{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.duality.claim-dictionary-check-failure-matrix",
  "artifact_type": "semantic table",
  "title": "SQCD duality claims, dependencies, checks, and failure ceilings",
  "evidence_cutoff": "2026-08-24",
  "reader_question": "Which bounded checks support the canonical SQCD duality claim, which inputs do they share, and what failure or untested sector limits the surviving conclusion?",
  "takeaway": "Continuous anomalies, chiral data, an exact index identity, and a one-flavor flow test different but correlated sectors; none alone proves full QFT equivalence, and the global background/extended sector remains explicitly not tested in this bounded dossier.",
  "scope": "The canonical four-dimensional N=1 simply connected SU(Nc) SQCD pair for Nc>=3, with generic and special color/flavor domains stated row by row; no claim is made about every Seiberg-like duality or the enhanced-flavor Nc=2 case.",
  "convention_snapshot": {
    "spacetime": "four-dimensional Lorentzian QFT with the site (+---) metric convention",
    "supersymmetry": "N=1",
    "electric_theory": "Simply connected SU(Nc) SQCD with Nc>=3, Nf fundamental-antifundamental pairs, and W_tree=0",
    "magnetic_theory": "SU(Nf-Nc) with Nf magnetic flavors, singlet M, and W_mag=M^i_j q_i qtilde^j/mu",
    "generic_rank_domain": "Nc>=3 throughout; Nf>=Nc+2 defines the generic magnetic theory card, and the simple one-flavor flow with a nontrivial residual magnetic gauge group uses Nf>=Nc+3",
    "generator_normalization": "T(fundamental)=1/2",
    "baryon_normalization": "B(Q)=+1, B(Qtilde)=-1, B(q)=Nc/(Nf-Nc), and B(qtilde)=-Nc/(Nf-Nc)",
    "anomaly_free_R": "R(Q)=R(Qtilde)=1-Nc/Nf, R(q)=R(qtilde)=Nc/Nf, and R(M)=2(1-Nc/Nf)",
    "holomorphic_scale_relation": "Lambda_e^(3Nc-Nf) Lambda_m^(3(Nf-Nc)-Nf)=(-1)^(Nf-Nc) mu^Nf in one fixed finite convention",
    "global_scope": "The dynamical fundamentals break the gauge-center one-form symmetry; faithful flavor quotients, nontrivial background bundles, and allowed counterterms remain separate global data",
    "evidence_rule": "A row supports only its stated observable and coverage layer; correlated rows sharing a dependency are not independent proofs",
    "sha256": "39da21f034feed2e9c82170a69cf522743bc58c75d13cb4453363101b4e260cb"
  },
  "dependency_role_legend": [
    {
      "code": "C",
      "label": "computation input",
      "meaning": "Input used to calculate the row's observable or endpoint."
    },
    {
      "code": "T",
      "label": "tested datum",
      "meaning": "Dictionary or theory datum whose correctness the row probes."
    },
    {
      "code": "H",
      "label": "interpretive hypothesis",
      "meaning": "Assumption needed to interpret agreement as evidence for the claim."
    },
    {
      "code": "N",
      "label": "normalization",
      "meaning": "Convention, contour, scheme, or counterterm lattice needed for comparison."
