{
  "model": {
    "dynamics": "x'(t)=A x(t), A real fixed skew-symmetric",
    "observer": "y_j(t)=P_j x(t), y'_j(t)=P_j A x(t)",
    "observer_output_dimension": 3,
    "observation_back_action": false
  },
  "written_results": {
    "frequency_capacity": "floor(d/2)",
    "single_observer_nullity_lower_bound": "d-3 for d>=3",
    "bank_identifiability": "rank(stack(P_j))=d",
    "observer_invariance": "changing P_j does not change the eigenfrequencies of A"
  },
  "dimension_table": [
    {
      "latent_dimension": 4,
      "positive_frequency_capacity": 2,
      "balmer_feature_count": 4,
      "capacity_deficit": 2,
      "capacity_gate_passed": false,
      "single_observer_rank": 3,
      "single_observer_nullity": 1,
      "covering_observer_count": 2,
      "covering_stack_rank": 4,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    },
    {
      "latent_dimension": 5,
      "positive_frequency_capacity": 2,
      "balmer_feature_count": 4,
      "capacity_deficit": 2,
      "capacity_gate_passed": false,
      "single_observer_rank": 3,
      "single_observer_nullity": 2,
      "covering_observer_count": 2,
      "covering_stack_rank": 5,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    },
    {
      "latent_dimension": 6,
      "positive_frequency_capacity": 3,
      "balmer_feature_count": 4,
      "capacity_deficit": 1,
      "capacity_gate_passed": false,
      "single_observer_rank": 3,
      "single_observer_nullity": 3,
      "covering_observer_count": 2,
      "covering_stack_rank": 6,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    },
    {
      "latent_dimension": 7,
      "positive_frequency_capacity": 3,
      "balmer_feature_count": 4,
      "capacity_deficit": 1,
      "capacity_gate_passed": false,
      "single_observer_rank": 3,
      "single_observer_nullity": 4,
      "covering_observer_count": 3,
      "covering_stack_rank": 7,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    },
    {
      "latent_dimension": 8,
      "positive_frequency_capacity": 4,
      "balmer_feature_count": 4,
      "capacity_deficit": 0,
      "capacity_gate_passed": true,
      "single_observer_rank": 3,
      "single_observer_nullity": 5,
      "covering_observer_count": 3,
      "covering_stack_rank": 8,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    },
    {
      "latent_dimension": 9,
      "positive_frequency_capacity": 4,
      "balmer_feature_count": 4,
      "capacity_deficit": 0,
      "capacity_gate_passed": true,
      "single_observer_rank": 3,
      "single_observer_nullity": 6,
      "covering_observer_count": 3,
      "covering_stack_rank": 9,
      "instantaneous_state_identifiable_from_bank": true,
      "claim_ceiling": "Passing the line-count gate is necessary only; it does not predict wavelengths, intensities, selection rules, or physical dimension."
    }
  ],
  "single_observer_counterexample": {
    "latent_dimension": 5,
    "observer_rank": 3,
    "observer_nullity": 2,
    "latent_separation": 1.0,
    "projected_separation": 0.0
  },
  "reconstruction_checks": [
    {
      "latent_dimension": 4,
      "observer_count": 2,
      "stack_rank": 4,
      "state_error": 2.2215299868541707e-16,
      "measurement_residual": 3.141717836627692e-16
    },
    {
      "latent_dimension": 5,
      "observer_count": 2,
      "stack_rank": 5,
      "state_error": 2.220446049250313e-16,
      "measurement_residual": 3.1401849173675503e-16
    },
    {
      "latent_dimension": 6,
      "observer_count": 2,
      "stack_rank": 6,
      "state_error": 0.0,
      "measurement_residual": 0.0
    },
    {
      "latent_dimension": 8,
      "observer_count": 3,
      "stack_rank": 8,
      "state_error": 2.2887833992611187e-16,
      "measurement_residual": 3.188872858294072e-16
    }
  ],
  "hydrogen_scope": "Dimensions 8 and 9 first pass the four-feature capacity count, but this is not a spectral prediction and does not establish eight or nine physical dimensions.",
  "evidence_type": "written linear algebra plus finite synthetic computation"
}