{
  "schema_version": "1.0.0",
  "benchmark_id": "qft.quantum-information.corner-angular-free-fields.v1",
  "generated_by": "scripts/generate-corner-angular-benchmark.mjs",
  "generator_sha256": "5804c1c98b7ad8e8efa06f2cbd2218d29aa3ead1806032217b471598ce25fdee",
  "quantity": {
    "name": "von Neumann entanglement corner function",
    "symbol": "a_1(theta)",
    "entropy_convention": "S(A)=alpha_R L/epsilon-a_1(theta) log(L/epsilon)+O(1) for one corner in a vacuum 2+1-dimensional CFT",
    "angle_convention": "theta is the physical interior opening angle in radians, with 0<theta<=pi; purity supplies a(2pi-theta)=a(theta)",
    "logarithm_base": "natural",
    "theories": "massless complex scalar and massless two-component Dirac fermion"
  },
  "primary_source": {
    "citation": "Johannes Helmes, Lauren E. Hayward Sierens, Anushya Chandran, William Witczak-Krempa, and Roger G. Melko, Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice, Physical Review B 94, 125142 (2016).",
    "doi": "https://doi.org/10.1103/PhysRevB.94.125142",
    "open_pdf": "https://arxiv.org/pdf/1606.03096",
    "locators": [
      "Eq. (22), preprint pp. 5-6: smooth--sharp interpolation ansatz",
      "Tables I-II, preprint p. 10: field-theory ansatz values and lattice extrapolations",
      "Tables III-IV, preprint pp. 13-14: smooth-limit coefficients",
      "Sec. IV.D.2, preprint pp. 9-11: finite-size extraction and systematic precision statement",
      "Sec. IV.D.3, preprint p. 12: sharp-limit coefficients kappa_b and kappa_f"
    ]
  },
  "method": {
    "status": "Reproducible evaluation of the published high-precision interpolation ansatz, not a new direct Green-function integration or lattice simulation.",
    "equation": "a_tilde(theta)=sum_{p=1}^M sigma^(p-1)(theta-pi)^(2p)+(2 kappa/pi^(2M+1))(theta-pi)^(2(M+1))/(theta(2pi-theta))",
    "orders": {
      "massless_complex_scalar_M": 8,
      "massless_two_component_dirac_M": 7
    },
    "exact_inputs": "sigma=1/128 and the quartic coefficients shown as closed forms in the theory records",
    "rounded_inputs": "higher smooth coefficients and kappa are transcribed at the significant digits printed in Helmes et al. Tables III-IV and Sec. IV.D.3",
    "plotting_grid": "40 points theta/pi=0.025,0.050,...,1.000; the complete CSV contains the ansatz plus smooth and sharp asymptotes"
  },
  "theories": {
    "massless-complex-scalar": {
      "label": "Massless complex scalar",
      "degrees_of_freedom": "one complex scalar, equivalently two independent real scalars",
      "lattice_method": "numerical linked-cluster expansion on square-lattice clusters",
      "C_T": 0.018997721932938333,
      "sigma": 0.0078125,
      "sigma_over_C_T": 0.4112335167120566,
      "kappa": 0.0794,
      "M": 8,
      "smooth_coefficients_sigma_superscript_p_minus_1": [
        0.0078125,
        0.0005454018742238233,
        0.0000534655497,
        0.00000540160621,
        5.45758486e-7,
        5.51156763e-8,
        5.57181927e-9,
        5.63580458e-10
      ],
      "coefficient_source_strings": [
        "1/128",
        "(20+3*pi^2)/(9216*pi^2)",
        "5.34655497e-5",
        "5.40160621e-6",
        "5.45758486e-7",
        "5.51156763e-8",
        "5.57181927e-9",
        "5.63580458e-10"
      ],
      "endpoint_round_trip_fits": {
        "interpretation": "These are round-trip endpoint diagnostics of an ansatz constructed to contain sigma and kappa. They test implementation and fit-window convergence; they are not independent determinations or statistical confidence intervals.",
        "smooth": [
          {
            "delta_maximum": 0.4,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.007812500692394376,
            "difference_from_input_sigma": 6.923943762088269e-10,
            "maximum_absolute_residual": 7.751737533118197e-10,
            "root_mean_square_residual": 5.14823706330907e-10
          },
          {
            "delta_maximum": 0.2,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.007812500010594456,
            "difference_from_input_sigma": 1.0594455868151442e-11,
            "maximum_absolute_residual": 1.1800475527290466e-11,
            "root_mean_square_residual": 7.851679275795691e-12
          },
          {
            "delta_maximum": 0.1,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.007812500000164603,
            "difference_from_input_sigma": 1.646027064650113e-13,
            "maximum_absolute_residual": 1.8317812544577095e-13,
            "root_mean_square_residual": 1.2193303238519015e-13
          }
        ],
        "sharp": [
          {
            "theta_maximum": 0.4,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.07945602154445074,