    }
  ],
  "status_legend": [
    "exact-observable-identity",
    "established-conditional-result",
    "strong-evidence",
    "conjecture",
    "failed-as-stated",
    "open",
    "not-tested"
  ],
  "primary_sources": [
    {
      "source_id": "seiberg-1995",
      "citation": "Nathan Seiberg, Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories, Nuclear Physics B 435 (1995) 129-146",
      "publication_doi": "10.1016/0550-3213(94)00023-8",
      "url": "https://arxiv.org/html/hep-th/9411149v1",
      "source_version": "arXiv HTML v1",
      "source_revision_date": "1994-11-20",
      "allowed_locators": [
        "§2",
        "§3.2, “Mass terms in the magnetic theory”"
      ]
    },
    {
      "source_id": "dolan-osborn-2009",
      "citation": "F. A. Dolan and Hugh Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories, Nuclear Physics B 818 (2009) 137-178",
      "publication_doi": "10.1016/j.nuclphysb.2009.01.028",
      "url": "https://arxiv.org/abs/0801.4947v3",
      "source_version": "arXiv v3",
      "source_revision_date": "2008-02-27",
      "allowed_locators": [
        "§§3 and 6"
      ]
    },
    {
      "source_id": "aharony-seiberg-tachikawa-2013",
      "citation": "Ofer Aharony, Nathan Seiberg, and Yuji Tachikawa, Reading between the Lines of Four-Dimensional Gauge Theories, JHEP 08 (2013) 115",
      "publication_doi": "10.1007/JHEP08(2013)115",
      "url": "https://arxiv.org/abs/1305.0318v5",
      "source_version": "arXiv v5",
      "source_revision_date": "2022-12-26",
      "allowed_locators": [
        "§§1–2"
      ]
    }
  ],
  "dependency_nodes": [
    {
      "dependency_id": "dep.theory.electric-sqcd.v1",
      "dependency_type": "theory-card",
      "label": "Electric SU(Nc) SQCD theory card",
      "depends_on": []
    },
    {
      "dependency_id": "dep.theory.magnetic-sqcd.v1",
      "dependency_type": "theory-card",
      "label": "Magnetic SU(Nf-Nc) SQCD theory card",
      "depends_on": []
    },
    {
      "dependency_id": "dep.normalization.anomaly.v1",
      "dependency_type": "normalization",
      "label": "T(fundamental)=1/2 and fermion R-charge convention",
      "depends_on": []
    },
    {
      "dependency_id": "dep.normalization.composites.v1",
      "dependency_type": "normalization",
      "label": "Meson, baryon, epsilon-tensor, and matching-scale normalization",
      "depends_on": []
    },
    {
      "dependency_id": "dep.normalization.index.v1",
      "dependency_type": "normalization",
      "label": "Index fugacity, balancing, Haar measure, and contour convention",
      "depends_on": []
    },
    {
      "dependency_id": "dep.normalization.holomorphic-scales.v1",
      "dependency_type": "normalization",
      "label": "Shared holomorphic scale and magnetic Yukawa convention",
      "depends_on": []
    },
    {
      "dependency_id": "dep.normalization.global-counterterms.v1",
      "dependency_type": "normalization",
      "label": "Allowed local-counterterm lattice for fixed backgrounds and tangential structure",
      "depends_on": []
    },
    {
      "dependency_id": "dep.input.fermion-charge-tables.v1",
      "dependency_type": "computation-input",
      "label": "Electric and magnetic Weyl-fermion charge tables",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1",
        "dep.normalization.anomaly.v1"
      ]
    },
    {
      "dependency_id": "dep.input.fterm-chiral-data.v1",
      "dependency_type": "computation-input",
      "label": "Magnetic F-terms and electric gauge-invariant chiral generators",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1",
        "dep.normalization.composites.v1"
      ]
    },
    {
      "dependency_id": "dep.input.index-integrals.v1",
      "dependency_type": "computation-input",
      "label": "Electric and magnetic elliptic-hypergeometric index integrals",
      "depends_on": [
        "dep.input.fermion-charge-tables.v1",
        "dep.normalization.index.v1"
      ]
    },
    {
      "dependency_id": "dep.input.mass-flow-equations.v1",
      "dependency_type": "computation-input",
      "label": "Linear meson F-term, D-flat Higgs branch, and threshold exponents",
      "depends_on": [
        "dep.input.fterm-chiral-data.v1",
        "dep.normalization.holomorphic-scales.v1"
      ]
    },
    {
      "dependency_id": "dep.input.broken-group-instanton.v1",
      "dependency_type": "computation-input",
      "label": "Completely broken magnetic-group instanton contribution at Nf=Nc+2",
      "depends_on": [
