{
  "schema_id": "qft.supersymmetry-duality.agt-one-instanton-benchmark.v1",
  "schema_version": "1.0.0",
  "benchmark_id": "qft.benchmark.supersymmetry-duality.agt-u2-nf4-one-instanton.v1",
  "generated_on": "2026-09-04",
  "deterministic": true,
  "status": "PASS",
  "claim_scope": {
    "supports": "The exact q^1 coefficient identity for the frozen four-punctured-sphere U(2), Nf=4 AGT fixture in the declared mass, Omega-background, block-ordering, U(1), and UV-coupling conventions.",
    "does_not_support": "Higher instanton orders, convergence or analytic continuation of the formal series, equality of full S4 integrals, other puncture types, higher rank, or a proof of the complete AGT correspondence."
  },
  "source_locators": [
    {
      "citation": "Luis F. Alday, Davide Gaiotto, and Yuji Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Letters in Mathematical Physics 91 (2010) 167-197",
      "arxiv": "0906.3219v2",
      "url": "https://arxiv.org/abs/0906.3219v2",
      "locators": [
        "Eqs. (10)-(16): scale, Q, U(2) Nf=4 orientation, mass map, U(1) factor, and Liouville momenta",
        "Eqs. (17)-(18) and (28): external dimensions and insertions at infinity, 1, q, and 0",
        "Eq. (56): four-point sewing order",
        "Eqs. (74)-(84): UV q weight and Nekrasov fixed-point factors",
        "Eq. (105): four-punctured-sphere U(1) factor"
      ]
    }
  ],
  "conventions": {
    "dimensionless_scale": "sqrt(epsilon1 epsilon2)=1 with epsilon1=b=2 and epsilon2=1/b=1/2",
    "background_charge_definition": "Q=epsilon1+epsilon2",
    "uv_coupling_definition": "q=exp(2 pi i tau_UV)",
    "instanton_weight": "q^|Y|; one box contributes at q^1",
    "gauge_group_for_fixed_point_sum": "U(2) localization with Coulomb eigenvalues (a,-a); the a-independent U(1) factor is retained explicitly",
    "matter_orientation_id": "agt-two-antifund-two-fund",
    "matter_orientation": "mu1,mu2 are antifundamentals and mu3,mu4 are fundamentals, exactly as in AGT Eq. (12)",
    "conformal_block_order": "<V_alpha0(infinity) V_m0(1) V_m1(q) V_alpha1(0)>; (Delta1,Delta2,Delta3,Delta4)=(Delta_infinity,Delta_at_1,Delta_at_q,Delta_at_0). No two-by-two array convention is stored.",
    "normalized_series": "Z_inst(q)=1+z1 q+O(q^2), F(q)=1+F1 q+O(q^2)",
    "U1_relation": "Z_inst(q)=(1-q)^nu F(q), so z1=F1-nu"
  },
  "formulas": {
    "mass_map": "mu1=m0+m0_tilde, mu2=m0-m0_tilde, mu3=m1+m1_tilde, mu4=m1-m1_tilde",
    "one_box_matter_numerator": "(a_r+mu1)(a_r+mu2)(a_r+Q-mu3)(a_r+Q-mu4)",
    "one_box_vector_denominator": "epsilon1 epsilon2 product_(s != r) (a_r-a_s+Q)(a_s-a_r)",
    "dimensions": "Delta=Q^2/4-a^2; Delta1=Delta_infinity=Q^2/4-m0_tilde^2; Delta2=Delta_at_1=m0(Q-m0); Delta3=Delta_at_q=m1(Q-m1); Delta4=Delta_at_0=Q^2/4-m1_tilde^2",
    "virasoro_level_one": "F1=((Delta+Delta2-Delta1)(Delta+Delta3-Delta4))/(2 Delta)",
    "u1_exponent": "nu=2 m0 (Q-m1)",
    "acceptance_identity": "z1+nu=F1"
  },
  "frozen_parameters": {
    "epsilon1": "2",
    "epsilon2": "1/2",
    "Q": "5/2",