            "difference_from_input_kappa": 0.000056021544450737815,
            "maximum_absolute_residual": 0.000019257510805364975,
            "root_mean_square_residual": 0.00001153312600369399
          },
          {
            "theta_maximum": 0.2,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.07940877977617689,
            "difference_from_input_kappa": 0.000008779776176887077,
            "maximum_absolute_residual": 0.00000309497495652733,
            "root_mean_square_residual": 0.0000018916445112010924
          },
          {
            "theta_maximum": 0.1,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.07940124402153743,
            "difference_from_input_kappa": 0.0000012440215374359687,
            "maximum_absolute_residual": 4.4465961940298726e-7,
            "root_mean_square_residual": 2.752111340189625e-7
          }
        ],
        "smooth_window_spread": 6.922297735023619e-10,
        "sharp_window_spread": 0.000054777522913301846
      },
      "ansatz_order_stability": {
        "interpretation": "Difference between the published M-term ansatz and the same construction with its last supplied smooth coefficient omitted. This is an internal interpolation-stability indicator, not a rigorous error bar on the exact corner function.",
        "rows": [
          {
            "angle_id": "atan-one-half",
            "full_order_value": 0.1557474104542686,
            "one_fewer_smooth_coefficient_value": 0.1557324303839079,
            "absolute_difference": 0.00001498007036068727,
            "relative_difference": 0.00009618182618250208
          },
          {
            "angle_id": "pi-over-four",
            "full_order_value": 0.08101879816273792,
            "one_fewer_smooth_coefficient_value": 0.08101686590318,
            "absolute_difference": 0.000001932259557926863,
            "relative_difference": 0.000023849521367197297
          },
          {
            "angle_id": "atan-two",
            "full_order_value": 0.04819450553364747,
            "one_fewer_smooth_coefficient_value": 0.04819432110902321,
            "absolute_difference": 1.8442462425966033e-7,
            "relative_difference": 0.000003826673232094942
          },
          {
            "angle_id": "pi-over-two",
            "full_order_value": 0.023666848771018156,
            "one_fewer_smooth_coefficient_value": 0.023666845829271154,
            "absolute_difference": 2.941747002244144e-9,
            "relative_difference": 1.242982126900872e-7
          },
          {
            "angle_id": "pi-minus-atan-two",
            "full_order_value": 0.010508266535399992,
            "one_fewer_smooth_coefficient_value": 0.010508266524486025,
            "absolute_difference": 1.0913967646297706e-11,
            "relative_difference": 1.0386078055339573e-9
          },
          {
            "angle_id": "three-pi-over-four",
            "full_order_value": 0.005040053676315704,
            "one_fewer_smooth_coefficient_value": 0.005040053676270816,
            "absolute_difference": 4.488770466437586e-14,
            "relative_difference": 8.906195756468395e-12
          },
          {
            "angle_id": "pi-minus-atan-one-half",
            "full_order_value": 0.0017051930150667912,
            "one_fewer_smooth_coefficient_value": 0.0017051930150667814,
            "absolute_difference": 9.75781955236954e-18,
            "relative_difference": 5.72241351339768e-15
          }
        ],
        "maximum_absolute_difference": 0.00001498007036068727,
        "maximum_relative_difference": 0.00009618182618250208
      }
    },
    "massless-two-component-dirac": {
      "label": "Massless two-component Dirac fermion",
      "degrees_of_freedom": "one two-component complex Dirac spinor",
      "lattice_method": "finite-size correlation-matrix calculation on a square lattice",
      "C_T": 0.018997721932938333,
      "sigma": 0.0078125,
      "sigma_over_C_T": 0.4112335167120566,
      "kappa": 0.0722,
      "M": 7,
      "smooth_coefficients_sigma_superscript_p_minus_1": [
        0.0078125,
        0.0005014256660457252,
        0.00004812997,
        0.0000048552317,
        4.9173353e-7,
        4.9777097e-8,
        5.0411447e-9
      ],
      "coefficient_source_strings": [
        "1/128",
        "(16+3*pi^2)/(9216*pi^2)",
        "4.8129970e-5",
        "4.8552317e-6",
        "4.9173353e-7",
        "4.9777097e-8",
        "5.0411447e-9"
      ],
      "endpoint_round_trip_fits": {
        "interpretation": "These are round-trip endpoint diagnostics of an ansatz constructed to contain sigma and kappa. They test implementation and fit-window convergence; they are not independent determinations or statistical confidence intervals.",
        "smooth": [