        "dep.input.mass-flow-equations.v1"
      ]
    },
    {
      "dependency_id": "dep.input.screening-backgrounds.v1",
      "dependency_type": "computation-input",
      "label": "Screening, genuine probes, faithful quotients, and background bundles",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1",
        "dep.normalization.global-counterterms.v1"
      ]
    },
    {
      "dependency_id": "dep.dictionary.symmetry-map.v1",
      "dependency_type": "tested-dictionary-datum",
      "label": "Continuous Lie-algebra symmetry and charge map",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1"
      ]
    },
    {
      "dependency_id": "dep.dictionary.chiral-map.v1",
      "dependency_type": "tested-dictionary-datum",
      "label": "Meson, baryon, and chiral-relation map",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1"
      ]
    },
    {
      "dependency_id": "dep.dictionary.protected-index-map.v1",
      "dependency_type": "tested-dictionary-datum",
      "label": "Protected charge and fugacity map entering the index",
      "depends_on": [
        "dep.dictionary.symmetry-map.v1"
      ]
    },
    {
      "dependency_id": "dep.dictionary.mass-deformation-map.v1",
      "dependency_type": "tested-dictionary-datum",
      "label": "Electric mass to magnetic linear-meson and Higgs-flow map",
      "depends_on": [
        "dep.dictionary.chiral-map.v1"
      ]
    },
    {
      "dependency_id": "dep.dictionary.global-data-map.v1",
      "dependency_type": "tested-dictionary-datum",
      "label": "Faithful global group, background, counterterm, and extended-probe map",
      "depends_on": [
        "dep.dictionary.symmetry-map.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.common-ir.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "The two UV theory cards reach a common infrared theory",
      "depends_on": [
        "dep.theory.electric-sqcd.v1",
        "dep.theory.magnetic-sqcd.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.unbroken-symmetry.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "The compared symmetry is unbroken and coupled to corresponding backgrounds",
      "depends_on": [
        "dep.dictionary.symmetry-map.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.chiral-protection.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "The protected chiral data are compared in corresponding supersymmetric vacua",
      "depends_on": [
        "dep.dictionary.chiral-map.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.index-prescription.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "The integral computes the same protected trace in the stated fugacity domain",
      "depends_on": [
        "dep.dictionary.protected-index-map.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.flow-branch-completeness.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "Corresponding D-flat vacua and all light, topological, and instanton sectors are retained",
      "depends_on": [
        "dep.dictionary.mass-deformation-map.v1"
      ]
    },
    {
      "dependency_id": "dep.hypothesis.global-completeness.v1",
      "dependency_type": "interpretive-hypothesis",
      "label": "The claimed equivalence includes the specified nontrivial backgrounds and extended sectors",
      "depends_on": [
        "dep.dictionary.global-data-map.v1"
      ]
    }
  ],
  "rows": [
    {
      "row_id": "sqcd-check.continuous-anomalies.v1",
      "claim": {
        "claim_id": "sqcd-claim.continuous-anomaly-match",
        "version": "1.0.0",
        "claim_scope": "protected-local-data",
        "statement": "The electric and magnetic charge tables give identical continuous 't Hooft anomaly coefficients in the mapped SU(Nf)_L x SU(Nf)_R x U(1)_B x U(1)_R charge basis.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Nf>=Nc+2, with special color/flavor cases kept separate"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.symmetry-and-charges",
        "version": "1.0.0",
        "content": "Map the continuous Lie-algebra symmetry generators in the displayed charge basis and the charges of Q, Qtilde, q, qtilde, M, and the gauginos."
      },
      "check": {
        "check_id": "check.anomaly-polynomial-continuous",
        "kind": "exact algebraic consistency check",
        "procedure": "Compute cubic nonabelian, mixed nonabelian-R and nonabelian-B, baryon-squared-R, gravitational-R, and cubic-R coefficients from Weyl fermions, and verify the remaining odd-B Abelian combinations vanish, all in one normalization."
