{
  "schema_version": 1,
  "metadata": {
    "title": "Fixed-window mean absolute autocorrelation with matched centered RMS energy",
    "state": "beta=0; rho=I16/16",
    "beta": 0,
    "normalization": "psi_i=gamma_i/sqrt(2); {psi_i,psi_j}=delta_ij I; hbar=1",
    "majorana_spin_mapping": "gamma_(2j-1)=Z^(tensor j-1) tensor X tensor I^(tensor 4-j); gamma_(2j) replaces X with Y; j=1..4, first qubit leftmost",
    "hamiltonian_terms": [
      {
        "coefficient": -0.36,
        "indices": [
          1,
          2,
          4,
          5
        ]
      },
      {
        "coefficient": 0.19,
        "indices": [
          1,
          3,
          4,
          7
        ]
      },
      {
        "coefficient": -0.71,
        "indices": [
          1,
          3,
          5,
          6
        ]
      },
      {
        "coefficient": 0.22,
        "indices": [
          2,
          3,
          4,
          6
        ]
      },
      {
        "coefficient": 0.49,
        "indices": [
          2,
          3,
          5,
          7
        ]
      }
    ],
    "perturbation_indices": [
      1,
      2,
      3,
      5
    ],
    "matching": "H_matched(lambda)=[H_raw(lambda)-Tr(H_raw(lambda))/16 I] * RMS(H0)/RMS(H_raw(lambda)); RMS(H)=sqrt(Tr[(H-Tr(H)/16 I)^2]/16)",
    "target_centered_RMS_energy": 0.24473199627347456,
    "time_units": "Time remains in inverse baseline coefficient-energy units, hbar=1; H0 is unchanged. No physical seconds assigned.",
    "observable": "G_i(t)=(2/16) Re Tr[exp(+iHt) psi_i exp(-iHt) psi_i]",
    "spectator": "psi8 commutes with both models and G8=1; it is plotted in supplied traces but excluded from D.",
    "array_layout": "traces[operator_index_zero_based][time_index]; mean_absolute_active7[time_index]",
    "numpy_version": "2.0.2",
    "sources": [
      {
        "url": "https://arxiv.org/abs/2302.07897",
        "use": "Published learned H0 coefficients and normalization; critical analysis"
      },
      {
        "url": "https://arxiv.org/abs/2303.15423",
        "use": "Authors response, including the quartic perturbation with coefficient 0.3"
      }
    ],
    "generator_sha256": "bfbd575aa8bc60aaf083f1a14159e97123fd2194284bd98a0464c5b31b634081"
  },
  "fixed_protocol": {
    "declared_before_this_calculation": true,
    "not_preregistered": true,
    "prior_exposure": "Earlier unscaled lambda=0 and .3 calculations were inspected. This is a newly stated diagnostic hypothesis, not blind validation or a novelty claim.",
    "active_operator_indices": [
      1,
      2,
      3,
      4,
      5,
      6,
      7
    ],
    "lambdas": [
      0.0,
      0.1,
      0.2,
      0.3
    ],
    "window": [
      12.0,
      24.0
    ],
    "step": 0.1,
    "endpoints": "Both included, 121 samples",
    "D_definition": "Arithmetic mean across all 121 selected times and seven active operators of abs(G_i(t)); no postselection",
    "contrast": "D(lambda)-D(0)",
    "directional_hypothesis": "D(lambda)<D(0) for every tested nonzero lambda",
    "fixed_choices": "No tuning of window, operators, lambdas or matching rule after obtaining these results",
    "refinement": "Same endpoints and operators, step .05 (241 samples). Refinement changes are numerical discretization sensitivity, not statistical confidence intervals.",
    "quadrature_note": "Primary D is a finite-grid arithmetic mean. Separately supplied normalized trapezoidal integrals approximate a continuous-time average and are not substituted for primary D."
  },
  "times": [
    12.0,
    12.1,
    12.2,
    12.3,
    12.4,
    12.5,
    12.6,
    12.7,
    12.8,
    12.9,
    13.0,
    13.1,
    13.2,
    13.3,
    13.4,
    13.5,
    13.6,
    13.7,
    13.8,
    13.9,
    14.0,
    14.1,
    14.2,
    14.3,
    14.4,
    14.5,
    14.6,
    14.7,
    14.8,
    14.9,
    15.0,
    15.1,
    15.2,
    15.3,
    15.4,
    15.5,
    15.6,
    15.7,
    15.8,
    15.9,
    16.0,
    16.1,
    16.2,
    16.3,
    16.4,
    16.5,
    16.6,
    16.7,
    16.8,
    16.9,
    17.0,
    17.1,
    17.2,
    17.3,
    17.4,
    17.5,
    17.6,
    17.7,
    17.8,
    17.9,
    18.0,
    18.1,
    18.2,
    18.3,
    18.4,
    18.5,
    18.6,
    18.7,
    18.8,
    18.9,
    19.0,
    19.1,
    19.2,
    19.3,
    19.4,
    19.5,
    19.6,
    19.7,
    19.8,
    19.9,
    20.0,
    20.1,
    20.2,
    20.3,
    20.4,
    20.5,
    20.6,
    20.7,
    20.8,
    20.9,
    21.0,
    21.1,
    21.2,
    21.3,
    21.4,
    21.5,
    21.6,
    21.7,
    21.8,
    21.9,
    22.0,
    22.1,
    22.2,
    22.3,
    22.4,
    22.5,
    22.6,
    22.7,
    22.8,
    22.9,
    23.0,
    23.1,
    23.2,
    23.3,
    23.4,
    23.5,
    23.6,
    23.7,
    23.8,
    23.9,
    24.0
  ],
  "cases": [
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      "rescaling_factor": 1.0,
      "matched_centered_RMS_energy": 0.24473199627347456,
      "matched_spectral_width": 0.8049999999999995,
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        0.19150470406974,
        0.19119658214059,
        0.19171710089893,
        0.1917564672127,
        0.19131856608605,
        0.1904094954678,
        0.18903750071773,
        0.18721289734956,
        0.18494798260863,
        0.18225693650748,
        0.17915571300366,
        0.17566192206133,
        0.17179470338964,
        0.16757459269831,
        0.16302338135212,
