or mass-gap theorem.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-direct-indirect-equations-seed-20261023",
      "claim_ids": [
        "claim-direct-indirect-equations"
      ],
      "version_ids": [
        "version-liu-yang-2026-loop-equations-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Abstract",
        "Version history"
      ],
      "finding": "The abstract describes systematic construction of direct loop equations and indirect relations obtained through elimination of longer intermediate loops, with SU(2) lattice examples in 2D, 3D, and 4D.",
      "limits": [
        "This record does not assert convergence of truncated hierarchies, a continuum construction, or a spectral-gap result. Detailed completeness and independence claims require checking the text.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-continuum-glueball-spectrum-seed-20261023",
      "claim_ids": [
        "claim-continuum-glueball-spectrum"
      ],
      "version_ids": [
        "version-athenodorou-teper-2021-spectrum-v3"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Abstract",
        "Version history and publication metadata"
      ],
      "finding": "The abstract reports Wilson-action lattice simulations of SU(N) theories for sampled values of N between 2 and 12, with continuum extrapolations of low-lying spectra and further large-N extrapolations.",
      "limits": [
        "The data support numerical spectroscopy, not a rigorous lower bound on the entire spectrum. Verify exact sampled values of N, tables, scale setting, and statistical/systematic errors before transcribing benchmark values.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-deformed-small-circle-seed-20261023",
      "claim_ids": [
        "claim-deformed-small-circle"
      ],
      "version_ids": [
        "version-unsal-yaffe-2008-center-stabilized-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Abstract",
        "Version history and journal reference"
      ],
      "finding": "The abstract studies a double-trace-deformed SU(N) model, controlled small-circle nonperturbative dynamics at suitably small N, and large-N volume independence under the stated center-stabilization conditions.",
      "limits": [
        "Deformation, geometry, hierarchy of scales, and large-N assumptions must remain visible. This does not itself establish the undeformed finite-N R4 target.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-twisted-center-vortices-seed-20261023",
      "claim_ids": [
        "claim-twisted-center-vortices"
      ],
      "version_ids": [
        "version-hayashi-tanizaki-unsal-2026-center-vortex-v2"
      ],
      "date": "2026-10-23",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Inherited webmaster-kit planning inspection; not a new reading on import.",
      "read_depth": "abstract",
      "examined": [
        "Abstract",
        "Version history",
        "Journal reference"
      ],
      "finding": "The abstract treats SU(N) on small R2 x T2 with coprime non-minimal 't Hooft twist, constructing center-vortex semiclassics and discussing center stabilization at large N.",
      "limits": [
        "The dilute-gas regime NL Lambda much less than 1 and twist assumptions are essential. No continuation to ordinary undeformed R4 is established by this register entry. First submission, v2 revision, and journal year are distinct dates.",
        "No independent expert proof verification or computational reproduction is recorded."