      },
      "evidence": "The magnetic quarks and meson reproduce the electric continuous anomaly coefficients; for example A[SU(Nf)_L^3]=-Ntilde_c+Nf=Nc.",
      "result": {
        "outcome": "match",
        "classification": "exact-protected-algebraic-match",
        "result_scope": "Necessary protected local consistency for the mapped unbroken continuous symmetry."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.fermion-charge-tables.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.symmetry-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.unbroken-symmetry.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.common-ir.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.anomaly.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "local",
        "protected",
        "anomaly"
      ],
      "global_form_scope": "Continuous anomaly coefficients in the displayed charge basis; faithful quotients, torsion backgrounds, and extended operators are not settled by this row.",
      "primary_source": {
        "source_id": "seiberg-1995",
        "locator": "§2",
        "source_version": "arXiv HTML v1",
        "source_revision_date": "1994-11-20",
        "checked_at": "2026-08-24"
      },
      "falsifier": "Any continuous anomaly coefficient that remains unequal after a fixed symmetry-basis translation and inclusion of named free fields falsifies this dictionary row.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The continuous-symmetry dictionary and any full infrared claim using it fail until the charge map or missing sector is corrected.",
        "repair_rule": "Revise the theory card or add an independently justified sector, then recompute every dependent row."
      },
      "status": "established-conditional-result",
      "confidence_ceiling": "Exact anomaly equality given the proposed charge dictionary; necessary but not sufficient for full infrared equivalence.",
      "unresolved_work": "Complete the faithful-group, torsion-anomaly, and extended-operator comparison on nontrivial backgrounds."
    },
    {
      "row_id": "sqcd-check.chiral-ring.v1",
      "claim": {
        "claim_id": "sqcd-claim.chiral-ring-map",
        "version": "1.0.0",
        "claim_scope": "protected-chiral-data",
        "statement": "The proposed meson and baryon map, their continuous quantum numbers, and the magnetic M F-term pass a bounded chiral-ring test in the stated generic rank domain.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Nf>=Nc+2 away from the separately described Nf=Nc and Nf=Nc+1 endpoints"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.chiral-generators-and-relations",
        "version": "1.0.0",
        "content": "Map Q Qtilde to M, map electric and magnetic baryons with epsilon tensors and scale factors, and impose dW/dM=q qtilde/mu=0 in the chiral ring."
      },
      "check": {
        "check_id": "check.chiral-ring-and-moduli-strata",
        "kind": "protected algebraic comparison",
        "procedure": "Compare representations, B and R charges, the displayed meson and baryon maps, and the magnetic M F-term that removes q qtilde from the chiral ring."
      },
      "evidence": "M has the electric meson quantum numbers, the magnetic composite q qtilde is removed by the M F-term, and epsilon-tensor baryons have the mapped flavor and baryon charges.",
      "result": {
        "outcome": "the displayed generators, charges, maps, and F-term relation match",
        "classification": "exact-protected-algebraic-match",
        "result_scope": "A bounded protected operator-dictionary check after fixed composite normalization; not a complete presentation of every branch or ideal."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.fterm-chiral-data.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.chiral-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.chiral-protection.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.common-ir.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.composites.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "local",
        "protected",
        "chiral-ring",
        "operator-dictionary"
      ],
      "global_form_scope": "Gauge-invariant local chiral operators in the simply connected theory card; genuine lines and flavor-background quotients remain outside this row.",
      "primary_source": {
        "source_id": "seiberg-1995",
        "locator": "§2",
        "source_version": "arXiv HTML v1",
        "source_revision_date": "1994-11-20",
        "checked_at": "2026-08-24"
      },
      "falsifier": "A displayed representation, charge, operator map, or F-term relation that cannot be matched after the predeclared normalization falsifies this protected dictionary row.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The claimed protected chiral-ring equivalence and any stronger duality claim containing it fail on the affected branch.",
        "repair_rule": "Identify an omitted generator, relation, free field, or special color/flavor description and repeat all dependent protected checks."
