{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n1-gauge.sqcd-phase-conformal-window",
  "artifact_type": "semantic figure record",
  "title": "Massless SU(Nc) SQCD: rank and evidence map",
  "evidence_cutoff": "2026-08-24",
  "reader_question": "Which massless simply connected SU(Nc) SQCD statements are exact, which rank intervals use Seiberg's proposed magnetic description, and what happens at each endpoint and at small rank?",
  "takeaway": "The low-flavor ladder is fixed by exact chiral data, while the free-magnetic and interacting interiors are duality-supported; beta signs and the meson unitarity bound locate the same lower boundary but share theoretical inputs and are not independent proofs.",
  "conventions": {
    "theory": "four-dimensional N=1 SU(Nc) SQCD with Nc >= 2",
    "global_form": "simply connected SU(Nc)",
    "matter": "Nf massless fundamental-antifundamental pairs",
    "tree_superpotential": "zero",
    "baryon_number": "B(Q)=+1 and B(Qtilde)=-1",
    "beta_sign": "b0>0 means asymptotically free; b0<0 means infrared free",
    "endpoint_rule": "strict inequalities define interiors; equality is a separately labeled endpoint"
  },
  "formulas": {
    "electric_one_loop": "b_e=3*Nc-Nf",
    "magnetic_one_loop": "b_m=3*(Nf-Nc)-Nf=2*Nf-3*Nc",
    "anomaly_free_quark_R": "R(Q)=R(Qtilde)=1-Nc/Nf",
    "meson_dimension_at_candidate_scft": "Delta(M)=3*(1-Nc/Nf)",
    "meson_unitarity_bound": "Delta(M)>=1",
    "lower_endpoint_equivalence": "b_m=0 and Delta(M)=1 each give Nf=3*Nc/2 after the magnetic dictionary and exact R assignment are assumed"
  },
  "regimes": [
    {
      "domain": "Nf=0",
      "label": "pure SYM",
      "status": "protected branch count plus dynamical input"
    },
    {
      "domain": "0<Nf<Nc",
      "label": "ADS",
      "status": "exact Wilsonian F-term on each local branch"
    },
    {
      "domain": "Nf=Nc",
      "label": "quantum-modified moduli",
      "status": "exact chiral constraint"
    },
    {
      "domain": "Nf=Nc+1",
      "label": "s-confining",
      "status": "exact Wilsonian F-term and anomaly-matched infrared description"
    },
    {
      "domain": "Nc+2<=Nf<3*Nc/2",
      "label": "free magnetic",
      "status": "infrared-free magnetic theory after accepting the duality dictionary"
    },
    {
      "domain": "Nf=3*Nc/2",
      "label": "lower endpoint",
      "status": "b_m=0 and Delta(M)=1; an endpoint, not either open interior"
    },
    {
      "domain": "3*Nc/2<Nf<3*Nc",
      "label": "interacting non-Abelian Coulomb phase",
      "status": "duality-supported; perturbative only parametrically near fractional edges"
    },
    {
      "domain": "Nf=3*Nc",
      "label": "upper endpoint",
      "status": "electric b_e=0"
    },
    {
      "domain": "Nf>3*Nc",
      "label": "electric infrared free",
      "status": "perturbative beta-function statement"
    }
  ],
  "endpoint_checks": {
    "lower": {
      "Nf_over_Nc": "3/2",
      "b_m_over_Nc": 0,
      "Delta_M": 1
    },
    "upper": {
      "Nf_over_Nc": "3",
      "b_e_over_Nc": 0,
      "Delta_M": 2
    },
    "sign_table": [
      {
        "domain": "Nf<3*Nc/2",
        "magnetic_b_m": "negative"
      },
      {
        "domain": "3*Nc/2<Nf<3*Nc",
        "electric_b_e": "positive",
        "magnetic_b_m": "positive"
      },
      {
        "domain": "Nf>3*Nc",
        "electric_b_e": "negative"
      }
    ]
  },
  "integer_rank_fixtures": [
    {
      "Nc": 2,
      "exact_low_flavor_values": [
        0,
        1,
        2,
        3
      ],
      "free_magnetic_integer_interior": [],
      "lower_endpoint_Nf": 3,
      "lower_endpoint_is_integer": true,
      "lower_endpoint_collides_with_s_confining_rank": true,
      "interacting_integer_interior": [
        4,
        5
      ],
      "upper_endpoint_Nf": 6,
      "first_electric_IR_free_Nf": 7
    },
    {
      "Nc": 3,
      "exact_low_flavor_values": [
        0,
        1,
        2,
        3,
        4
      ],
      "free_magnetic_integer_interior": [],
      "lower_endpoint_Nf": null,
      "lower_endpoint_is_integer": false,
      "lower_endpoint_collides_with_s_confining_rank": false,
      "interacting_integer_interior": [
        5,
        6,
        7,
        8
      ],
      "upper_endpoint_Nf": 9,
      "first_electric_IR_free_Nf": 10
    },
    {
      "Nc": 4,
      "exact_low_flavor_values": [
        0,
        1,
        2,
        3,
        4,
        5
      ],
      "free_magnetic_integer_interior": [],
      "lower_endpoint_Nf": 6,
      "lower_endpoint_is_integer": true,
      "lower_endpoint_collides_with_s_confining_rank": false,
      "interacting_integer_interior": [
        7,
        8,
        9,
        10,
        11
      ],
      "upper_endpoint_Nf": 12,
      "first_electric_IR_free_Nf": 13
    },
    {
      "Nc": 5,
      "exact_low_flavor_values": [
        0,
        1,
        2,
        3,
        4,
        5,
        6
      ],
      "free_magnetic_integer_interior": [
        7
      ],
      "lower_endpoint_Nf": null,
      "lower_endpoint_is_integer": false,
      "lower_endpoint_collides_with_s_confining_rank": false,
      "interacting_integer_interior": [
        8,
        9,
        10,
        11,
        12,
        13,
        14
      ],
      "upper_endpoint_Nf": 15,
      "first_electric_IR_free_Nf": 16
    }
  ],
  "shared_input_caveat": "The magnetic beta coefficient and meson unitarity calculation reuse Nc, Nf, the exact candidate R symmetry, and the proposed magnetic dictionary. Their agreement is a consistency loop, not logically independent evidence for the duality or fixed point.",
  "exceptions": [
    "For SU(2), pseudoreality enhances the connected flavor symmetry to SU(2Nf) and baryons join the meson multiplet.",
    "At Nc=2, Nf=3 is simultaneously Nc+1 and 3Nc/2; the SU(2) s-confining Pfaffian description is the appropriate special-rank record.",
    "The open free-magnetic integer interval is empty for Nc=2, Nc=3, and Nc=4, and contains only Nf=7 for Nc=5."
  ],
  "primary_sources": [
    {
      "citation": "Seiberg 1995, Electric-magnetic duality in supersymmetric non-Abelian gauge theories",
      "url": "https://arxiv.org/abs/hep-th/9411149"
    },
    {
      "citation": "Intriligator and Seiberg 1996, Lectures on supersymmetric gauge theories and electric-magnetic duality",
      "url": "https://arxiv.org/abs/hep-th/9509066"
    }
  ],
  "governance_snapshot": {
    "registry_status": "prototype",
    "registry_public_route": null,
    "integration_state": "complete"
  }
}