{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n1-scft.rg-extremization-conformal-manifolds",
  "artifact_class": "a-maximization decision flow with protected SCFT export",
  "title": "Protected N=1 data require a checked extremization loop",
  "evidence_date": "2026-08-24",
  "reader_question": "How does a candidate anomaly-free R-current become a qualified protected-data or conformal-manifold output?",
  "dominant_point": "Constrained a-maximization must pass a local-maximum test and an iterative accidental-field check before exact R-data are exported; conformal-manifold counting remains local, and a protected export remains upstream of crossing.",
  "conditional_export_extremization_fixture": {
    "theory": "candidate SU(3) SQCD fixed point with Nf=6",
    "allowed_mixing": "R_t(Q)=1/2+s, R_t(Q-tilde)=1/2-s",
    "trial_a": "a_t(s)=123/64-(243/16)s^2",
    "stationary_points": [
      {
        "s": "0",
        "a": "123/64",
        "hessian": "-243/8",
        "status": "unique local maximum on the baryon-mixing line"
      }
    ],
    "trial_c": "c_t(s)=163/64-(243/16)s^2",
    "chiral_unitarity_check": [
      {
        "operator": "M",
        "R": "1",
        "status": "passes"
      },
      {
        "operator": "B and B-tilde",
        "R": "3/2",
        "status": "passes"
      }
    ],
    "conclusion_ceiling": "the anomaly calculation identifies the protected candidate R-current conditional on the SCFT; it does not prove fixed-point existence"
  },
  "operator_decoupling_fixture": {
    "scope": "leading Nf^2 Veneziano sector of SU(Nc) adjoint SQCD with one adjoint X, Nf pairs Q,Q-tilde, W=0 and x=Nc/Nf=6",
    "constraint": "R(Q)=R(Q-tilde)=y; R(X)=(1-y)/6",
    "trial_anomaly_per_Nf_squared": "a_tilde_0/Nf^2=36(y-1)^3-12(y-1)+108((1-y)/6-1)^3-36((1-y)/6-1)+72",
    "iteration_0": {
      "derivative": "(3/2)(43-154y+71y^2)",
      "accepted_root": "y_0=(77-2 sqrt(719))/71=0.329178158998",
      "hessian": "-6 sqrt(719)<0",
      "discarded_root": "(77+2 sqrt(719))/71=1.839835925509; positive Hessian",
      "unitarity_check": [
        {
          "operator": "M1=Q-tilde Q",
          "multiplicity": "Nf^2",
          "trial_R": "2y_0=0.658356317995<2/3",
          "action": "add to free set"
        }
      ]
    },
    "correction_per_Nf_squared": "[a_chi(2/3)-a_chi(2y)]*32/3=(1/9)(2-6y)^2(5-6y)",
    "iteration_1": {
      "derivative": "(97-318y+69y^2)/2",
      "accepted_root": "y_1=(159-2 sqrt(4647))/69=0.328437453084",
      "hessian": "-2 sqrt(4647)<0",
      "discarded_root": "(159+2 sqrt(4647))/69=4.280258199090; positive Hessian",
      "stable_free_set": [
        "M1=Q-tilde Q, multiplicity Nf^2"
      ],
      "next_extensive_operator": "R(M2)=2y_1+(1-y_1)/6=0.768801997320>2/3",
      "status": "leading extensive meson sector terminates after one re-extremization"
    },
    "finite_rank_limitation": "single-trace Tr X^j and other O(1) sectors are suppressed in this leading-Nf^2 fixture and can themselves violate unitarity; this table is not a complete finite-rank spectrum analysis",
    "polynomial_deformation_boundary": "adding W=g_k Tr X^(k+1) imposes R(X)=2/(k+1), changes the mixing space and requires a new complete free-set analysis; the W=0 benchmark is not mislabeled as that deformed theory",
    "arithmetic_unit_test": "for one trial chiral with R=1/2, a_chi(2/3)-a_chi(1/2)=7/768"
  },
  "conformal_manifold_output": {
    "local_definition": "solutions of the moment-map equations for marginal chiral couplings modulo the continuous global group, equivalently a complexified quotient on a declared stable stratum",
    "smooth_stratum_dimension": "compute from the kernel of the linearized constraints modulo the tangent to the complexified orbit; dim V-dim G+dim H is usable only when the slice, stability and constant-rank hypotheses are verified",
    "singular_locus_warning": "a stabilizer alone does not determine local component dimension or Zariski tangent dimension",
    "global_unknowns": [