          {
            "delta_maximum": 0.4,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.00781250062240036,
            "difference_from_input_sigma": 6.224003604171635e-10,
            "maximum_absolute_residual": 6.968230922832497e-10,
            "root_mean_square_residual": 4.627851652864564e-10
          },
          {
            "delta_maximum": 0.2,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.007812500009522954,
            "difference_from_input_sigma": 9.522953606233564e-12,
            "maximum_absolute_residual": 1.0607093328673933e-11,
            "root_mean_square_residual": 7.057616350999258e-12
          },
          {
            "delta_maximum": 0.1,
            "sample_count": 12,
            "fit_model": "a(pi-delta)/delta^2 = sigma_fit + b2 delta^2 + b4 delta^4",
            "sigma_fit": 0.007812500000147942,
            "difference_from_input_sigma": 1.4794242220173004e-13,
            "maximum_absolute_residual": 1.6466342178667048e-13,
            "root_mean_square_residual": 1.095985676132569e-13
          }
        ],
        "sharp": [
          {
            "theta_maximum": 0.4,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.07223002698483531,
            "difference_from_input_kappa": 0.000030026984835312964,
            "maximum_absolute_residual": 0.00001073603250446864,
            "root_mean_square_residual": 0.000006651489049258827
          },
          {
            "theta_maximum": 0.2,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.0722040065452308,
            "difference_from_input_kappa": 0.000004006545230805791,
            "maximum_absolute_residual": 0.00000144137317303894,
            "root_mean_square_residual": 8.977826925258668e-7
          },
          {
            "theta_maximum": 0.1,
            "sample_count": 12,
            "fit_model": "theta a(theta) = kappa_fit + c1 theta + c2 theta^2",
            "kappa_fit": 0.07220051972283573,
            "difference_from_input_kappa": 5.197228357262107e-7,
            "maximum_absolute_residual": 1.8767422904630848e-7,
            "root_mean_square_residual": 1.1730569170940262e-7
          }
        ],
        "smooth_window_spread": 6.222524179949618e-10,
        "sharp_window_spread": 0.000029507261999586754
      },
      "ansatz_order_stability": {
        "interpretation": "Difference between the published M-term ansatz and the same construction with its last supplied smooth coefficient omitted. This is an internal interpolation-stability indicator, not a rigorous error bar on the exact corner function.",
        "rows": [
          {
            "angle_id": "atan-one-half",
            "full_order_value": 0.14648647368972612,
            "one_fewer_smooth_coefficient_value": 0.14648409571150353,
            "absolute_difference": 0.0000023779782225841206,
            "relative_difference": 0.00001623343208889669
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            "angle_id": "pi-over-four",
            "full_order_value": 0.07759704443372688,
            "one_fewer_smooth_coefficient_value": 0.07759664820995071,
            "absolute_difference": 3.962237761639953e-7,
            "relative_difference": 0.0000051061709766845204
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            "angle_id": "atan-two",
            "full_order_value": 0.04679124593050869,
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            "absolute_difference": 5.072531999300578e-8,
            "relative_difference": 0.0000010840771384531995
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            "angle_id": "pi-over-two",
            "full_order_value": 0.023292180893144325,
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            "absolute_difference": 1.357259551276746e-9,
            "relative_difference": 5.8271037714473245e-8
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            "angle_id": "pi-minus-atan-two",
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            "one_fewer_smooth_coefficient_value": 0.01043096030423069,
            "absolute_difference": 1.0136036107666335e-11,
            "relative_difference": 9.717260733612238e-10
          },
          {
            "angle_id": "three-pi-over-four",
            "full_order_value": 0.005021983961295865,
            "one_fewer_smooth_coefficient_value": 0.005021983961213025,
            "absolute_difference": 8.283998487179645e-14,
            "relative_difference": 1.649546982034179e-11
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            "angle_id": "pi-minus-atan-one-half",
            "full_order_value": 0.0017031066027421264,
            "one_fewer_smooth_coefficient_value": 0.0017031066027420748,