      ],
      "external_source_urls": [],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-uv-stability-20261024",
      "claim_ids": [
        "claim-uv-stability"
      ],
      "version_ids": [
        "version-balaban-1989-large-field-ii-1989"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "abstract",
      "examined": [
        "Publisher abstract referring to Theorem 1; bibliographic record"
      ],
      "finding": "Author abstract says bounds for the R operation complete the stated four-dimensional pure-gauge ultraviolet stability result.",
      "limits": [
        "Theorem 1 group, regulator, finite-volume and coupling hypotheses were not obtained. Do not publish a detailed theorem digest from this abstract. UV stability is not the full continuum existence-and-gap conclusion.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-center-divergent-potential-20261024",
      "claim_ids": [
        "claim-center-divergent-potential"
      ],
      "version_ids": [
        "version-chatterjee-2021-confinement-v5"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2.1 model",
        "Definition 2.1 and Theorem 2.2",
        "Definition 2.3 and Theorem 2.4"
      ],
      "finding": "For pure lattice theory with closed connected G subgroup of U(n), d>=2, the slab-center condition implies divergent static potential for center-charged irreducible finite-dimensional unitary representations. The stronger arbitrary-boundary exponential-correlation hypothesis yields rectangular area law for every Gibbs measure.",
      "limits": [
        "Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-uniform-decay-area-law-20261024",
      "claim_ids": [
        "claim-uniform-decay-area-law"
      ],
      "version_ids": [
        "version-chatterjee-2021-confinement-v5"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2.1 model",
        "Definition 2.1 and Theorem 2.2",
        "Definition 2.3 and Theorem 2.4"
      ],
      "finding": "For pure lattice theory with closed connected G subgroup of U(n), d>=2, the slab-center condition implies divergent static potential for center-charged irreducible finite-dimensional unitary representations. The stronger arbitrary-boundary exponential-correlation hypothesis yields rectangular area law for every Gibbs measure.",
      "limits": [
        "Definition 2.3 constants are uniform over cubes and every boundary condition and bounded edge-local observables. Decay in one vacuum is insufficient; no weak-coupling continuum bound is supplied.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-dynamical-area-law-20261024",
      "claim_ids": [
        "claim-dynamical-area-law"
      ],
      "version_ids": [
        "version-cao-nissim-sheffield-2025-dynamical-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 1, action (1.2), Definition 1.4, Remark 1.5, Theorem 1.6 and Remark 1.7",
        "Section 2, slab criterion and dependency discussion",
        "Printed page 2 visually inspected in local Poppler rendering"
      ],
      "finding": "Finite periodic d-dimensional lattice, d>=2, pure gauge action N beta sum ReTr Qp. Theorem 1.6 bounds rectangular loops with sides at most L/2 by C exp(-c area), C,c depending on beta,d,N but independent of L, in the small-beta regime for U(N), SU(N), and even orthogonal groups.",
      "limits": [
        "Definition 1.4 gives beta*=1/[8(d-1)] for U/SU and beta*=1/[16(d-1)]-1/[8M(d-1)] for SO(M); beta<beta* and the stated group-parameter restrictions matter. Odd SO is excluded. Finite-regulator result; stochastic estimates are not a physical continuum gap.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-stochastic-counterterm-uniqueness-20261024",
      "claim_ids": [
        "claim-stochastic-counterterm-uniqueness"
      ],
      "version_ids": [
        "version-chevyrev-shen-2026-renormalisation-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Sections 1.1-1.2",
        "Theorem 1.6 and Remark 1.7, printed pages 7-8, in PDF"
      ],
      "finding": "Uniqueness of gauge-covariant mass renormalization in local stochastic Yang-Mills-Higgs on T3, compact Lie group and the specified representation, nonanticipative mollifier class. Finite counterterm shifts in the stated equivariant classes distinguish gauge-related regularizations.",
      "limits": [
        "Local possibly explosive dynamics and counterterm uniqueness. No invariant 3D Yang-Mills measure or physical Hamiltonian mass-gap theorem is established. Complete dependency proofs and all function-space estimates not audited. HTML conversion truncated; PDF used.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-bootstrap-loop-constraints-20261024",
      "claim_ids": [
        "claim-bootstrap-loop-constraints"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2 setup and strategy",
        "Section 3.2.2, especially equation (43)"
      ],