      "connectedness",
      "completeness",
      "topology",
      "metric distance",
      "all cusps",
      "all discrete duality identifications"
    ]
  },
  "protected_export": {
    "canonical_record_route": "/data/supersymmetry-duality/protected-scft/n1-sqcd-su3-nf6-v1.0.0.json",
    "schema_route": "/data/schemas/qft-protected-scft-export-1.0.0.schema.json",
    "record_tuple": {
      "schema_version": "1.0.0",
      "record_version": "1.0.0",
      "content_hash": "68a67e66c43e4d0a258f0d26b0b9c08c6164922be256a41112bb5761ee582910",
      "convention_hash": "81124c40cb39131147e94235022cbcea42222795e2a4e4c4842936fc212bfd95"
    },
    "payload": {
      "$schema": "/data/schemas/qft-protected-scft-export-1.0.0.schema.json",
      "schema_id": "qft.org/protected-scft-export/n1",
      "schema_version": "1.0.0",
      "record_version": "1.0.0",
      "profile": {
        "kind": "four-dimensional N=1 specialization",
        "generic_envelope_owner": "/supersymmetry-duality/protected-operators-defects-correspondences/protected-data-versioned-exports/",
        "profile_owner": "/supersymmetry-duality/n1-duality-rg-scft/protected-scft-data-bootstrap-export/",
        "relation": "this profile specializes, rather than competes with, the generic content-addressed protected-data envelope"
      },
      "theory": {
        "id": "qft.scft-candidate.n1-sqcd.su3-nf6-origin",
        "name": "candidate interacting SU(3) SQCD fixed point with Nf=6 at the origin",
        "spacetime_dimension": 4,
        "supersymmetry": "N=1",
        "superconformal_algebra": "SU(2,2|1)",
        "gauge_global_form": "simply connected SU(3)",
        "faithful_zero_form_flavor_group": "[SU(6)_L x SU(6)_R x U(1)_B]/[Z_6 x Z_3]",
        "faithful_kernel": "(z,z,exp(i beta))=(z,z,omega*z^(-1)), z^6=1, omega^3=1; its matter action equals the SU(3) gauge-center action",
        "deformation_point": "massless origin with vanishing superpotential",
        "free_factors": [],
        "topological_factors": "none asserted",
        "claim_status": "all algebraic values are exact conditional on the standard interacting fixed-point identification and its baryon-unmixed superconformal R-current"
      },
      "observable": {
        "kind": "protected_theory_card",
        "sectors": [
          "Weyl and flavor anomalies",
          "N=1 chiral primaries"
        ],
        "arguments": "origin of the massless SQCD moduli space"
      },
      "values": {
        "anomaly_traces": {
          "TrR": "-10",
          "TrR3": "7/2"
        },
        "central_charges": {
          "a": "123/64",
          "c": "163/64"
        },
        "two_point_coefficients": {
          "C_T": "815/(8*pi^4)",
          "tau_SU6_L": "9/4",
          "tau_SU6_R": "9/4",
          "tau_B_for_B_of_Q_equal_1": "54"
        }
      },
      "protected_operators": [
        {
          "identity": "meson M^i_jtilde",
          "uv_expression": "Q^i Q-tilde_jtilde",
          "lorentz_CDI": "[0;0]",
          "multiplet_CDI": "L\\bar B_1[0;0]_{Delta=3/2}^{(R=1)}",
          "R": "1",
          "Delta": "3/2",
          "flavor_representation": "(6,6-bar)_0",
          "representation_dimension": 36,
          "copy_multiplicity": 1,
          "recombination": {
            "status": "absolutely protected",
            "reason": "CDI scalar chiral class with 2/3<R<2; no long-multiplet recombination partner at fixed R"
          },
          "mixing": {
            "basis": [
              "M^i_jtilde"
            ],
            "gram_matrix": "<M^i_jtilde(x) (M^dagger)_k^ltilde(0)>=delta^i_k delta^ltilde_jtilde/|x|^3",
            "status": "one protected operator species in this representation; the displayed unit Gram matrix is a chosen basis normalization, not a dynamical prediction"
          }
        },
        {
          "identity": "baryon B",
          "uv_expression": "epsilon Q^3",
          "lorentz_CDI": "[0;0]",
          "multiplet_CDI": "L\\bar B_1[0;0]_{Delta=9/4}^{(R=3/2)}",