    "a": "1/3",
    "m0": "2/3",
    "m0_tilde": "1/5",
    "m1": "3/4",
    "m1_tilde": "1/7",
    "mu1": "13/15",
    "mu2": "7/15",
    "mu3": "25/28",
    "mu4": "17/28"
  },
  "conformal_dimensions": {
    "Delta": "209/144",
    "Delta1": "609/400",
    "Delta2": "11/9",
    "Delta3": "21/16",
    "Delta4": "1209/784"
  },
  "fixed_point_terms": [
    {
      "record_id": "one_box_in_Y_plus",
      "gauge_eigenvalue": "1/3",
      "antifundamental_product": "24/25",
      "fundamental_product": "30481/7056",
      "matter_numerator": "30481/7350",
      "vector_denominator": "-19/9",
      "fixed_point_contribution": "-91443/46550"
    },
    {
      "record_id": "one_box_in_Y_minus",
      "gauge_eigenvalue": "-1/3",
      "antifundamental_product": "16/225",
      "fundamental_product": "14017/7056",
      "matter_numerator": "14017/99225",
      "vector_denominator": "11/9",
      "fixed_point_contribution": "14017/121275"
    }
  ],
  "exact_values": {
    "z1": "-1217173/658350",
    "u1_exponent_nu": "7/3",
    "virasoro_level_one_F1": "318977/658350",
    "acceptance_residual": "0"
  },
  "nonpolar_checks": {
    "epsilon1_epsilon2": "1",
    "vector_denominators": [
      "-19/9",
      "11/9"
    ],
    "level_one_gram_denominator_2Delta": "209/72",
    "all_nonzero": true
  },
  "negative_controls": {
    "omitted_u1_exponent": {
      "mutation": "compare z1 directly with F1",
      "computed_residual": "-7/3",
      "rejected": true
    },
    "sign_flipped_u1_exponent": {
      "mutation": "replace nu by -nu in z1 + nu = F1",
      "computed_residual": "-14/3",
      "rejected": true
    },
    "swapped_Delta1_Delta2": {
      "mutation": "replace Delta + Delta2 - Delta1 by Delta + Delta1 - Delta2",
      "wrong_F1": "9060671/12289200",
      "computed_residual": "-9319301/36867600",
      "rejected": true
    },
    "all_fundamental_orientation": {
      "mutation": "treat mu1 and mu2 as fundamentals at the same tabulated masses, omitting Eq. (83) and its Q-minus-mu conversion",
      "wrong_z1": "-7810871/1316700",
      "computed_residual": "-49/12",
      "rejected": true
    },
    "wrong_q_square_root_convention": {
      "mutation": "replace q=exp(2 pi i tau_UV) by x=exp(pi i tau_UV), so q=x^2",
      "first_one_box_power_in_x": 2,
      "x_linear_coefficients": {
        "Z_inst": "0",
        "U1_factor": "0",
        "conformal_block": "0"
      },
      "rejection_basis": "the x^1 identity is vacuous and is not the q^1 benchmark defined by AGT Eqs. (14) and (74)",
      "rejected": true
    },
    "coincident_eigenvalue_chart_input": {
      "mutation": "set a=0 so the two U(2) Coulomb eigenvalues coincide",
      "rejected": true,
      "direct_chart_outcome": "rejected because the separate one-box fixed-point denominators vanish",
      "summed_meromorphic_continuation": "-98947/55125",
      "continuation_finite": true,
      "interpretation": "This is a removable singularity of the separated fixed-point terms, not a pole of their summed meromorphic continuation."
