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      "imaginary_negative_index": 4,
      "real_nonpositive_index_upper_excluding_selected_phase": 2,
      "gyroscopic_index_deficit": 2,
      "full_spatial_real_growth_established": true,
      "growth_rate_interval": "0 < sigma <= omega; no sharper rate certified"
    }
  ],
  "interval_decimal_precision": 85,
  "limits": [
    "proof uses the source-projection and gyroscopic spectral-flow arguments in the companion note",
    "the actual full-matrix zonal carrier differs from minimum-column particle candidates",
    "only the two displayed excited backgrounds receive a full spatial certificate here",
    "the index deficit certifies existence; no precise growth multiplicity is asserted",
    "finite source support is exact polynomial algebra, not a truncation assumption",
    "no empirical particle decay or lifetime is identified",
    "the envelope source remains a separate action with a separate persistence theorem"
  ]
}