        0.15816397034201,
        0.15302021952057,
        0.14761679307391,
        0.14197900221975,
        0.13613264613347,
        0.13276955493398,
        0.13088248599495,
        0.12868491176341,
        0.12617671370767,
        0.12335911323828,
        0.12023467969205,
        0.11680733149972,
        0.11424872564611,
        0.11601446666411,
        0.11759894786081,
        0.11899940135123,
        0.12021396429906,
        0.12124166975283,
        0.12208243221933,
        0.12273702811788
      ],
      "primary": {
        "D": 0.1534471413586158,
        "per_operator_mean_absolute": [
          0.07281271783252974,
          0.060619062603922115,
          0.1877997786977498,
          0.27249937751787773,
          0.2618476539102057,
          0.11683803342748146,
          0.10171336552054422
        ],
        "normalized_trapezoidal_integral": 0.15330651282581095
      },
      "refinement_step_0_05": {
        "D": 0.1533677954708126,
        "per_operator_mean_absolute": [
          0.07242317698540349,
          0.06040764009190471,
          0.18758336172670578,
          0.27323856890768083,
          0.2618547396007698,
          0.11693513385239167,
          0.10113194713083186
        ],
        "normalized_trapezoidal_integral": 0.15329715059654433
      },
      "refinement_changes": {
        "D_refined_minus_primary": -7.934588780320051e-05,
        "trapezoid_refined_minus_coarse": -9.36222926661845e-06
      },
      "validation": {
        "hermiticity": {
          "passed": true,
          "max_absolute_error": 0.0,
          "tolerance": 1e-11
        },
        "centered_trace": {
          "passed": true,
          "max_absolute_error": 0.0,
          "tolerance": 1e-11
        },
        "matched_RMS_matrix": {
          "passed": true,
          "max_absolute_error": 0.0,
          "tolerance": 1e-11
        },
        "matched_RMS_eigenvalues": {
          "passed": true,
          "max_absolute_error": 2.7755575615628914e-17,
          "tolerance": 1e-11
        },
        "initial_all_one": {
          "passed": true,
          "max_absolute_error": 2.220446049250313e-16,
          "tolerance": 1e-11
        },
        "spectator_commutes": {
          "passed": true,
          "max_absolute_error": 0.0,
          "tolerance": 1e-11
        },
        "spectator_conserved": {
          "passed": true,
          "max_absolute_error": 2.220446049250313e-16,
          "tolerance": 1e-11
        },
        "direct_U_vs_spectral_all_times": {
          "passed": true,
          "max_absolute_error": 6.661338147750939e-16,
          "tolerance": 1e-11
        },
        "unitarity_all_times": {
          "passed": true,
          "max_absolute_error": 1.5543122344752192e-15,
          "tolerance": 1e-11
        },
        "independent_exponential_U": {
          "passed": true,
          "max_absolute_error": 2.236391618786659e-15,
          "tolerance": 1e-11
        },
        "independent_exponential_correlations": {
          "passed": true,
          "max_absolute_error": 3.885780586188048e-15,
          "tolerance": 1e-11
        },
        "nested_grid_agreement": {
          "passed": true,
          "max_absolute_error": 0.0,
          "tolerance": 1e-11
        }
      },
      "contrast_D_minus_baseline": -0.03483843252325555,
      "refined_contrast_D_minus_baseline": -0.03460714605127757,
      "contrast_refinement_change": 0.0002312864719779817,
      "hypothesis_comparison": "supports_direction",
      "refined_direction_matches_primary": true
    }
  ],
  "validation": {
    "majorana_algebra": {
      "passed": true,
      "max_absolute_error": 2.220446049250313e-16,
      "tolerance": 1e-11
    },
    "baseline_term_commutation": {
      "passed": true,
      "max_absolute_error": 0.0,
      "tolerance": 1e-11
    },
    "baseline_centering": {
      "passed": true,
      "max_absolute_error": 0.0,
      "tolerance": 1e-11
    },
    "zero_hamiltonian": {
      "passed": true,
      "max_absolute_error": 2.220446049250313e-16,
      "tolerance": 1e-11
    }
  },
  "result": {
    "hypothesis_status": "supported_on_tested_grid",
    "direction_supported_lambdas": [
      0.1,
      0.2,
      0.3
    ],
    "direction_failed_lambdas": [],
    "all_refined_directions_match": true,
    "interpretation": "The fixed-grid directional hypothesis holds for all tested nonzero lambdas. These deterministic finite-system results quantify one chosen window and observable. Refinement is a numerical sensitivity check, not statistical uncertainty.",
    "limitations": [
      "Matching centered RMS removes this particular overall energy-scale difference, not every confound; spectral width and spectrum shape may still differ.",
      "Adding a quartic term also changes detailed couplings, conserved quantities and operator dynamics. This is not causal proof isolating noncommutation.",
      "No conclusion about gravitation, emergent geometry, thermalization, teleportation or experimental replication follows.",
      "The tested window and operators were fixed for this calculation, but prior unscaled data were inspected. No preregistration, blind test or novel-physics claim is made.",
      "No ensemble sampling or statistical confidence intervals are used. Numerical refinement changes are not rigorous bounds on continuum error."
    ]
  }
}