{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n2.sw-singularities-monodromies-bps-chambers",
  "title": "One frozen convention ties singular fibers to chambers and confinement",
  "evidence_date": "2026-08-24",
  "record_kind": "exact rank-one convention with schematic path, chamber and deformation layout",
  "reader_question": "How do one exact pure-SU(2) curve, cycle and charge convention connect singular fibers, ordered monodromies, chamber-dependent BPS data and the controlled small-N=1 confinement model?",
  "dominant_point": "With one frozen rank-one convention, the two finite discriminant points determine integral monodromies and strong-chamber charges, wall crossing reorganizes those charges into the weak towers and W vector, and a small adjoint mass produces two local condensate vacua without licensing a large-deformation or QCD claim.",
  "governance": {
    "registry_status_observed": "prototype",
    "registry_public_route_observed": null,
    "target_owner_page_ids": [
      "qft.topic.seiberg-witten.singularities-monodromies-global-geometry",
      "qft.topic.seiberg-witten.pure-su-two-solution",
      "qft.topic.seiberg-witten.bps-spectra-walls",
      "qft.topic.seiberg-witten.n-one-deformations-monopole-condensation"
    ],
    "registry_owner_page_ids_observed": [
      "qft.topic.seiberg-witten.singularities-monodromies-global-geometry",
      "qft.topic.seiberg-witten.pure-su-two-solution",
      "qft.topic.seiberg-witten.bps-spectra-walls",
      "qft.topic.seiberg-witten.n-one-deformations-monopole-condensation"
    ],
    "registry_scope_is_final": true,
    "retired_owners_still_observed": [],
    "owner_routes": [
      {
        "id": "qft.topic.seiberg-witten.singularities-monodromies-global-geometry",
        "route": "/supersymmetry-duality/n2-seiberg-witten/singularities-monodromies-global-geometry/"
      },
      {
        "id": "qft.topic.seiberg-witten.pure-su-two-solution",
        "route": "/supersymmetry-duality/n2-seiberg-witten/pure-su-two-solution/"
      },
      {
        "id": "qft.topic.seiberg-witten.bps-spectra-walls",
        "route": "/supersymmetry-duality/n2-seiberg-witten/bps-spectra-walls/"
      },
      {
        "id": "qft.topic.seiberg-witten.n-one-deformations-monopole-condensation",
        "route": "/supersymmetry-duality/n2-seiberg-witten/n-one-deformations-monopole-condensation/"
      }
    ]
  },
  "frozen_convention": {
    "theory_and_regime": "pure four-dimensional N=2 SU(2) Yang-Mills; Lambda>0",
    "base_point": "u_b=2 Lambda^2",
    "curve": "y^2=(x-u)(x^2-Lambda^4)",
    "differential": "lambda=sqrt(2)/(2 pi) (x-u) dx/y",
    "differential_derivative": "partial_u lambda=-sqrt(2)/(4 pi) dx/y",
    "x_plane_cuts_at_base_point": [
      "[-Lambda^2,+Lambda^2]",
      "[u_b,infinity]"
    ],
    "cycle_orientation": "A intersect B=+1",
    "periods": "Pi=(a_D,a)^T with a>0 and a_D in i R_(>0) at u_b",
    "charge_order": "gamma=(n_m,n_e)^T",
    "central_charge": "Z_gamma=n_m a_D+n_e a",
    "coordinate_pairing": "omega(gamma,gamma')=n_m n'_e-n_e n'_m",
    "physical_pairing": "<gamma,gamma'>=2 omega(gamma,gamma')",
    "transport": "Pi'=M Pi and gamma'=M^-T gamma",
    "monodromy_action": "left column action; the rightmost factor acts first"
  },
  "singularity_and_monodromy_table": [
    {
      "locus": "+Lambda^2",
      "vanishing_charge": [
        1,
        0
      ],
      "particle_name": "monopole",
      "monodromy": [
        [
          1,
          0
        ],
        [
          -2,
          1
        ]
      ],
      "local_period": "a_D=i(u-Lambda^2)/(2 Lambda)+higher orders"
    },
    {
      "locus": "-Lambda^2",
      "vanishing_charge": [
        1,
        -1
      ],
      "particle_name": "dyon",
      "monodromy": [
        [
          -1,
          2
        ],
        [
          -2,
          3
        ]
      ],
      "local_period": "Z_(1,-1)=a_D-a vanishes"
    },
    {
      "locus": "infinity",
      "vanishing_charge": null,
      "particle_name": null,
      "monodromy": [
        [
          -1,
          2
        ],
        [
          0,
          -1
        ]
      ],
      "based_product": "M_infinity=M_m M_d"
    }
  ],
  "discriminant": {
    "formula": "Disc_x[(x-u)(x^2-Lambda^4)]=4 Lambda^4 (u^2-Lambda^4)^2",
    "finite_roots": [
      "-Lambda^2",
      "+Lambda^2"
    ],
    "path_data": "positive based loops from u_b; dyon loop is the rightmost factor in M_m M_d and is therefore applied first"
  },
  "chamber_table": {
    "half_plane": "chosen so gamma1 and gamma2 below are particles; antiparticles are implicit",
    "strong": [
      {
        "charge": [
          1,
          0
        ],
        "label": "gamma1=gamma_m",
        "multiplet": "hypermultiplet",
        "Omega": 1
      },
      {
        "charge": [
          -1,
          1
        ],