    },
    "virasoro_pole_input": {
      "mutation": "set a=Q/2=5/4 so Delta=0",
      "rejected": true,
      "outcome": "genuine pole",
      "reason": "the level-one Virasoro Gram denominator and one one-box vector denominator vanish while the corresponding numerators remain nonzero"
    }
  },
  "csv_fields": [
    "schema_id",
    "benchmark_id",
    "record_type",
    "record_id",
    "gauge_eigenvalue",
    "antifundamental_product",
    "fundamental_product",
    "matter_numerator",
    "vector_denominator",
    "fixed_point_contribution",
    "z1",
    "u1_exponent_nu",
    "virasoro_level_one_F1",
    "acceptance_residual"
  ],
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      "benchmark_id": "qft.benchmark.supersymmetry-duality.agt-u2-nf4-one-instanton.v1",
      "record_type": "fixed_point",
      "record_id": "one_box_in_Y_plus",
      "gauge_eigenvalue": "1/3",
      "antifundamental_product": "24/25",
      "fundamental_product": "30481/7056",
      "matter_numerator": "30481/7350",
      "vector_denominator": "-19/9",
      "fixed_point_contribution": "-91443/46550",
      "z1": "",
      "u1_exponent_nu": "",
      "virasoro_level_one_F1": "",
      "acceptance_residual": ""
    },
    {
      "schema_id": "qft.supersymmetry-duality.agt-one-instanton-benchmark.v1",
      "benchmark_id": "qft.benchmark.supersymmetry-duality.agt-u2-nf4-one-instanton.v1",
      "record_type": "fixed_point",
      "record_id": "one_box_in_Y_minus",
      "gauge_eigenvalue": "-1/3",
      "antifundamental_product": "16/225",
      "fundamental_product": "14017/7056",
      "matter_numerator": "14017/99225",
      "vector_denominator": "11/9",
      "fixed_point_contribution": "14017/121275",
      "z1": "",
      "u1_exponent_nu": "",
      "virasoro_level_one_F1": "",
      "acceptance_residual": ""
    },
    {
      "schema_id": "qft.supersymmetry-duality.agt-one-instanton-benchmark.v1",
      "benchmark_id": "qft.benchmark.supersymmetry-duality.agt-u2-nf4-one-instanton.v1",
      "record_type": "acceptance_identity",
      "record_id": "z1_plus_nu_equals_F1",
      "gauge_eigenvalue": "",
      "antifundamental_product": "",
      "fundamental_product": "",
      "matter_numerator": "",
      "vector_denominator": "",
      "fixed_point_contribution": "",
      "z1": "-1217173/658350",
      "u1_exponent_nu": "7/3",
      "virasoro_level_one_F1": "318977/658350",
      "acceptance_residual": "0"
    }
  ],
  "integrity": {
    "hash_algorithm": "SHA-256 over exact file bytes",
    "generator": {
      "path": "figures-src/supersymmetry-duality/generate-agt-one-instanton-benchmark.mjs",
      "sha256": "82fac54f56df73c1d356877cd692b263127fb77fa9e2c52e150a099a26464b59",
      "bytes": 25671,
      "use": "closed-form exact-rational one-box calculation and deterministic serialization"
    },
    "independent_checker": {
      "path": "scripts/verify-agt-one-instanton-benchmark.mjs",
      "sha256": "8cfe6ed4422a92b1a8c970894bd5b83a520f14d4773fb9be312df24d3f48f03a",
      "bytes": 43631,
      "use": "independent Young-diagram reconstruction, artifact audit, and adversarial mutations"
    },
    "csv": {
      "source_path": "figures-src/supersymmetry-duality/agt-one-instanton-benchmark.csv",
      "public_path": "public/data/supersymmetry-duality/agt-one-instanton-benchmark.csv",
      "sha256": "17fd02373cd3a44fd26fbb62587eb0ab5677f2166fc0044471ff0130a9f51713",
      "bytes": 846,
      "records": 3,
      "byte_identical": true
    },
    "json": {
      "source_path": "figures-src/supersymmetry-duality/agt-one-instanton-benchmark.json",
      "public_path": "public/data/supersymmetry-duality/agt-one-instanton-benchmark.json",
      "byte_identical": true,
      "self_hash_policy": "The JSON does not embed its own SHA-256 or byte count because that would be circular. The independent verifier computes both externally and requires source and public copies to be byte-identical."
    }
  },
  "reproduction": {
    "generate": "node figures-src/supersymmetry-duality/generate-agt-one-instanton-benchmark.mjs",
    "exact_regeneration_check": "node figures-src/supersymmetry-duality/generate-agt-one-instanton-benchmark.mjs --check",
    "independent_check": "node scripts/verify-agt-one-instanton-benchmark.mjs"
  },
  "limitations": [
    "This is an exact one-instanton, level-one formal-series fixture at one nonpolar rational parameter point.",
    "Agreement at q^1 is a convention-sensitive diagnostic, not evidence that all higher coefficients or analytically continued functions agree.",
    "The calculation checks the instanton/conformal-block relation only; classical, one-loop, DOZZ, S4 contour, and Weyl-measure factors are outside this artifact.",
    "The U(2) fixed-point sum and its a-independent U(1) factor are not relabeled as a direct SU(2) instanton sum."