      "finding": "Finite-N SU(3) Wilson-loop bootstrap uses exact loop identities and ordinary positivity with numerical truncated applications in 2D/3D/4D. The extra twist-reflection argument is justified in two dimensions after the stated gauge fixing.",
      "limits": [
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-twist-reflection-two-dimensions-20261024",
      "claim_ids": [
        "claim-twist-reflection-two-dimensions"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2 setup and strategy",
        "Section 3.2.2, especially equation (43)"
      ],
      "finding": "Finite-N SU(3) Wilson-loop bootstrap uses exact loop identities and ordinary positivity with numerical truncated applications in 2D/3D/4D. The extra twist-reflection argument is justified in two dimensions after the stated gauge fixing.",
      "limits": [
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-bootstrap-numerical-bounds-20261024",
      "claim_ids": [
        "claim-bootstrap-numerical-bounds"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2 setup and strategy",
        "Section 3.2.2, especially equation (43)"
      ],
      "finding": "Finite-N SU(3) Wilson-loop bootstrap uses exact loop identities and ordinary positivity with numerical truncated applications in 2D/3D/4D. The extra twist-reflection argument is justified in two dimensions after the stated gauge fixing.",
      "limits": [
        "Twist positivity cannot be advertised as a general 3D/4D result. Numerical solver output is separate from certified inequalities; no hierarchy-convergence or continuum spectrum result checked.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-direct-indirect-equations-20261024",
      "claim_ids": [
        "claim-direct-indirect-equations"
      ],
      "version_ids": [
        "version-liu-yang-2026-loop-equations-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 1",
        "Section 5.1, Table 6",
        "Section 5.2 improved filtering and Section 5.3 application examples"
      ],
      "finding": "Direct loop equations and indirect relations obtained by elimination of longer intermediate loops, with finite SU(2) loop sets in 2D/3D/4D. Vertex-support filtering is a computational heuristic tested against the stated finite sets.",
      "limits": [
        "Completeness in reported finite comparisons is not convergence of the infinite hierarchy. Auxiliary loop-space enlargement, truncation and projected relations remain explicit; no continuum or spectral result.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-continuum-glueball-spectrum-20261024",
      "claim_ids": [
        "claim-continuum-glueball-spectrum"
      ],
      "version_ids": [
        "version-athenodorou-teper-2021-spectrum-v3"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Sections 5.3-5.7; continuum extrapolation equation (27)",
        "Tables 23-30 (sampled groups)",
        "Table 24, printed page 74, A1++ ground state",
        "Table 34, printed page 83, 0++ gs SU(3)",
        "Table 34 PDF page 84 visually inspected; 0++ gs SU(3) cell 3.405(21), no star"
      ],
      "finding": "Pure Wilson-action SU(N) lattice spectroscopy in 3+1 dimensions for N=2,3,4,5,6,8,10,12. Finite-spacing variational masses are extrapolated in a^2 sigma and further in 1/N^2 where reported.",
      "limits": [
        "Quoted numerical uncertainty is not a rigorous spectral lower bound or a guarantee that every systematic effect is covered. Spin assignment, poor continuum fits and multiparticle/ditorelon contamination require the paper's caveats. No simulation reproduced.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-deformed-small-circle-20261024",
      "claim_ids": [
        "claim-deformed-small-circle"
      ],
      "version_ids": [
        "version-unsal-yaffe-2008-center-stabilized-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 1",
        "Sections 3.1-3.2, equations (3.31)-(3.37) and (3.44)-(3.47)",
        "Section 3.3, equation (3.48) and large-N limits"
      ],
      "finding": "Double-trace-deformed pure SU(N) on R3 x S1 has calculable dual-photon masses and nonzero string tensions in the dilute-monopole regime NL Lambda <<1, with the stated theta choice and perturbative corrections.",
      "limits": [
        "At fixed circle length, N to infinity exits the Abelian semiclassical regime. Section 3.3 requires correlated scaling to retain control; no extension to undeformed finite-N R4 is established.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-twisted-center-vortices-20261024",
      "claim_ids": [
        "claim-twisted-center-vortices"
      ],
      "version_ids": [
        "version-hayashi-tanizaki-unsal-2026-center-vortex-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex editorial source inspection; inherited and current actions remain distinct.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 2.3 center-vortex semiclassics",
        "Section 2.3.2, equations (47)-(51)",