      },
      "status": "established-conditional-result",
      "confidence_ceiling": "Exact agreement for the displayed protected generators, charges, maps, and F-term relation; not a complete chiral-ring presentation or equality of unprotected data.",
      "unresolved_work": "Complete the branch-by-branch chiral ideal and specify operator mixing or accidental free fields near every unitarity-bound crossing."
    },
    {
      "row_id": "sqcd-check.superconformal-index.v1",
      "claim": {
        "claim_id": "sqcd-claim.dolan-osborn-index-identity",
        "version": "1.0.0",
        "claim_scope": "protected-observable",
        "statement": "The Dolan–Osborn electric and magnetic SQCD index integrals are equal in their common fugacity, balancing, and contour domain.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Generic SU(Nc) Seiberg pair in the color/flavor and convergence domain stated in the source"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.protected-charges-and-fugacities",
        "version": "1.0.0",
        "content": "Map the protected charges, exact candidate R symmetry, flavor fugacities, gauge measures, and meson contribution entering the two integrals."
      },
      "check": {
        "check_id": "check.elliptic-hypergeometric-index-identity",
        "kind": "exact protected-observable identity",
        "procedure": "Apply the elliptic-hypergeometric transformation identity to the fully specified electric and magnetic index integrals."
      },
      "evidence": "Dolan and Osborn reduce the generic electric-magnetic equality to a proved elliptic-hypergeometric integral identity after matching the protected fugacity data.",
      "result": {
        "outcome": "exact equality of the named index expressions",
        "classification": "exact-observable-identity",
        "result_scope": "One protected supertrace with the stated contour and balancing conditions."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.index-integrals.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.protected-index-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.index-prescription.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.common-ir.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.index.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "local",
        "protected",
        "superconformal-index"
      ],
      "global_form_scope": "The source identity uses the stated gauge integrals and flavor fugacities; it does not by itself enumerate every global form, discrete background, or genuine line sector.",
      "primary_source": {
        "source_id": "dolan-osborn-2009",
        "locator": "§§3 and 6",
        "source_version": "arXiv v3",
        "source_revision_date": "2008-02-27",
        "checked_at": "2026-08-24"
      },
      "falsifier": "Failure of the two correctly normalized integrals to agree at an admissible fugacity point, after decoupled factors and contour prescriptions are aligned, falsifies this index row.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The protected-index dictionary fails; a full duality claim using that dictionary is also inconsistent unless the input theory or contour was misstated.",
        "repair_rule": "Correct the fugacity map, contour, balancing condition, or missing protected factor and rerun the identity check."
      },
      "status": "exact-observable-identity",
      "confidence_ceiling": "Exact equality of this protected trace only; it cannot promote the claim to full QFT equivalence.",
      "unresolved_work": "Unprotected correlators, complete Hilbert-space equivalence, and global extended sectors require independent evidence."
    },
    {
      "row_id": "sqcd-check.one-flavor-mass-flow-generic.v1",
      "claim": {
        "claim_id": "sqcd-claim.one-flavor-mass-flow-generic",
        "version": "1.0.0",
        "claim_scope": "dynamical-flow-compatibility",
        "statement": "For Nf>=Nc+3, an electric one-flavor mass maps to a magnetic linear meson term, a rank-one D-flat Higgs vacuum, and the adjacent Nf-1 dual pair with matched holomorphic thresholds.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Nf>=Nc+3; m and mu are nonzero, and the endpoint is compared below both the electric decoupling scale |m| and the magnetic Higgs scale sqrt(|m mu|)"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.relevant-mass-and-higgs-flow",
        "version": "1.0.0",
        "content": "Map m Q^Nf Qtilde_Nf to m M^Nf_Nf, impose q_Nf qtilde^Nf=-m mu, and reduce SU(Nf-Nc) to SU(Nf-Nc-1)."
      },
      "check": {
        "check_id": "check.mass-flow-rank-fields-and-scales",
        "kind": "dynamical deformation consistency check",
        "procedure": "Verify F- and D-flatness, the residual gauge group and light fields, electric decoupling, magnetic Higgs threshold matching, and the lower-flavor scale relation in one convention."