          "R": "3/2",
          "Delta": "9/4",
          "flavor_representation": "(wedge^3 6,1)_{+3}",
          "representation_dimension": 20,
          "copy_multiplicity": 1,
          "recombination": {
            "status": "absolutely protected",
            "reason": "CDI scalar chiral class with 2/3<R<2; no long-multiplet recombination partner at fixed R"
          },
          "mixing": {
            "basis": [
              "B"
            ],
            "gram_matrix": null,
            "status": "stored separately from the antibaryon; unit two-point normalization is not supplied"
          }
        },
        {
          "identity": "antibaryon B-tilde",
          "uv_expression": "epsilon Q-tilde^3",
          "lorentz_CDI": "[0;0]",
          "multiplet_CDI": "L\\bar B_1[0;0]_{Delta=9/4}^{(R=3/2)}",
          "R": "3/2",
          "Delta": "9/4",
          "flavor_representation": "(1,wedge^3 6-bar)_{-3}",
          "representation_dimension": 20,
          "copy_multiplicity": 1,
          "recombination": {
            "status": "absolutely protected",
            "reason": "CDI scalar chiral class with 2/3<R<2; no long-multiplet recombination partner at fixed R"
          },
          "mixing": {
            "basis": [
              "B-tilde"
            ],
            "gram_matrix": null,
            "status": "stored separately from the baryon; unit two-point normalization is not supplied"
          }
        }
      ],
      "protection": {
        "anomaly_values": "RG-invariant 't Hooft anomalies of the candidate exact R-current",
        "chiral_dimensions": "Delta=3R/2 from N=1 superconformal shortening",
        "multiplet_status": "CDI four-dimensional N=1 labels and recombination rules, arXiv:1612.00809v1, Sections 2.2.1 and 4.5"
      },
      "conventions": {
        "lorentz_labels": {
          "stored": "CDI integer Dynkin labels [j_CDI;jbar_CDI]",
          "forward": "j_CDI=2*j_spin",
          "inverse": "j_spin=j_CDI/2"
        },
        "chiral_dimension": {
          "forward": "Delta=3*R/2",
          "inverse": "R=2*Delta/3"
        },
        "generator_trace": "tr_fund(T^a T^b)=delta^{ab}/2",
        "anomaly_trace": "sum over left-handed Weyl fermions; a chiral superfield of scalar charge R_Phi contributes fermion charge R_Phi-1",
        "stress_tensor_two_point": {
          "definition": "<T_mn(x) T_rs(0)>=C_T I_mn,rs(x)/|x|^8, I_mn,rs=(I_mr I_ns+I_ms I_nr)/2-delta_mn delta_rs/4, I_mn=delta_mn-2 x_m x_n/x^2",
          "forward": "C_T=40*c/pi^4",
          "inverse": "c=pi^4*C_T/40"
        },
        "flavor_current_two_point": {
          "definition": "<J_m^a(x) J_n^b(0)>=3*tau^{ab} I_mn(x)/(4*pi^4*|x|^6)",
          "forward": "tau^{ab}=-3*Tr(R T^a T^b)",
          "inverse": "Tr(R T^a T^b)=-tau^{ab}/3"
        }
      },
      "convention_hash": {
        "algorithm": "sha256",
        "canonicalization": "RFC 8785",
        "value": "81124c40cb39131147e94235022cbcea42222795e2a4e4c4842936fc212bfd95"
      },
      "normalization_maps": [
        {
          "id": "chiral_R_Delta",
          "domain": "declared chiral primary",
          "forward": "Delta=3*R/2",
          "inverse": "R=2*Delta/3",
          "test": {
            "R": "1",
            "Delta": "3/2"
          }
        },
        {
          "id": "spin_to_CDI",
          "domain": "four-dimensional Lorentz spin",
          "forward": "j_CDI=2*j_spin",
          "inverse": "j_spin=j_CDI/2",
          "test": {
            "j_spin": "0",
            "j_CDI": "0"
          }
        },
        {
          "id": "c_to_C_T",
          "domain": "stored stress-tensor convention",
          "forward": "C_T=40*c/pi^4",
          "inverse": "c=pi^4*C_T/40",
          "test": {
            "c": "163/64",
            "C_T": "815/(8*pi^4)"
          }
        },
        {
          "id": "anomaly_to_tau",
          "domain": "stored flavor-current convention",
          "forward": "tau=-3*Tr(R*T*T)",
          "inverse": "Tr(R*T*T)=-tau/3",
