{
  "round": "RUN-101",
  "gate": "src103",
  "C9": "PASS",
  "detail": {
    "residual_graph_complete": "YES",
    "components": {
      "I": 1,
      "II": 1
    },
    "beta_1": {
      "I": 0,
      "II": 2
    },
    "fundamental_cycle_products": {
      "I": [],
      "II": [
        "1",
        "1"
      ]
    },
    "residual_monodromy": "trivial"
  },
  "representations_agree": true,
  "full_graph": {
    "vertices": [
      "L0",
      "L1",
      "L2",
      "L2'",
      "L3",
      "L4"
    ],
    "edges": {
      "Psi_1": {
        "from": "L1",
        "to": "L0",
        "theorem": "LR C1.9 (LZ16 regulator)",
        "labels": {
          "actual map": true,
          "same map under theorem": null,
          "frame change": false,
          "lattice change": false,
          "partner factor": true,
          "exact comparison": true
        },
        "divisors_attached": [
          "X^n",
          "L_E",
          "L_p(Ad g)"
        ],
        "note": "partner enters through eta_g and L_p^g; all partner factors are P-slots"
      },
      "Psi_2": {
        "from": "L1",
        "to": "L2",
        "theorem": "LR C1.6-C1.8",
        "labels": {
          "actual map": true,
          "same map under theorem": null,
          "frame change": false,
          "lattice change": false,
          "partner factor": false,
          "exact comparison": true
        },
        "divisors_attached": [
          "X^n"
        ]
      },
      "Psi_3": {
        "from": "L2",
        "to": "L3",
        "theorem": "LR C1.13 (Prop C1.1 lift)",
        "labels": {
          "actual map": true,
          "same map under theorem": null,
          "frame change": false,
          "lattice change": false,
          "partner factor": true,
          "exact comparison": true
        },
        "divisors_attached": [
          "log^{[r+1]}",
          "L_p(Ad g)"
        ]
      },
      "Psi_4": {
        "from": "L3",
        "to": "L4",
        "theorem": "Kato's reciprocity (through LR)",
        "labels": {
          "actual map": true,
          "same map under theorem": null,
          "frame change": true,
          "lattice change": false,
          "partner factor": false,
          "exact comparison": true
        },
        "note": "the period frame gamma^+- enters and cancels (RUN-099); eta_pdR = 1"
      },
      "A'": {
        "from": "L2",
        "to": "L4",
        "theorem": "the regulator on the leading term (C1.9 with C1.7/C1.8)",
        "labels": {
          "actual map": true,
          "same map under theorem": true,
          "frame change": false,
          "lattice change": false,
          "partner factor": true,
          "exact comparison": true
        },
        "same_as": "Psi_4 o Psi_3",
        "witness": "C1.13's proof + Eq. 11 (RUN-100)"
      },
      "id": {
        "from": "L2",
        "to": "L2'",
        "theorem": "PC-001 section 1 (a definition naming LR's object)",
        "labels": {
          "actual map": false,
          "same map under theorem": true,
          "frame change": false,
          "lattice change": false,
          "partner factor": false,
          "exact comparison": true
        },
        "same_as": "identity of L2",
        "witness": "dictionary identification: PC-001 section 1 names LR's object; slots matched RUN-094/095"
      },
      "Psi_3^PC": {
        "from": "L2'",
        "to": "L3",
        "theorem": "PC-001 (1) = C1.13 in six slots",
        "labels": {
          "actual map": true,
          "same map under theorem": true,
          "frame change": false,
          "lattice change": false,
          "partner factor": true,
          "exact comparison": true
        },
        "same_as": "Psi_3 o id^{-1}",
        "witness": "the six-slot dictionary (C/q_f -> C_pi; C_d = S_5(0) = 26 numerically; log <-> j; L_p(g x tau)/L_p(Ad g) <-> lambda_i/D_i; kappa <-> z_i), RUN-094/095"
      }
    },
    "loops": {
      "b_f+ <-> omega_f": {
        "at": "L0",
        "tag": "G",
        "value": "u (RUN-096 gauge law)"
      },
      "X uniformiser": {
        "at": "L0",
        "tag": "G",