        "Section 2.4 opening anomaly-matching hypotheses"
      ],
      "finding": "Pure SU(N) on R2 x T2 with coprime twist p, inverse q modulo N, in the small-torus dilute regime NL Lambda <<1. Effective dilute-vortex calculation gives N-ality and twist-dependent string tensions; equations (49)-(51) select the vacuum and theta range.",
      "limits": [
        "Controlled semiclassical approximation, not a rigorous full R4 construction. Generalized-anomaly statements have separate geometric hypotheses. No decompactification or undeformed-target conclusion inferred.",
        "No independent proof certification or numerical simulation reproduction."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/current-review-sources.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-bootstrap-numerical-bounds-cross-review-20261024",
      "claim_ids": [
        "claim-bootstrap-numerical-bounds"
      ],
      "version_ids": [
        "version-guo-li-yang-zhu-2025-bootstrap-v2"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex cross-review of exact numerical-results sections; separate from the earlier setup/positivity inspection.",
      "read_depth": "selected-sections",
      "examined": [
        "Section 6.2 (2D SU(3) numerical results)",
        "Section 7.2, Figure 14 (3D SU(3))",
        "Section 7.4, Figure 16 (4D SU(3))",
        "Section 8, Figure 17 (summary)"
      ],
      "finding": "The 2D/3D/4D SU(3) numerical-result claims are supported by their results sections and figures, rather than only by setup or twist-reflection passages.",
      "limits": [
        "No solver output was reproduced or independently certified; no hierarchy convergence or continuum spectral theorem follows."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/research-cross-review.json"
      ],
      "workflow_visibility": "public"
    },
    {
      "id": "assessment-balaban-selected-sections-followup-20261024",
      "claim_ids": [
        "claim-uv-stability"
      ],
      "version_ids": [
        "version-balaban-1989-large-field-ii-1989",
        "version-balaban-1987-rg-i-1987",
        "version-balaban-1988-convergent-expansions-1988"
      ],
      "date": "2026-10-24",
      "action": "scope-checked",
      "actor_type": "codex",
      "attribution": "Codex follow-up inspection of the published Part II scan and selected [I]/[III] support passages; prior abstract/access assessments retained.",
      "read_depth": "selected-sections",
      "examined": [
        "Printed page 355/PDF page 1: Theorem 1 and following qualification",
        "Printed pages 355-356: inequality (0.1) and stated independence of constants",
        "Printed page 392/PDF page 38: concluding representation (1.104) discussion and references",
        "Introduction, printed pp251-252: torus, lattice scales, compact semisimple matrix group and Wilson action (0.2)",
        "Printed p259: Theorems 1 and 2; p260 opening qualification",
        "Introduction pp243-245: Wilson initial density, E normalization, RT iteration to unit lattice",
        "Section 2 pp254-264, especially density (2.18), effective action (2.23), localized terms (2.25)-(2.42), Theorems 1-2 and Corollary 3 (2.50)"
      ],
      "finding": "Conditional on that running-coupling hypothesis, effective densities have the form and satisfy the conditions and bounds specified in [III] section 2. Part II then states ultraviolet-stability bounds (0.1), with constants independent of renormalization step, torus and background field as specified immediately after the displayed bound. Part II page 355 explicitly retains the coupling assumption: the proposed removal relies on the second-order perturbative result called Theorem 2 in [I], whose proof the article says had not yet been published. This review does not assert that the proof remains unpublished today; that later-history question was not resolved.",
      "limits": [
        "No claim of full-proof review, reconstruction of a nontrivial continuum QFT, infinite-volume control of all physical observables, or physical Hamiltonian mass gap.",
        "The full [I] and [III] proofs, [II] cluster-expansion proof and [IV] first R-operation paper were not audited. Background-field/propagator/cluster-expansion dependencies in earlier references remain unread here.",
        "No assertion is made that all effective couplings stay small for a prescribed continuum tuning; the cited theorem assumes this.",
        "Equations have poor OCR. Exact stability-bound formulas were not retypeset from OCR. Identified objects, assumptions and locators are supported by readable text; no claimed pixel verification for these scans.",
        "Canonical source identity is the journal article/DOI; the readable scan is third-party hosted and is not redistributed.",
        "Web PDF text was readable; local scan download returned HTTP 403, so no local checksum or pixel-inspection pass is asserted."