        "label": "gamma2=-gamma_d",
        "multiplet": "hypermultiplet",
        "Omega": 1
      }
    ],
    "strong_pairing": "+2",
    "wall": "Im(a_D/a)=0 together with the sign selecting aligned, rather than anti-aligned, central charges",
    "weak": {
      "plus_tower": {
        "charge": "gamma_n^+=(1,n)",
        "range": "n>=0",
        "multiplet": "hypermultiplet",
        "Omega": 1
      },
      "minus_tower": {
        "charge": "gamma_n^-=(-1,n+1)",
        "range": "n>=0",
        "multiplet": "hypermultiplet",
        "Omega": 1
      },
      "W": {
        "charge": [
          0,
          1
        ],
        "label": "gamma_W=gamma1+gamma2",
        "multiplet": "vector",
        "Omega": -2
      }
    },
    "protected_data_ceiling": "Omega is a protected index, not a complete unprotected degeneracy or a multiparticle-state count"
  },
  "wall_crossing": {
    "pullback_convention": "rightmost factor acts first",
    "strong_product": "K_gamma2 K_gamma1",
    "weak_product": "(product n=0..infinity K_gamma^+_n) K_gammaW^-2 (product n=infinity..0 K_gamma^-_n)",
    "finite_formal_series_verification": {
      "status": "passed",
      "formal_variables": "x=X_gamma1, y=X_gamma2 in the positive cone",
      "pairing": "<gamma1,gamma2>=2",
      "pullback_convention": "rightmost factor acts first",
      "checked_total_degree": 8,
      "included_tower_indices": "0 through 3; omitted factors first contribute above the checked degree",
      "strong_order": [
        "K_gamma2",
        "K_gamma1"
      ],
      "weak_order": [
        "K_gamma^+_n for n=0 through 3",
        "K_gammaW^-2",
        "K_gamma^-_n for n=3 down to 0"
      ],
      "common_action": {
        "x_image_terms": [
          {
            "x_power": 1,
            "y_power": 0,
            "coefficient": "1"
          },
          {
            "x_power": 1,
            "y_power": 1,
            "coefficient": "-2"
          },
          {
            "x_power": 1,
            "y_power": 2,
            "coefficient": "1"
          }
        ],
        "y_image_terms": [
          {
            "x_power": 0,
            "y_power": 1,
            "coefficient": "1"
          },
          {
            "x_power": 1,
            "y_power": 1,
            "coefficient": "2"
          },
          {
            "x_power": 1,
            "y_power": 2,
            "coefficient": "-4"
          },
          {
            "x_power": 2,
            "y_power": 1,
            "coefficient": "3"
          },
          {
            "x_power": 1,
            "y_power": 3,
            "coefficient": "2"
          },
          {
            "x_power": 2,
            "y_power": 2,
            "coefficient": "-12"
          },
          {
            "x_power": 3,
            "y_power": 1,
            "coefficient": "4"
          },
          {
            "x_power": 2,
            "y_power": 3,
            "coefficient": "18"
          },
          {
            "x_power": 3,
            "y_power": 2,
            "coefficient": "-24"
          },
          {
            "x_power": 4,
            "y_power": 1,
            "coefficient": "5"
          },
          {
            "x_power": 2,
            "y_power": 4,
            "coefficient": "-12"
          },
          {
            "x_power": 3,
            "y_power": 3,
            "coefficient": "60"
          },
          {
            "x_power": 4,
            "y_power": 2,
            "coefficient": "-40"
          },
          {
            "x_power": 5,
            "y_power": 1,
            "coefficient": "6"
          },
          {
            "x_power": 2,
            "y_power": 5,
            "coefficient": "3"
          },
          {
            "x_power": 3,
            "y_power": 4,
            "coefficient": "-80"
          },
          {
            "x_power": 4,
            "y_power": 3,
            "coefficient": "140"
          },
          {
            "x_power": 5,
            "y_power": 2,
            "coefficient": "-60"
          },
          {
            "x_power": 6,
            "y_power": 1,
            "coefficient": "7"
          },
          {
            "x_power": 3,
            "y_power": 5,
            "coefficient": "60"
          },
          {
            "x_power": 4,
            "y_power": 4,
            "coefficient": "-280"
          },
          {
            "x_power": 5,
            "y_power": 3,
            "coefficient": "270"
          },
          {
            "x_power": 6,
            "y_power": 2,
            "coefficient": "-84"
          },
          {
            "x_power": 7,
            "y_power": 1,
            "coefficient": "8"
          }
        ]
      },
      "evidence_ceiling": "This is an exact coefficient comparison in the declared finite quotient, not by itself a proof of the complete infinite-product identity."