        "value": "inside c_lead / q_f (RUN-096/097)"
      },
      "Kummer <-> ES realisation of Theta_aux": {
        "at": "L0",
        "tag": "G",
        "value": "C4 PASS by theorem (A2.1(3), A6.3(i)); a realisation of the same element, not a second vertex"
      },
      "period frame Omega^+- / lambda^+-": {
        "at": "L4",
        "tag": "G",
        "value": "cancels along the route (RUN-099); lambda measured 1, 6"
      },
      "Gauss(chi_8)": {
        "at": "L4",
        "tag": "G",
        "value": "G(chi_8)^2 = 8"
      },
      "gamma_Kato <-> gamma_tree": {
        "at": "L3/L4",
        "tag": "L",
        "value": "q (RUN-099, bucket q_D, v_11 UNMEASURED)"
      },
      "lattice indices": {
        "at": "L4",
        "tag": "L",
        "value": "2, 2 / 2, 1 (RUN-080/096)"
      },
      "partner factors": {
        "at": "L3/L4",
        "tag": "P",
        "value": "E_adj_i, Pi_i, lambda_i(0), kappa_E, ell_0 (net 0 in delta)"
      }
    },
    "E_labels": {
      "eta_pdR": {
        "on": "Psi_4",
        "status": "resolved",
        "value": 1,
        "by": "RUN-099"
      },
      "kappa_det": {
        "on": "the composite",
        "status": "resolved",
        "value": 1,
        "by": "RUN-100"
      },
      "id(LR class, PC-001 projected class)": {
        "on": "id",
        "status": "resolved",
        "value": 1,
        "by": "this round: dictionary identification: PC-001 section 1 names LR's object; slots matched RUN-094/095"
      },
      "C_pi": {
        "on": "Psi_1 / Psi_2 (an edge label, invariant under both frames)",
        "status": "a label, not a cycle",
        "value": "C = c*_f(r), unmeasured numerically",
        "by": "RUN-096"
      }
    },
    "merges": {
      "A' = Psi_4 o Psi_3": "C1.13's proof + Eq. 11 (RUN-100)",
      "Psi_3^PC o id = Psi_3": "the six-slot dictionary (RUN-094/095)"
    },
    "divisors": {
      "L_E": "Psi_1 (RUN-098)",
      "X^n": "Psi_2 (the named extraction)",
      "log^{[r+1]}": "Psi_3 (C1.8)"
    }
  },
  "residual": {
    "I": {
      "V": [
        "L0",
        "L1",
        "L2",
        "L3",
        "L4"
      ],
      "E": {
        "Psi_1": {
          "from": "L1",
          "to": "L0",
          "label": "5/9"
        },
        "Psi_2": {
          "from": "L1",
          "to": "L2",
          "label": "5/8"
        },
        "Psi_3": {
          "from": "L2",
          "to": "L3",
          "label": "7/4"
        },
        "Psi_4": {
          "from": "L3",
          "to": "L4",
          "label": "-4/9"
        }
      },
      "removed_as": {
        "b_f+ <-> omega_f": "G",
        "X uniformiser": "G",
        "Kummer <-> ES realisation of Theta_aux": "G",
        "period frame Omega^+- / lambda^+-": "G",
        "Gauss(chi_8)": "G",
        "gamma_Kato <-> gamma_tree": "L",
        "lattice indices": "L",
        "partner factors": "P"
      },
      "holonomy": {
        "components": 1,
        "beta_1": 0,
        "beta_1_check": 0,
        "spanning_tree": [
          "Psi_1",
          "Psi_2",
          "Psi_3",
          "Psi_4"
        ],
        "fundamental_cycles": [],
        "trivial": true
      }
    },
    "II": {
      "V": [
        "L0",
        "L1",
        "L2",
        "L3",
        "L4",
        "L2'"
      ],
      "E": {
        "Psi_1": {
          "from": "L1",
          "to": "L0",
          "label": "5/9"
        },
        "Psi_2": {
          "from": "L1",
          "to": "L2",
          "label": "5/8"
        },
        "Psi_3": {
          "from": "L2",
          "to": "L3",
          "label": "7/4"
        },
        "Psi_4": {
          "from": "L3",
          "to": "L4",
          "label": "-4/9"
        },
        "A'": {
          "from": "L2",
          "to": "L4",
          "label": "-7/9"
        },
        "id": {
          "from": "L2",
          "to": "L2'",
          "label": "1"
        },
        "Psi_3^PC": {
          "from": "L2'",
          "to": "L3",
          "label": "7/4"
        }
      },
      "removed_as": {
        "b_f+ <-> omega_f": "G",
        "X uniformiser": "G",
        "Kummer <-> ES realisation of Theta_aux": "G",