      },
      "evidence": "The residual magnetic group is SU(Nf-Nc-1)=SU((Nf-1)-Nc), while the electric and magnetic threshold exponents change by +1 and -2 and reproduce the adjacent dual-pair scale relation.",
      "result": {
        "outcome": "generic flow square closes after the complete Higgsed sector is integrated out",
        "classification": "nontrivial-dynamical-consistency",
        "result_scope": "One relevant deformation and its selected vacuum, not a proof of every deformation or of the undeformed full duality."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.mass-flow-equations.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.mass-deformation-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.flow-branch-completeness.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.common-ir.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.holomorphic-scales.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "dynamical",
        "deformation",
        "vacuum",
        "threshold"
      ],
      "global_form_scope": "Simply connected electric and magnetic gauge groups with fundamental matter; background contact terms and any endpoint topological factor must still be checked separately.",
      "primary_source": {
        "source_id": "seiberg-1995",
        "locator": "§3.2, “Mass terms in the magnetic theory”",
        "source_version": "arXiv HTML v1",
        "source_revision_date": "1994-11-20",
        "checked_at": "2026-08-24"
      },
      "falsifier": "A residual gauge group, light multiplet, branch, or threshold relation different from the Nf-1 theory after fixed-convention matching falsifies the proposed deformation map.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The one-flavor flow square and any claim that the dictionary transports this relevant operator fail.",
        "repair_rule": "Restore an omitted branch, field, instanton, topological sector, or threshold factor and recompute both paths."
      },
      "status": "strong-evidence",
      "confidence_ceiling": "Nontrivial dynamical support for this mass deformation, conditional on branch completeness and the fixed scale convention.",
      "unresolved_work": "Repeat the endpoint comparison with faithful flavor backgrounds and possible local contact terms turned on."
    },
    {
      "row_id": "sqcd-check.one-flavor-mass-flow-nc-plus-2.v1",
      "claim": {
        "claim_id": "sqcd-claim.one-flavor-mass-flow-nc-plus-2-exception",
        "version": "1.0.0",
        "claim_scope": "special-color-number-flow-exception",
        "statement": "At Nf=Nc+2, the magnetic gauge group is completely broken by the one-flavor mass; the generic residual-group argument is incomplete unless broken-group instanton effects and the s-confining Nf=Nc+1 endpoint are included.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Nf=Nc+2 flowing to Nf=Nc+1"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.special-color-number-mass-flow",
        "version": "1.0.0",
        "content": "Replace the generic residual SU(Nf-Nc-1) card by the complete-breaking endpoint and its instanton-generated interaction."
      },
      "check": {
        "check_id": "check.complete-breaking-instanton-exception",
        "kind": "special-color-number endpoint completeness check",
        "procedure": "Set Nf=Nc+2, verify SU(2) is fully Higgsed, and retain the broken-group instanton contribution required for the Nf=Nc+1 composite endpoint."
      },
      "evidence": "The residual-group formula gives SU(1), the trivial group, so no nonabelian magnetic gauge dynamics remains; Seiberg explicitly identifies the generic mass-flow discussion as incomplete here and requires instanton contributions from the broken group.",
      "result": {
        "outcome": "generic Higgs-only row excluded; corrected special-color-number endpoint required",
        "classification": "established-domain-exception",
        "result_scope": "Boundary of applicability of the generic mass-flow mechanism."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.broken-group-instanton.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.mass-deformation-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.flow-branch-completeness.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.holomorphic-scales.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "dynamical",
        "deformation",
        "special-color-number",
        "instanton"
      ],
      "global_form_scope": "Simply connected SU(Nc) electric theory and completely broken SU(2) magnetic gauge group; the composite endpoint must be compared as a complete infrared theory.",
      "primary_source": {
        "source_id": "seiberg-1995",
        "locator": "§3.2, “Mass terms in the magnetic theory”",
        "source_version": "arXiv HTML v1",
        "source_revision_date": "1994-11-20",
        "checked_at": "2026-08-24"
      },
      "falsifier": "A treatment that applies the generic residual nonabelian card at Nf=Nc+2 or omits the broken-group instanton fails this endpoint-completeness check.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The generic mass-flow claim fails at this color/flavor boundary; it does not invalidate the separately corrected endpoint duality statement.",
        "repair_rule": "Use the s-confining Nf=Nc+1 endpoint and include the required broken-group instanton interaction before rechecking the square."