    }
  },
  "controlled_n1_deformation": {
    "regime": "|m_Phi| << |Lambda|",
    "local_superpotential": "W_loc=sqrt(2) A_gamma M Mtilde+m_Phi u(A_gamma)",
    "f_term_solution": "A_gamma=0; M Mtilde=-(m_Phi/sqrt(2))(du/dA_gamma)_0",
    "local_frames": [
      {
        "vacuum": "+Lambda^2",
        "light_charge": [
          1,
          0
        ],
        "coordinate": "A_gamma=a_D",
        "du_dA": "-2 i Lambda",
        "condensate": "M Mtilde=i sqrt(2) m_Phi Lambda",
        "D_flat_magnitude": "|M|=|Mtilde|=2^(1/4)|m_Phi Lambda|^(1/2)"
      },
      {
        "vacuum": "-Lambda^2",
        "light_charge": [
          1,
          -1
        ],
        "coordinate": "A_gamma=a_D-a",
        "statement": "solve in the separate dyonic frame; no single local Abelian Lagrangian contains both light hypermultiplets"
      }
    ],
    "vacuum_superpotentials": [
      "W_+=m_Phi Lambda^2",
      "W_-=-m_Phi Lambda^2"
    ],
    "domain_wall_bound": "T_wall>=2|W_+-W_-|=4|m_Phi| Lambda^2; equality requires an existing BPS-saturating wall, and the profile is not controlled by either endpoint patch alone",
    "local_vortex": "in canonical local BPS normalization T_string=2 pi |M Mtilde|=2 pi sqrt(2)|m_Phi Lambda|; other charge and field normalizations redistribute the coefficient, and higher-order terms correct it away from the small-deformation limit",
    "confinement_claim": "only genuine line probes with nontrivial center charge are forced onto flux tubes; adjoint-screenable probes are not",
    "inference_stops": [
      "no claim at large |m_Phi|/|Lambda|",
      "no derivation of nonsupersymmetric QCD confinement",
      "no globally local simultaneous monopole-dyon Lagrangian"
    ]
  },
  "scientific_verification": {
    "status": "passed",
    "discriminant": {
      "formula": "4 Lambda^4 (u^2-Lambda^4)^2",
      "exact_integer_fixtures": [
        {
          "u_over_Lambda_squared": 2,
          "normalized_discriminant": 36
        },
        {
          "u_over_Lambda_squared": 3,
          "normalized_discriminant": 256
        },
        {
          "u_over_Lambda_squared": -2,
          "normalized_discriminant": 36
        },
        {
          "u_over_Lambda_squared": -3,
          "normalized_discriminant": 256
        }
      ]
    },
    "monodromy": {
      "formula": "M_(nm,ne)=[[1+2 nm ne,2 ne^2],[-2 nm^2,1-2 nm ne]]",
      "monopole": [
        [
          1,
          0
        ],
        [
          -2,
          1
        ]
      ],
      "dyon": [
        [
          -1,
          2
        ],
        [
          -2,
          3
        ]
      ],
      "based_product": [
        [
          -1,
          2
        ],
        [
          0,
          -1
        ]
      ],
      "convention": "left column action; rightmost loop factor first"
    },
    "charges": {
      "coordinate_pairing": "omega(gamma,gamma')=nm ne'-ne nm'",
      "physical_pairing": "<gamma,gamma'>=2 omega(gamma,gamma')",
      "gamma1": [
        1,
        0
      ],
      "gamma2": [
        -1,
        1
      ],
      "physical_pairing_gamma1_gamma2": 2,
      "weak_towers_checked_primitive_through_n": 16
    },
    "periods": {
      "status": "passed",
      "elliptic_integral_algorithm": "arithmetic-geometric mean with the exact complementary-parameter formulas",
      "base_point": {
        "u_over_Lambda_squared": 2,
        "m": 0.6666666666666666,
        "a_over_Lambda": 1.96668530155033,
        "aD_over_Lambda": "0.4734344365355367 i",
        "tau": "0.8545844432787435 i"
      },
      "local_monopole_limit": {
        "formula": "a_D=i(u-Lambda^2)/(2 Lambda)+higher orders",
        "epsilon_over_Lambda_squared": 1e-7,
        "relative_error": 2.1447543741004438e-9
      },
      "exact_checks": [
        "for the displayed representative, d lambda/du=-sqrt(2) dx/(4 pi y) exactly",
        "det M=1 and M^T J M=J for the frozen rank-one transitions",