      ],
      "external_source_urls": [],
      "artifacts": [
        "plans/operations/balaban-followup-20261024.json"
      ],
      "workflow_visibility": "public"
    }
  ],
  "benchmarks": [
    {
      "id": "benchmark-su3-scalar-continuum-2021",
      "claim_id": "claim-continuum-glueball-spectrum",
      "quantity": "m_0++ / sqrt(sigma)",
      "value": 3.405,
      "units": "dimensionless",
      "normalization": "Mass divided by the square root of the confining string tension, m/sqrt(sigma), within pure SU(3) gauge theory.",
      "method": "continuum-extrapolated numerical estimate, pure gauge Wilson-action ensembles",
      "statistical_error": null,
      "systematic_error": null,
      "source_locator": {
        "version_id": "version-athenodorou-teper-2021-spectrum-v3",
        "label": "Table 34, printed page 83 (PDF page 84), 0++ gs row, SU(3) column; cross-check Table 24 A1++ ground state",
        "url": "https://arxiv.org/abs/2106.00364v3"
      },
      "limits": [],
      "notes": "Source notation 3.405(21); quoted uncertainty 0.021. Parenthesized uncertainty as reported; do not relabel as an all-systematics bound. Statistical/systematic decomposition is not asserted by this transcription. No GeV scale or experimental resonance identification is implied.",
      "reproduction_url": null,
      "workflow_visibility": "public"
    }
  ],
  "events": [
    {
      "id": "event-uv-stability-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-uv-stability"
      ],
      "summary": "Separated provenance and claim type for Ultraviolet stability in the stated four-dimensional lattice construction.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Publisher abstract (refers to Theorem 1)",
          "url": "https://link.springer.com/article/10.1007/BF01238433"
        },
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Bibliographic record",
          "url": "https://link.springer.com/article/10.1007/BF01238433"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-center-divergent-potential-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-center-divergent-potential"
      ],
      "summary": "Separated provenance and claim type for Unbroken slab center symmetry implies a divergent potential.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Section 2.1; Definition 2.1; Theorem 2.2",
          "url": "https://arxiv.org/abs/2006.16229v5"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-uniform-decay-area-law-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-uniform-decay-area-law"
      ],
      "summary": "Separated provenance and claim type for Uniform boundary-condition decay implies an area law.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Definition 2.3; Theorem 2.4",
          "url": "https://arxiv.org/abs/2006.16229v5"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-dynamical-area-law-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-dynamical-area-law"
      ],
      "summary": "Separated provenance and claim type for Volume-uniform lattice area law at strong coupling.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 1, especially Theorem 1.6 and definitions preceding it",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 2, slab criterion",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 3, Proposition 3.2 and semigroup estimates",
          "url": "https://arxiv.org/abs/2509.04688v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-stochastic-counterterm-uniqueness-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-stochastic-counterterm-uniqueness"
      ],
      "summary": "Separated provenance and claim type for Gauge covariance selects the stochastic mass counterterm.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2503.03060v2"
        },
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Version history and journal reference",
          "url": "https://arxiv.org/abs/2503.03060v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-bootstrap-loop-constraints-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-bootstrap-loop-constraints"
      ],
      "summary": "Separated provenance and claim type for Multiple-trace loop equations and positivity constraints.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 2 (setup and strategy)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2 (twist reflection positivity)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 4 (loop equations)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-twist-reflection-two-dimensions-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-twist-reflection-two-dimensions"
      ],
      "summary": "Separated provenance and claim type for The additional twist-reflection positivity argument is two-dimensional.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2 (two-dimensional twist reflection positivity)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-bootstrap-numerical-bounds-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-bootstrap-numerical-bounds"
      ],