            "absolute_difference": 5.160802341031001e-17,
            "relative_difference": 3.030228602673333e-14
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        "maximum_absolute_difference": 0.0000023779782225841206,
        "maximum_relative_difference": 0.00001623343208889669
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  "benchmark_angles": {
    "source_angle_order": "tan(theta)=1/2,1,2,infinity,-2,-1,-1/2, mapped to 0<theta<=pi",
    "rows": [
      {
        "angle_id": "atan-one-half",
        "label": "arctan(1/2)",
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      {
        "angle_id": "three-pi-over-four",
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        "theta_over_pi": 0.75,
        "values": {
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          "massless-two-component-dirac": {
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            "source_table_rounding_tolerance": 5e-7,
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            "ansatz_minus_lattice": 0.00012198396129586495,
            "relative_difference_ansatz_minus_lattice": 0.024289994200696014
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        }
      },
      {
        "angle_id": "pi-minus-atan-one-half",
        "label": "pi-arctan(1/2)",
        "tan_theta": "-1/2",
        "theta_radians": 2.677945044588987,
        "theta_over_pi": 0.8524163823495667,
        "values": {
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    ],
    "source_precision": "Helmes et al. report robust displayed digits under fit-range variations and estimate systematic uncertainty in roughly the third significant digit of the lattice-extracted corner coefficient; strict confidence intervals were not available."
  },
  "angular_grid": [
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  ],
  "independent_checks": {
    "checks": [
      {
        "label": "massless-complex-scalar: coefficient count equals M",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: coefficient-source count equals M",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: coefficient count equals M",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: coefficient-source count equals M",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: angle lies in (0,2*pi)",
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      },
      {
        "label": "massless-complex-scalar: valid ansatz order",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at arctan(1/2)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: angle lies in (0,2*pi)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: valid ansatz order",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at arctan(1/2)",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at pi/4",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at pi/4",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at arctan(2)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at arctan(2)",
        "passed": true
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      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at pi/2",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at pi/2",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at pi-arctan(2)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at pi-arctan(2)",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at 3pi/4",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at 3pi/4",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: reproduce Helmes Table I ansatz at pi-arctan(1/2)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: reproduce Helmes Table II ansatz at pi-arctan(1/2)",
        "passed": true
      },
      {
        "label": "polynomial fit has enough rows",
        "passed": true
      },
      {
        "label": "endpoint fit matrix is nonsingular",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: sigma/C_T equals pi^2/24",
        "passed": true
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      {
        "label": "massless-complex-scalar: sharp endpoint approaches kappa",
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      {
        "label": "massless-complex-scalar: smooth endpoint approaches sigma",
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      {
        "label": "massless-complex-scalar: complement-angle symmetry at theta=0.2",
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      {
        "label": "massless-complex-scalar: complement-angle symmetry at theta=0.6",
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      {
        "label": "massless-complex-scalar: complement-angle symmetry at theta=1.1",
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      {