        "Pi -> M Pi, gamma -> M^-T gamma preserves gamma^T Pi",
        "a(Lambda^2)=4 Lambda/pi"
      ]
    },
    "wall_crossing_finite_quotient": {
      "status": "passed",
      "formal_variables": "x=X_gamma1, y=X_gamma2 in the positive cone",
      "pairing": "<gamma1,gamma2>=2",
      "pullback_convention": "rightmost factor acts first",
      "checked_total_degree": 8,
      "included_tower_indices": "0 through 3; omitted factors first contribute above the checked degree",
      "strong_order": [
        "K_gamma2",
        "K_gamma1"
      ],
      "weak_order": [
        "K_gamma^+_n for n=0 through 3",
        "K_gammaW^-2",
        "K_gamma^-_n for n=3 down to 0"
      ],
      "common_action": {
        "x_image_terms": [
          {
            "x_power": 1,
            "y_power": 0,
            "coefficient": "1"
          },
          {
            "x_power": 1,
            "y_power": 1,
            "coefficient": "-2"
          },
          {
            "x_power": 1,
            "y_power": 2,
            "coefficient": "1"
          }
        ],
        "y_image_terms": [
          {
            "x_power": 0,
            "y_power": 1,
            "coefficient": "1"
          },
          {
            "x_power": 1,
            "y_power": 1,
            "coefficient": "2"
          },
          {
            "x_power": 1,
            "y_power": 2,
            "coefficient": "-4"
          },
          {
            "x_power": 2,
            "y_power": 1,
            "coefficient": "3"
          },
          {
            "x_power": 1,
            "y_power": 3,
            "coefficient": "2"
          },
          {
            "x_power": 2,
            "y_power": 2,
            "coefficient": "-12"
          },
          {
            "x_power": 3,
            "y_power": 1,
            "coefficient": "4"
          },
          {
            "x_power": 2,
            "y_power": 3,
            "coefficient": "18"
          },
          {
            "x_power": 3,
            "y_power": 2,
            "coefficient": "-24"
          },
          {
            "x_power": 4,
            "y_power": 1,
            "coefficient": "5"
          },
          {
            "x_power": 2,
            "y_power": 4,
            "coefficient": "-12"
          },
          {
            "x_power": 3,
            "y_power": 3,
            "coefficient": "60"
          },
          {
            "x_power": 4,
            "y_power": 2,
            "coefficient": "-40"
          },
          {
            "x_power": 5,
            "y_power": 1,
            "coefficient": "6"
          },
          {
            "x_power": 2,
            "y_power": 5,
            "coefficient": "3"
          },
          {
            "x_power": 3,
            "y_power": 4,
            "coefficient": "-80"
          },
          {
            "x_power": 4,
            "y_power": 3,
            "coefficient": "140"
          },
          {
            "x_power": 5,
            "y_power": 2,
            "coefficient": "-60"
          },
          {
            "x_power": 6,
            "y_power": 1,
            "coefficient": "7"
          },
          {
            "x_power": 3,
            "y_power": 5,
            "coefficient": "60"
          },
          {
            "x_power": 4,
            "y_power": 4,
            "coefficient": "-280"
          },
          {
            "x_power": 5,
            "y_power": 3,
            "coefficient": "270"
          },
          {
            "x_power": 6,
            "y_power": 2,
            "coefficient": "-84"
          },
          {
            "x_power": 7,
            "y_power": 1,
            "coefficient": "8"
          }
        ]
      },
      "evidence_ceiling": "This is an exact coefficient comparison in the declared finite quotient, not by itself a proof of the complete infinite-product identity."
