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      "graph": 1,
      "vertex": "upper",
      "circuit_word": "JPJ",
      "P_circuits": 1,
      "J_circuits": 2,
      "edge_count": 6,
      "affine_coefficients": [
        "32812500000/25018065241",
        "-23656109493565/25018065241"
      ]
    },
    {
      "graph": 1,
      "vertex": "upper",
      "circuit_word": "JJP",
      "P_circuits": 1,
      "J_circuits": 2,
      "edge_count": 6,
      "affine_coefficients": [
        "32812500000/25018065241",
        "-23656109493565/25018065241"
      ]
    },
    {
      "graph": 1,
      "vertex": "upper",
      "circuit_word": "JJJ",
      "P_circuits": 0,
      "J_circuits": 3,
      "edge_count": 6,
      "affine_coefficients": [
        "421875000000/325234848133",
        "-293302860916345/325234848133"
      ]
    }
  ],
  "counts": {
    "three_operator_source_graphs": 2,
    "source_vertices": 4,
    "loop_generators": 8,
    "closed_word_trials": 56,
    "commuting_generator_pairs": 4
  },
  "multipliers": {
    "P_circuit": "1750/1587",
    "J_circuit": "7500/6877"
  },
  "prime_valuation_certificate": {
    "prime7": [
      1,
      0
    ],
    "prime13": [
      0,
      -1
    ]
  },
  "general_form": "A loop based at sourcec with p P-circuits and j J-circuits is t -> c+(1750/1587)^p(7500/6877)^j(t-c). Circuit order does not matter; different count pairs have different multipliers.",
  "interpretation": "Each source vertex supports an exact two-generator commutative monoid of positive circuits. No nonempty positive circuit is the identity, and E/P/J are not individually commuting or inverse operators. Repeated paths use the same admitted source vertices and do not create additional Gears."
}