          "test": {
            "anomaly": "-3/4",
            "tau": "9/4"
          }
        },
        {
          "id": "R_traces_to_a_c",
          "domain": "four-dimensional N=1 SCFT",
          "forward": [
            "a=3*(3*TrR3-TrR)/32",
            "c=(9*TrR3-5*TrR)/32"
          ],
          "inverse": [
            "TrR=16*(a-c)",
            "TrR3=16*(5*a-3*c)/9"
          ],
          "test": {
            "TrR": "-10",
            "TrR3": "7/2",
            "a": "123/64",
            "c": "163/64"
          }
        }
      ],
      "scope": {
        "includes": [
          "R, Delta, a, c, C_T, tau_F",
          "three selected chiral multiplet identities and exact flavor representations"
        ],
        "excludes": [
          "unit-normalized non-Ward OPE coefficients",
          "index coefficients or unsigned multiplicities",
          "partition functions",
          "long-multiplet gaps",
          "a complete external-correlator or crossing problem",
          "fixed-point existence or uniqueness proof"
        ],
        "allowed_downstream_claim": "initialize only the stored protected numbers and representation identities after reproducing the schema, content hash, convention hash and inverse-map tests"
      },
      "sources": [
        {
          "id": "seiberg-1995",
          "revision": "arXiv:hep-th/9411149v1",
          "url": "https://arxiv.org/pdf/hep-th/9411149v1",
          "locator": "Sections 2-3, printed pp. 4-12",
          "use": "SQCD fixed-point charges and duality evidence",
          "citation_identity_sha256": "4f96fc0270e08d7b1b3af5a3af48d6e958a17abb0d9b4113f07816ae10c84baf",
          "content_sha256": null,
          "content_hash_status": "the external bytes are not vendored; the versioned citation identity, locator and record content hash are frozen"
        },
        {
          "id": "anselmi-et-al-1998",
          "revision": "arXiv:hep-th/9708042v1",
          "url": "https://arxiv.org/pdf/hep-th/9708042v1",
          "locator": "Sections 2-4",
          "use": "central and flavor-current anomaly formulas",
          "citation_identity_sha256": "c83df6c6fb7e0ca948f75f618ec46cbd4a5f5abdac6de4e4342ae44dad6bbdd3",
          "content_sha256": null,
          "content_hash_status": "the external bytes are not vendored; the versioned citation identity, locator and record content hash are frozen"
        },
        {
          "id": "intriligator-wecht-2003",
          "revision": "arXiv:hep-th/0304128v3",
          "url": "https://arxiv.org/pdf/hep-th/0304128v3",
          "locator": "Section 1, equations (1.5)-(1.8)",
          "use": "exact R-current and local maximum",
          "citation_identity_sha256": "89ebf024a1b0476e0a72b10063cee3ced0259d16348a10fce2339e6ec1a6b254",
          "content_sha256": null,
          "content_hash_status": "the external bytes are not vendored; the versioned citation identity, locator and record content hash are frozen"
        },
        {
          "id": "cordova-dumitrescu-intriligator-2016",
          "revision": "arXiv:1612.00809v1",
          "url": "https://arxiv.org/pdf/1612.00809v1",
          "locator": "Sections 2.2.1 and 4.5",
          "use": "multiplet labels, shortening and recombination",
          "citation_identity_sha256": "561e0ea27a71a9b68a685ce29993fda689c85340e14bccbe012e003c687f620a",
          "content_sha256": null,
          "content_hash_status": "the external bytes are not vendored; the versioned citation identity, locator and record content hash are frozen"
        },
        {
          "id": "qft-profile-owner",
          "revision": "2026-08-24",
          "route": "/supersymmetry-duality/n1-duality-rg-scft/protected-scft-data-bootstrap-export/",
          "citation_identity_sha256": "128010019a798b0a3beb031823f122a5246104024937f75d619bfb568da97956"
        },
        {
          "id": "qft-generic-envelope-owner",
          "revision": "2026-08-24",