        "label": "massless-complex-scalar: complement-angle symmetry at theta=2",
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      {
        "label": "massless-complex-scalar: complement-angle symmetry at theta=2.8",
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      {
        "label": "massless-complex-scalar: sampled corner function is nonnegative",
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      },
      {
        "label": "massless-complex-scalar: sampled corner function is nonincreasing on (0,pi)",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: sampled first derivative is nonpositive",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: sampled corner function is convex",
        "passed": true
      },
      {
        "label": "massless-complex-scalar: sampled strong-subadditivity differential inequality",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sigma/C_T equals pi^2/24",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sharp endpoint approaches kappa",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: smooth endpoint approaches sigma",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: complement-angle symmetry at theta=0.2",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: complement-angle symmetry at theta=0.6",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: complement-angle symmetry at theta=1.1",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: complement-angle symmetry at theta=2",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: complement-angle symmetry at theta=2.8",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sampled corner function is nonnegative",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sampled corner function is nonincreasing on (0,pi)",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sampled first derivative is nonpositive",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sampled corner function is convex",
        "passed": true
      },
      {
        "label": "massless-two-component-dirac: sampled strong-subadditivity differential inequality",
        "passed": true
      }
    ],
    "published_table_ansatz_rounding_reproduced": true,
    "complement_angle_symmetry_sampled": true,
    "positivity_and_monotonicity_sampled_on_open_half_domain": true,
    "sigma_over_C_T": {
      "value": 0.4112335167120566,
      "target": 0.4112335167120566,
      "passes": true
    },
    "endpoint_spot_checks": {
      "massless-complex-scalar": {
        "theta_0_01_times_a": 0.07933945235825966,
        "target_kappa": 0.0794,
        "a_pi_minus_0_01_over_delta_squared": 0.00781255454072175,
        "target_sigma": 0.0078125
      },
      "massless-two-component-dirac": {
        "theta_0_01_times_a": 0.07219883376837355,
        "target_kappa": 0.0722,
        "a_pi_minus_0_01_over_delta_squared": 0.007812550143047577,
        "target_sigma": 0.0078125
      }
    }
  },
  "uncertainty_and_evidence_boundary": {
    "fit_window_diagnostics": "The endpoint tables vary fit windows and expose numerical residuals, but they round-trip inputs built into Eq. (22); they are implementation controls rather than independent error estimates.",
    "ansatz_stability": "M versus M-1 changes are recorded as an interpolation-stability indicator only.",
    "source_rounding": "The supplied higher smooth coefficients and kappa inherit the printed precision of the source.",
    "lattice_cross_check": "The listed lattice values are independent finite-size extrapolations from the source, but their displayed precision is not a strict confidence interval.",
    "excluded_claims": [
      "The record does not reproduce the original cut-sphere Green-function integrations.",
      "The record does not rerun the square-lattice NLCE or direct fermion correlation-matrix simulations.",
      "The ansatz is not asserted to be the exact full angular function.",
      "No rigorous error bar on the difference between the ansatz and the exact continuum curve is claimed.",
      "No interacting-CFT, nonconformal, mixed-state, anisotropic-regulator, or higher-dimensional universality is inferred."
    ]
  },
  "downloadable_files": {
    "json": "/data/quantum-information/corner-angular-free-fields.json",
    "csv": "/data/quantum-information/corner-angular-free-fields.csv"
  },
  "strongest_supported_claim": "Using the published M=8 complex-scalar and M=7 Dirac smooth--sharp inputs, Eq. (22) reproducibly generates the cited finite-angle estimates, satisfies the built-in sigma=1/128 and kappa endpoint limits, and agrees with the independent lattice values to their reported precision. This is high-precision interpolation evidence, not a direct recomputation of the exact continuum functions."
}