      },
      "status": "established-conditional-result",
      "confidence_ceiling": "Established exception to the generic mechanism and a required repair rule; not independent evidence for the full duality.",
      "unresolved_work": "Bind the explicit composite-endpoint superpotential and its normalization when the endpoint owner page is integrated."
    },
    {
      "row_id": "sqcd-check.global-lines-and-backgrounds.v1",
      "claim": {
        "claim_id": "sqcd-claim.global-data-completeness",
        "version": "1.0.0",
        "claim_scope": "complete-global-qft-data",
        "statement": "A complete Seiberg-duality claim must match faithful flavor quotients, allowed background bundles, counterterms, and extended probes after fundamental-matter screening is included.",
        "source_theory": {
          "theory_id": "sqcd.electric.su-nc",
          "version": "1.0.0",
          "definition": "4d N=1 simply connected SU(Nc), Nc>=3, with Nf Q plus Qtilde flavors and W_tree=0"
        },
        "target_theory": {
          "theory_id": "sqcd.magnetic.su-nf-minus-nc",
          "version": "1.0.0",
          "definition": "4d N=1 SU(Nf-Nc) with Nf q plus qtilde flavors, singlet M, and W=M q qtilde/mu"
        },
        "regime": {
          "version": "1.0.0",
          "spacetime": "four dimensions",
          "supersymmetry": "N=1",
          "energy_scope": "infrared duality claim and the protected or flow sector named by the row",
          "rank_domain": "Nc>=3; Generic Seiberg pair with global forms and background structures explicitly declared"
        }
      },
      "dictionary_field": {
        "field_id": "dictionary.global-form-lines-backgrounds",
        "version": "1.0.0",
        "content": "Record the simply connected gauge groups, broken gauge-center one-form symmetries, faithful flavor quotient, nontrivial bundles, allowed counterterms, and any surviving genuine extended probes."
      },
      "check": {
        "check_id": "check.global-data-and-background-response",
        "kind": "global completeness check",
        "procedure": "Compare screening classes, faithful-group bundles, background partition functions, torsion anomalies, and allowed counterterm equivalence on both sides."
      },
      "evidence": "Aharony, Seiberg, and Tachikawa establish that identical local Lie-algebra data can define distinct theories with different genuine lines and discrete theta data; the bounded source chain here does not complete that comparison for the SQCD pair.",
      "result": {
        "outcome": "required comparison identified but not completed in this bounded dossier",
        "classification": "not-tested-global-sector",
        "result_scope": "No positive full-global-theory conclusion is authorized by this row."
      },
      "dependency_roles": [
        {
          "dependency_id": "dep.input.screening-backgrounds.v1",
          "dependency_type": "computation-input",
          "roles": [
            "C"
          ]
        },
        {
          "dependency_id": "dep.dictionary.global-data-map.v1",
          "dependency_type": "tested-dictionary-datum",
          "roles": [
            "T"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.global-completeness.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.hypothesis.common-ir.v1",
          "dependency_type": "interpretive-hypothesis",
          "roles": [
            "H"
          ]
        },
        {
          "dependency_id": "dep.normalization.global-counterterms.v1",
          "dependency_type": "normalization",
          "roles": [
            "N"
          ]
        }
      ],
      "coverage_layer": [
        "global",
        "backgrounds",
        "extended-operators",
        "anomaly"
      ],
      "global_form_scope": "The dynamical fundamentals remove the gauge-center one-form symmetry, but this does not erase faithful flavor quotients, bundle sectors, torsion responses, or scheme-independent counterterm remainders.",
      "primary_source": {
        "source_id": "aharony-seiberg-tachikawa-2013",
        "locator": "§§1–2",
        "source_version": "arXiv v5",
        "source_revision_date": "2022-12-26",
        "checked_at": "2026-08-24"
      },
      "falsifier": "A background response, torsion anomaly, or extended-probe spectrum that differs by more than the predeclared allowed local counterterm lattice falsifies complete global equivalence.",
      "failure_disposition": {
        "if_triggered": "failed-as-stated",
        "propagation_scope": "The complete-global-QFT claim fails, although a narrower local or protected-sector equivalence may survive.",
        "repair_rule": "Correct the global form or tensor a separately justified invertible/topological sector, then repeat every global and flow check."