    },
    "n1_local_fixture": {
      "du_daD_at_monopole": "-2 i Lambda",
      "M_Mtilde_at_monopole": "i sqrt(2) m_Phi Lambda",
      "D_flat_magnitude": "2^(1/4) |m_Phi Lambda|^(1/2)",
      "vacuum_superpotentials": [
        "+m_Phi Lambda^2",
        "-m_Phi Lambda^2"
      ],
      "domain_wall_bound": "T_wall >= 2 |Delta W|=4 |m_Phi| Lambda^2; equality requires an existing wall that saturates the BPS bound",
      "leading_vortex_tension": "in canonical local BPS normalization T_string=2 pi |M Mtilde|=2 pi sqrt(2) |m_Phi Lambda|; other charge and field normalizations redistribute the coefficient, and higher-order terms correct the strict local limit"
    }
  },
  "sources": [
    {
      "id": "seiberg-witten-1994",
      "locator": "Sections 2-5 and Appendix B",
      "url": "https://arxiv.org/abs/hep-th/9407087",
      "use": "pure-SU(2) curve, differential, periods, singularities, monodromies and the small N=1 deformation"
    },
    {
      "id": "bilal-ferrari-1996",
      "locator": "Sections 3-6",
      "url": "https://arxiv.org/abs/hep-th/9602082",
      "use": "strong- and weak-coupling pure-SU(2) BPS spectra"
    },
    {
      "id": "kontsevich-soibelman-2008",
      "locator": "Section 2.3",
      "url": "https://arxiv.org/abs/0811.2435",
      "use": "ordered stability-data factorization"
    },
    {
      "id": "gaiotto-moore-neitzke-2010",
      "locator": "Sections 2.2 and 5",
      "url": "https://arxiv.org/abs/0807.4723",
      "use": "four-dimensional BPS wall crossing and mutually nonlocal corrections"
    },
    {
      "id": "douglas-shenker-1995",
      "locator": "Sections 3-4",
      "url": "https://arxiv.org/abs/hep-th/9503163",
      "use": "controlled N=1 deformation and Abelian confinement regime"
    }
  ],
  "visual_encodings": {
    "solid_and_dashed_cycles": "distinguish A and B projections without color",
    "solid_and_dashed_loops": "distinguish the two based u-plane loops without color",
    "solid_arrow": "licensed transformation or controlled implication",
    "dashed_box": "inference ceiling",
    "color_dependence": "none"
  },
  "accessibility": {
    "alt_text": "Five monochrome panels use one pure SU(2) convention. The frozen x-plane fiber shows branch cuts, A and B cycles and their intersection. The based u-plane shows the dyon and monopole singularities, a base point and ordered loops with matrices whose product is the infinity monodromy. A charge crosswalk defines the root-unit pairing. Strong and weak chamber cards list two hypermultiplets versus two infinite dyon towers and a W vector, with the correctly ordered Kontsevich-Soibelman product. A final flow uses one local frame at each singularity to turn a small adjoint mass into a monopole or dyon condensate and center-sensitive flux tubes, ending at a dashed scope ceiling.",
    "semantic_equivalent": "/figures/supersymmetry-duality/sw-singularities-monodromies-bps-chambers.json",
    "narrow_width": "the full-size path-based SVG can pan without page-wide overflow; this JSON preserves all cut, cycle, loop, matrix, charge, chamber and deformation data"
  },
  "limitations": [
    "The x-plane cycles and u-plane loops are schematic projections; their orientations and algebraic actions, not their drawn Euclidean shapes, are the exact data.",
    "The BPS spectrum is the standard protected pure-SU(2) spectrum in the declared half-plane and chamber convention; antiparticles and multiparticle continua are not separately listed.",
    "The finite formal-series wall-crossing check proves equality only in the declared bounded quotient, not the entire infinite-product identity by itself.",
    "The condensate and confinement mechanism are controlled only for a parametrically small adjoint mass in separate local frames.",
    "The audited final owner scope has four pages; retired owners still present during generation: none.",
    "The artifact registry lifecycle observed during generation was prototype."
  ]
}