      "summary": "Separated provenance and claim type for Finite bootstrap computations constrain plaquette values.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 2 (setup and strategy)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2 (twist reflection positivity)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 4 (loop equations)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-direct-indirect-equations-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-direct-indirect-equations"
      ],
      "summary": "Separated provenance and claim type for Elimination beyond a loop cutoff supplies indirect relations.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2601.04316v2"
        },
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Version history",
          "url": "https://arxiv.org/abs/2601.04316v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-continuum-glueball-spectrum-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-continuum-glueball-spectrum"
      ],
      "summary": "Separated provenance and claim type for Continuum-extrapolated pure SU(N) glueball spectra.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Version history and publication metadata",
          "url": "https://arxiv.org/abs/2106.00364v3"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-deformed-small-circle-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-deformed-small-circle"
      ],
      "summary": "Separated provenance and claim type for Confinement mechanisms in center-stabilized small-circle Yang–Mills.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/0803.0344v2"
        },
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Version history and journal reference",
          "url": "https://arxiv.org/abs/0803.0344v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-twisted-center-vortices-provenance-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-twisted-center-vortices"
      ],
      "summary": "Separated provenance and claim type for Center-vortex semiclassics with non-minimal twist.",
      "reason": "Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.",
      "source_locators": [
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2505.07467v2"
        },
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Version history",
          "url": "https://arxiv.org/abs/2505.07467v2"
        },
        {
          "version_id": "version-hayashi-tanizaki-unsal-2026-center-vortex-v2",
          "label": "Journal reference",
          "url": "https://arxiv.org/abs/2505.07467v2"
        }
      ],
      "previous_statement": "Source scope checked, 22 October 2026 (inherited undifferentiated label).",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-inherited-ultraviolet-stability",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-uv-stability"
      ],
      "summary": "Inherited dossier: Ultraviolet stability through constructive renormalization.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Publisher abstract (refers to Theorem 1)",
          "url": "https://link.springer.com/article/10.1007/BF01238433"
        },
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
          "label": "Bibliographic record",
          "url": "https://link.springer.com/article/10.1007/BF01238433"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-confinement-criterion",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-center-divergent-potential"
      ],
      "summary": "Inherited dossier: A precise criterion connecting center symmetry and confinement.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-chatterjee-2021-confinement-v5",
          "label": "Section 2.1; Definition 2.1; Theorem 2.2",
          "url": "https://arxiv.org/abs/2006.16229v5"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-lattice-area-law",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-dynamical-area-law"
      ],
      "summary": "Inherited dossier: A dynamical route to the lattice area law.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 1, especially Theorem 1.6 and definitions preceding it",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 2, slab criterion",
          "url": "https://arxiv.org/abs/2509.04688v2"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Section 3, Proposition 3.2 and semigroup estimates",
          "url": "https://arxiv.org/abs/2509.04688v2"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-stochastic-renormalization",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-stochastic-counterterm-uniqueness"
      ],
      "summary": "Inherited dossier: Uniqueness of gauge-covariant stochastic renormalization.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2503.03060v2"
        },
        {
          "version_id": "version-chevyrev-shen-2026-renormalisation-v2",
          "label": "Version history and journal reference",
          "url": "https://arxiv.org/abs/2503.03060v2"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-positivity-bootstrap",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-bootstrap-loop-constraints"
      ],
      "summary": "Inherited dossier: Positivity bounds for SU(3) lattice Yang–Mills.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 2 (setup and strategy)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 3.2.2 (twist reflection positivity)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        },
        {
          "version_id": "version-guo-li-yang-zhu-2025-bootstrap-v2",
          "label": "Section 4 (loop equations)",