      },
      "status": "not-tested",
      "confidence_ceiling": "No claim beyond the tested local, protected, and one-flow sectors until the global-data comparison is completed.",
      "unresolved_work": "Compute the faithful flavor quotient and background response for both SQCD descriptions, including torsion and counterterm quantization."
    }
  ],
  "status_propagation_fixture": {
    "source_row_id": "sqcd-check.continuous-anomalies.v1",
    "injected_result": "decisive-mismatch",
    "expected_status": "failed-as-stated",
    "purpose": "A decisive mismatch must propagate through the row's declared failure disposition rather than leaving a positive headline unchanged."
  },
  "validation_contract": {
    "required_dependency_roles_per_row": [
      "C",
      "T",
      "H",
      "N"
    ],
    "dependency_graph": "typed directed acyclic graph",
    "blank_evidence_falsifier_status_forbidden": true,
    "exact_observable_identity_scope_rule": "An exact observable identity must retain claim_scope=protected-observable and may not claim full QFT equivalence.",
    "surviving_ceiling_rule": "Every row has exactly one top-level confidence_ceiling field.",
    "decisive_mismatch_rule": "Every row maps a triggered falsifier to failed-as-stated for the scope named in failure_disposition."
  },
  "scientific_fixtures": {
    "anomaly": {
      "Nc": 3,
      "Nf": 7,
      "magnetic_color_number": 4,
      "SU_Nf_L_cubic": {
        "electric": "3",
        "magnetic": "-4+7=3"
      },
      "twice_SU_Nf_L_squared_R": {
        "electric": "-9/7",
        "magnetic": "-16/7+(7-6)=-9/7"
      },
      "twice_SU_Nf_L_squared_B": {
        "electric": "3",
        "magnetic": "4*(3/4)=3"
      },
      "U1_B_squared_R": {
        "electric": "-18",
        "magnetic": "-18"
      },
      "odd_B_abelian": {
        "gravitational_B": "0",
        "B_cubed": "0",
        "B_R_squared": "0"
      }
    },
    "chiral": {
      "Nc": 3,
      "Nf": 7,
      "R_M": "8/7",
      "R_q_qtilde": "6/7",
      "R_superpotential": "2",
      "fterm_relation": "q qtilde=0 in the chiral ring"
    },
    "mass_flow_generic": {
      "Nc": 3,
      "Nf": 7,
      "magnetic_color_number_before": 4,
      "magnetic_color_number_after": 3,
      "electric_beta_exponent_before": 2,
      "electric_beta_exponent_after": 3,
      "magnetic_beta_exponent_before": 5,
      "magnetic_beta_exponent_after": 3
    },
    "mass_flow_exception": {
      "Nc": 3,
      "Nf": 5,
      "magnetic_color_number_before": 2,
      "residual_group": "SU(1), the trivial group",
      "endpoint": "Nf=Nc+1 s-confining composite theory",
      "required_extra_sector": "broken-group instanton contribution"
    },
    "global": {
      "dynamical_fundamentals_present": true,
      "gauge_center_one_form_symmetry": "broken",
      "faithful_flavor_background_comparison": "not tested"
    }
  },
  "governance_snapshot": {
    "registry_status": "prototype",
    "registry_public_route": null,
    "integration_state": "complete"
  }
}