          "route": "/supersymmetry-duality/protected-operators-defects-correspondences/protected-data-versioned-exports/",
          "citation_identity_sha256": "580fdb11da2878cb6e6d553576df40d10a83c32bdaa6ce7ef792b3064509a714"
        }
      ],
      "source_cutoff": "2026-08-24",
      "uncertainty": {
        "kind": "exact algebraic values conditional on a duality-supported dynamical identification",
        "numerical_covariance": null,
        "dynamical_assumption": "existence and uniqueness of the proposed interacting fixed point are not established by the exported anomaly calculation"
      },
      "unresolved": [
        "unit-normalized baryon and antibaryon Gram matrices and non-Ward OPE coefficients",
        "superconformal-index decomposition and recombination inversion",
        "background, contour, counterterm, contact-term and global-sector data for partition functions",
        "long-multiplet spectrum, gaps and crossing feasibility"
      ],
      "content_hash": {
        "algorithm": "sha256",
        "canonicalization": "RFC 8785",
        "excluded_field": "content_hash",
        "value": "68a67e66c43e4d0a258f0d26b0b9c08c6164922be256a41112bb5761ee582910"
      }
    },
    "round_trip_checks": [
      "j_spin -> j_CDI=2 j_spin -> j_spin",
      "c -> C_T=40c/pi^4 -> c",
      "Tr(R T T) -> tau_F=-3 Tr(R T T) -> anomaly",
      "R -> Delta=3R/2 -> R for a declared chiral primary",
      "TrR and TrR3 -> a,c and back through the stored exact equations"
    ]
  },
  "sources": [
    {
      "id": "intriligator-wecht-2003",
      "locator": "Sections 1-2.4",
      "url": "https://arxiv.org/abs/hep-th/0304128",
      "use": "local a-maximization and Hessian/current relation"
    },
    {
      "id": "kutasov-parnachev-sahakyan-2003",
      "locator": "Sections 1-2",
      "url": "https://arxiv.org/abs/hep-th/0308071",
      "use": "accidental-current correction and iterative decoupling"
    },
    {
      "id": "green-et-al-2010",
      "locator": "Sections 2-4",
      "url": "https://arxiv.org/abs/1005.3546",
      "use": "local conformal-manifold quotient and broken-current criterion"
    },
    {
      "id": "cordova-dumitrescu-intriligator-2019",
      "locator": "Sections 2.2 and 4",
      "url": "https://arxiv.org/abs/1612.00809",
      "use": "multiplet labels, shortening and recombination"
    }
  ],
  "visual_encodings": {
    "solid_arrow": "licensed one-way step",
    "dashed_arrow": "accidental-field correction and re-extremization loop",
    "dashed_box": "explicit inference stop",
    "gray_box": "decision or accepted protected core",
    "color_dependence": "none"
  },
  "accessibility": {
    "alt_text": "A flowchart starts with anomaly traces, allowed flavor currents, gauge constraints and superpotential constraints, then extremizes the trial a function. Every gauge-invariant chiral generator is classified relative to R equals two thirds: equality is separated as a free factor, while a violation enters a dashed free-chiral correction loop and triggers re-extremization. A stable local maximum with every free factor separated yields a protected core that branches to a local conformal-manifold quotient and to a versioned protected-data export. Stop boxes forbid claims of global manifold geometry, fixed-point existence or solved crossing.",
    "semantic_equivalent": "/figures/supersymmetry-duality/rg-extremization-conformal-manifolds.json",
    "narrow_width": "the SVG remains unclipped and has a full-size link; this JSON preserves all constraints, fixtures, output fields and stop rules"
  },
  "limitations": [
    "The operator-crossing fixture is a genuine leading-Nf^2 adjoint-SQCD re-extremization, but its explicitly named O(1) sectors still require a finite-rank analysis.",
    "The SU(3), Nf=6 export is conditional on the standard interacting fixed-point claim.",
    "No index, partition function, non-Ward OPE coefficient, long spectrum or crossing result is exported.",
    "The artifact registry lifecycle observed during generation was prototype."
  ]
}