          "url": "https://arxiv.org/abs/2502.14421v2"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-loop-equations",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-direct-indirect-equations"
      ],
      "summary": "Inherited dossier: Direct and indirect loop equations.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2601.04316v2"
        },
        {
          "version_id": "version-liu-yang-2026-loop-equations-v2",
          "label": "Version history",
          "url": "https://arxiv.org/abs/2601.04316v2"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-glueball-spectrum",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-continuum-glueball-spectrum"
      ],
      "summary": "Inherited dossier: Continuum-extrapolated glueball spectra.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/2106.00364v3"
        },
        {
          "version_id": "version-athenodorou-teper-2021-spectrum-v3",
          "label": "Version history and publication metadata",
          "url": "https://arxiv.org/abs/2106.00364v3"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-inherited-semiclassical-confinement",
      "published_at": "2026-10-22T00:00:00Z",
      "event_type": "new-result",
      "affected_ids": [
        "claim-deformed-small-circle"
      ],
      "summary": "Inherited dossier: Controlled confinement on compactified spaces.",
      "reason": "Original editorial snapshot, retained during migration. This is not the date the source result first appeared.",
      "source_locators": [
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Abstract",
          "url": "https://arxiv.org/abs/0803.0344v2"
        },
        {
          "version_id": "version-unsal-yaffe-2008-center-stabilized-v2",
          "label": "Version history and journal reference",
          "url": "https://arxiv.org/abs/0803.0344v2"
        }
      ],
      "previous_statement": null,
      "workflow_visibility": "archived",
      "created_at": "2026-10-22T00:00:00Z"
    },
    {
      "id": "event-source-history-cao-nissim-sheffield-2025-dynamical-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "source-revision",
      "affected_ids": [
        "version-cao-nissim-sheffield-2025-dynamical-v2",
        "version-cao-nissim-sheffield-2025-dynamical-v1"
      ],
      "summary": "Recorded the v1/v2 source lineage without transferring a technical assessment to the earlier version.",
      "reason": "ArXiv reports minor revisions. Metadata and v2 selected sections were checked; a complete technical diff was not performed.",
      "source_locators": [
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v1",
          "label": "Version history (metadata only)",
          "url": "https://arxiv.org/abs/2509.04688v1"
        },
        {
          "version_id": "version-cao-nissim-sheffield-2025-dynamical-v2",
          "label": "Version history and v2 selected sections",
          "url": "https://arxiv.org/abs/2509.04688v2"
        }
      ],
      "previous_statement": "The inherited record retained only v2, without a machine-readable prior-version link.",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
    },
    {
      "id": "event-balaban-conditional-scope-followup-20261024",
      "published_at": "2026-10-24T00:00:00Z",
      "event_type": "scope-correction",
      "affected_ids": [
        "claim-uv-stability",
        "assessment-balaban-selected-sections-followup-20261024"
      ],
      "summary": "Recovered Bałaban’s exact conditional theorem and supporting model setup.",
      "reason": "The readable published scan explicitly assumes all effective couplings stay in (0,gamma], gamma sufficiently small. It describes the argument proposed to remove that assumption as not yet published at the time. The record now classifies the result as a conditional theorem, without asserting its later publication history or a physical mass gap.",
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        },
        {
          "version_id": "version-balaban-1989-large-field-ii-1989",
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          "label": "Introduction, printed pp251-252: torus, lattice scales, compact semisimple matrix group and Wilson action (0.2)",
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          "label": "Introduction pp243-245: Wilson initial density, E normalization, RT iteration to unit lattice",
          "url": "https://scispace.com/pdf/convergent-renormalization-expansions-for-lattice-gauge-29jyexd38e.pdf"
        },
        {
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          "label": "Section 2 pp254-264, especially density (2.18), effective action (2.23), localized terms (2.25)-(2.42), Theorems 1-2 and Corollary 3 (2.50)",
          "url": "https://scispace.com/pdf/convergent-renormalization-expansions-for-lattice-gauge-29jyexd38e.pdf"
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      "previous_statement": "The publisher abstract reports that bounds for the R operation complete the paper’s stated four-dimensional pure-gauge ultraviolet-stability result. Theorem 1 was not accessible in this review.",
      "workflow_visibility": "public",
      "created_at": "2026-10-24T00:00:00Z"
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