  ],
  "self_check": {
    "exact_arithmetic": "normalized BigInt rational arithmetic; no floating-point comparisons",
    "count": 35,
    "all_passed": true,
    "checks": [
      {
        "label": "Omega-background scale is unity",
        "status": "PASS",
        "exact_value": "1"
      },
      {
        "label": "Q equals epsilon1 plus epsilon2",
        "status": "PASS",
        "exact_value": "5/2"
      },
      {
        "label": "mu1 mass map",
        "status": "PASS",
        "exact_value": "13/15"
      },
      {
        "label": "mu2 mass map",
        "status": "PASS",
        "exact_value": "7/15"
      },
      {
        "label": "mu3 mass map",
        "status": "PASS",
        "exact_value": "25/28"
      },
      {
        "label": "mu4 mass map",
        "status": "PASS",
        "exact_value": "17/28"
      },
      {
        "label": "intermediate dimension",
        "status": "PASS",
        "exact_value": "209/144"
      },
      {
        "label": "external dimension Delta1",
        "status": "PASS",
        "exact_value": "609/400"
      },
      {
        "label": "external dimension Delta2",
        "status": "PASS",
        "exact_value": "11/9"
      },
      {
        "label": "external dimension Delta3",
        "status": "PASS",
        "exact_value": "21/16"
      },
      {
        "label": "external dimension Delta4",
        "status": "PASS",
        "exact_value": "1209/784"
      },
      {
        "label": "one-box Y_plus contribution",
        "status": "PASS",
        "exact_value": "-91443/46550"
      },
      {
        "label": "one-box Y_minus contribution",
        "status": "PASS",
        "exact_value": "14017/121275"
      },
      {
        "label": "U(2) one-instanton coefficient z1",
        "status": "PASS",
        "exact_value": "-1217173/658350"
      },
      {
        "label": "U(1) exponent nu",
        "status": "PASS",
        "exact_value": "7/3"
      },
      {
        "label": "Virasoro level-one coefficient F1",
        "status": "PASS",
        "exact_value": "318977/658350"
      },
      {
        "label": "AGT level-one acceptance identity",
        "status": "PASS",
        "exact_value": "0"
      },
      {
        "label": "both one-box vector denominators are nonzero",
        "status": "PASS"
      },
      {
        "label": "the level-one Virasoro Gram denominator is nonzero",
        "status": "PASS"
      },
      {
        "label": "the frozen fixture is nonpolar",
        "status": "PASS"
      },
      {
        "label": "finite summed continuation at coincident Coulomb eigenvalues",
        "status": "PASS",
        "exact_value": "-98947/55125"
      },
      {
        "label": "coincident-eigenvalue continuation agrees with F1-nu",
        "status": "PASS",
        "exact_value": "-98947/55125"
      },
      {
        "label": "omitted_u1_exponent is rejected",
        "status": "PASS"
      },
      {
        "label": "omitted_u1_exponent has nonzero residual",
        "status": "PASS"
      },
      {
        "label": "sign_flipped_u1_exponent is rejected",
        "status": "PASS"
      },
      {
        "label": "sign_flipped_u1_exponent has nonzero residual",
        "status": "PASS"
      },
      {
        "label": "swapped_Delta1_Delta2 is rejected",
        "status": "PASS"
      },
      {
        "label": "swapped_Delta1_Delta2 has nonzero residual",
        "status": "PASS"
      },
      {
        "label": "all_fundamental_orientation is rejected",
        "status": "PASS"
      },
      {
        "label": "all_fundamental_orientation has nonzero residual",
        "status": "PASS"
      },
      {
        "label": "wrong_q_square_root_convention is rejected",
        "status": "PASS"
      },
      {
        "label": "coincident_eigenvalue_chart_input is rejected",
        "status": "PASS"
      },
      {
        "label": "virasoro_pole_input is rejected",
        "status": "PASS"
      },
      {
        "label": "coincident-eigenvalue input is rejected by the separated fixed-point chart",
        "status": "PASS"
      },
      {
        "label": "Delta-zero input raises a genuine fixed-point division-by-zero failure",
        "status": "PASS"
      }
    ]
  }
}