{
  "collective_profiles": {
    "gas-like": {
      "features": {
        "discrete_spectral_entropy": 6.935120643225439,
        "low_frequency_fraction_0_to_1": 0.45994431963722227,
        "peak_frequency": 0.0,
        "spectral_centroid": 1.9317925350238627,
        "zero_frequency_density": 0.5453291259321279
      },
      "parameters": {
        "collision_rate": 1.25,
        "gas_fraction": 1.0,
        "name": "gas-like",
        "vibrational_center": 5.0,
        "vibrational_width": 1.2
      }
    },
    "liquid-like": {
      "features": {
        "discrete_spectral_entropy": 7.323290688376107,
        "low_frequency_fraction_0_to_1": 0.16098636102981922,
        "peak_frequency": 5.505,
        "spectral_centroid": 4.28013671184151,
        "zero_frequency_density": 0.19086519407624475
      },
      "parameters": {
        "collision_rate": 1.25,
        "gas_fraction": 0.35,
        "name": "liquid-like",
        "vibrational_center": 5.0,
        "vibrational_width": 1.2
      }
    },
    "solid-like": {
      "features": {
        "discrete_spectral_entropy": 6.111156360806851,
        "low_frequency_fraction_0_to_1": 1.2030080372677517e-21,
        "peak_frequency": 6.1000000000000005,
        "spectral_centroid": 6.099993113421942,
        "zero_frequency_density": 0.0
      },
      "parameters": {
        "collision_rate": 1.25,
        "gas_fraction": 0.0,
        "name": "solid-like",
        "vibrational_center": 6.0,
        "vibrational_width": 0.55
      }
    }
  },
  "continuum_checks": {
    "low_frequency_fraction_endpoints": [
      8.998702756079757e-06,
      0.45994431963722227
    ],
    "low_frequency_fraction_monotone": true,
    "maximum_fraction_recovery_error": 1.1102230246251565e-16,
    "states": 101
  },
  "evidence_class": {
    "finite_computation": "three constructed reference spectra and a 101-state mixture continuum",
    "measured_inputs": "published water modes and literature argon state parameters only",
    "not_established": [
      "a universal phase classifier",
      "that vibrations alone cause or identify an equilibrium phase",
      "a physical fourth or fifth dimension",
      "a universal fixed grid or tick",
      "a theological cause"
    ],
    "written_mathematics": "identical-descriptor lower bound and fixed-basis zero-frequency identity"
  },
  "next_test": "Run one validated intermolecular potential through solid, liquid, and gas trajectories, freeze spectral features, then score held-out state points against independently assigned thermodynamic phases.",
  "pairwise_squared_hellinger": {
    "liquid-like__gas-like": 0.1832326307708333,
    "solid-like__gas-like": 0.7512211851810795,
    "solid-like__liquid-like": 0.3077968074963826
  },
  "question": "Can molecule-level vibrations uniquely determine solid, liquid, and gas, and what extra information appears in a collective velocity spectrum?",
  "real_world_anchors": {
    "rahman_1964_liquid_argon": {
      "density_star": 0.8176998962538926,
      "mass_density_kg_m3": 1374.0,
      "note": "Literature state only; this release does not reproduce Rahman's 864-particle trajectory.",
      "number_density_m_minus_3": 2.0712980385100632e+28,
      "temperature_k": 94.4,
      "temperature_star": 0.7879799666110184
    },
    "sun_et_al_r2pt_representative_argon": {
      "conversion_replay": {
        "density_star": 0.8499999999999999,
        "number_density_m_minus_3": 2.1531167373254663e+28,
        "temperature_star": 1.1
      },
      "density_star": 0.85,
      "mass_density_kg_m3_from_frozen_lj_scale": 1428.2746094875026,
      "reported_gas_like_fraction": 0.35,
      "temperature_k_from_frozen_lj_scale": 131.78,
      "temperature_star": 1.1
    },
    "water_gas_phase_fundamentals_cm_minus_1": [
      1595.0,
      3657.0,
      3756.0
    ]
  },
  "release": "0.9.0",
  "water_counterexample": {
    "best_deterministic_accuracy_using_only_that_identical_descriptor": 0.3333333333333333,
    "isolated_modes_cm_minus_1": [
      1595.0,
      3657.0,
      3756.0
    ],
    "maximum_pairwise_descriptor_difference": 0.0,
    "minimum_error_rate_using_only_that_identical_descriptor": 0.6666666666666667,
    "states": [
      {
        "name": "ice control",
        "phase": "solid",
        "pressure_pa": 101325.0,
        "temperature_k": 260.0
      },
      {
        "name": "liquid control",
        "phase": "liquid",
        "pressure_pa": 101325.0,
        "temperature_k": 298.15
      },
      {
        "name": "vapor control",
        "phase": "gas",
        "pressure_pa": 101325.0,
        "temperature_k": 400.0
      }
    ]
  },
  "written_result": {
    "declared_collective_model": "F(w)=f_g F_g(w)+(1-f_g)F_s(w), with F_s(0)=0 and F_g(0)>0",
    "fraction_identity": "f_g=F(0)/F_g(0) only when the gas basis is fixed and declared",
    "isolated_mode_phase_identifiability": "rejected by exact descriptor collision",
    "reason": "The same isolated H2O mode vector is paired with solid, liquid, and gas thermodynamic states, so every deterministic descriptor-only classifier must assign the same label to all three."
  }
}