        "period frame Omega^+- / lambda^+-": "G",
        "Gauss(chi_8)": "G",
        "gamma_Kato <-> gamma_tree": "L",
        "lattice indices": "L",
        "partner factors": "P"
      },
      "holonomy": {
        "components": 1,
        "beta_1": 2,
        "beta_1_check": 2,
        "spanning_tree": [
          "A'",
          "Psi_1",
          "Psi_2",
          "Psi_3",
          "id"
        ],
        "fundamental_cycles": [
          {
            "cotree_edge": "Psi_4",
            "cycle": [
              "Psi_4^+",
              "A'^-",
              "Psi_3^+"
            ],
            "product": "1",
            "_product": "1",
            "component": 1
          },
          {
            "cotree_edge": "Psi_3^PC",
            "cycle": [
              "Psi_3^PC^+",
              "Psi_3^-",
              "id^+"
            ],
            "product": "1",
            "_product": "1",
            "component": 1
          }
        ],
        "trivial": true
      }
    }
  },
  "controls": {
    "tree_accepted": true,
    "genuine_cycle_product_1_accepted": true,
    "same_cycle_product_not_1_rejected": true,
    "bad_product": "6/5",
    "agrees": true
  },
  "completeness": {
    "from": {
      "R0": "the route realized: these arrows and no others (RUN-097)",
      "C5": {
        "records": 23,
        "unassigned": [],
        "unknown_slots": []
      },
      "C6": "PASS / canonical",
      "C7 routes": 1
    },
    "candidates_outside_the_graph": {
      "A (RUN-097's chain)": "the edges Psi_1..Psi_4",
      "A' (regulator on the leading term)": "an edge in II, merged into Psi_4 o Psi_3 in I (witness: RUN-100)",
      "PC-001's B-hat description": "the vertex L2' with id and Psi_3^PC in II, merged in I (witness: the dictionary)",
      "B (direct f-to-g comparison)": "no theorem induces it (RUN-100): not an edge",
      "C (Kato's class -> the tree's series)": "a change of generator by q: a loop, tag L, removed",
      "the Kummer / ES realisation of Theta_aux": "a realisation of L0's element (C4 PASS): a loop, tag G, removed",
      "the frame changes b_f+, X, Omega^+-, Gauss": "loops, tag G, removed",
      "the lattice indices": "loops, tag L, removed",
      "the partner factors": "loops, tag P, removed",
      "R0.5's divisors": "extraction / localization data on Psi_1..Psi_3, not edges"
    },
    "every_candidate_dispositioned": true,
    "II_carries_every_edge_disposition": true,
    "unknown_candidate": null,
    "complete": true,
    "agrees": true
  },
  "c10_boundary": {
    "components": 1,
    "claim": "one anchor fixes the one component; k components would need k anchors or canonical sections",
    "trivial_holonomy_is_not_canonical_normalization": true,
    "torsor_remaining": "the nonzero flat sections form a K^x-torsor (Round 030 (20.2)); C10 asks for a canonical section or one exact anchor",
    "C10_claimed": false,
    "consistent": true,
    "agrees": true
  },
  "not_claimed": [
    "C10: a canonical flat section or an exact anchor - trivial holonomy leaves the K^x-torsor of (20.2)",
    "v_11(q) (L, q_D, outside-certificate)",
    "the numerical value of C_pi (an edge label, not holonomy)"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "beta_1 and every fundamental-cycle product in both representations, on exact instances",
      "the three controls"
    ],
    "CITED_THEOREM": [
      "LR C1.13's proof + Eq. 11 (the A' merge)",
      "PC-001 section 1 (the id edge is a definition)",
      "Round 030 (20.1)-(20.2)"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-097's chain and slot table",
      "RUN-098's divisors",
      "RUN-099's q and eta_pdR",
      "RUN-100's route count and kappa_det"
    ],
    "UNMEASURED": [
      "C10: a canonical flat section or an exact anchor - trivial holonomy leaves the K^x-torsor of (20.2)",
      "v_11(q) (L, q_D, outside-certificate)",
      "the numerical value of C_pi (an edge label, not holonomy)"
    ],
    "FAILED": []
  },
  "status": {
    "C1-C7": "PASS",
    "C8": "PASS",
    "C9": "PASS",
    "C10": "UNMEASURED",
    "R0": "PASS (category C)",
    "BSD": "OPEN"
  }
}