{
  "audit": "NORMALIZATION AUDIT V2 — Round 030's certificate C1-C10, re-judged with RUN-091..094's closures",
  "round": "RUN-095",
  "gate": "src97",
  "verdicts": {
    "C1": "PASS",
    "C2": "PASS",
    "C3": "PASS",
    "C4": "PASS",
    "C5": "UNMEASURED",
    "C6": "UNMEASURED",
    "C7": "UNMEASURED",
    "C8": "UNMEASURED",
    "C9": "UNMEASURED",
    "C10": "UNMEASURED"
  },
  "v1_verdicts": {
    "C1": "UNMEASURED",
    "C2": "UNMEASURED",
    "C3": "PASS",
    "C4": "UNMEASURED",
    "C5": "UNMEASURED",
    "C6": "UNMEASURED",
    "C7": "UNMEASURED",
    "C8": "UNMEASURED",
    "C9": "UNMEASURED",
    "C10": "UNMEASURED"
  },
  "upgrades": [
    "C1",
    "C2",
    "C4"
  ],
  "downgrades": [],
  "first_blocker_certificate_order": "C5",
  "first_blocker_attack_order": "C6",
  "first_blocker_dependency_order": "C8",
  "frontier": {
    "attackable_now": [
      "C5",
      "C8"
    ],
    "blocked_by": {
      "C6": [
        "C5"
      ],
      "C7": [
        "C6"
      ],
      "C9": [
        "C5",
        "C6",
        "C7"
      ],
      "C10": [
        "C9"
      ]
    },
    "note": "C5's only in-certificate upstream is C4, which now passes, so the chain C4 opened has C5 next. C6 lists C5 as its upstream, C7 lists C6, C9 lists C5/C6/C7 and C10 lists C9 - so every other open item sits behind C5, directly or transitively, which is why the handoff's section 11 attack order (C6, C7, then C5) runs against the recorded dependencies. C8 has NO in-certificate upstream at all: it was never blocked by anything in the certificate and is open simply because nobody has attacked it, so it can be taken in parallel."
  },
  "dependency_order": {
    "recorded_by": "src92 (RUN-090), item['dependencies']",
    "upstream_within_the_certificate": {
      "C1": [],
      "C2": [
        "C1"
      ],
      "C3": [],
      "C4": [
        "C1",
        "C3"
      ],
      "C5": [
        "C4"
      ],
      "C6": [
        "C4",
        "C5"
      ],
      "C7": [
        "C4",
        "C6"
      ],
      "C8": [],
      "C9": [
        "C4",
        "C5",
        "C6",
        "C7"
      ],
      "C10": [
        "C9"
      ]
    },
    "layer": {
      "C1": 0,
      "C3": 0,
      "C8": 0,
      "C2": 1,
      "C4": 1,
      "C5": 2,
      "C6": 3,
      "C7": 4,
      "C9": 5,
      "C10": 6
    },
    "topological_order": [
      "C1",
      "C3",
      "C8",
      "C2",
      "C4",
      "C5",
      "C6",
      "C7",
      "C9",
      "C10"
    ],
    "unordered_because_of_a_cycle": []
  },
  "first_blocker_note": "'the first blocker' has no single-item answer any more, and the three orders show why. Round 030's numbering gives C5. A topological pass over RUN-090's recorded dependencies gives C8, because C8 has no in-certificate upstream at all. The handoff's section 11 attack order gives C6, and that one contradicts the recorded dependencies, since C6 lists C5 as its upstream. Round 030 states the certificate as a conjunction, so no order is part of the criterion; the useful answer is the FRONTIER - the open items whose upstreams have all passed - and that is exactly {C5, C8}.",
  "normalization_status": "NORMALIZATION-UNMEASURED",
  "why_each_item": {
    "C1": "LR Thm C1.9 + section A8.3 + Thm C1.10 + Remark C1.11 + Thm C1.13, hypotheses instantiated here",
    "C2": "LR Thm C1.13's explicit factorisation + the C_d commonality computed here",
    "C3": "RUN-090's coordinate audit on the actual series in one cyclotomic coordinate",
    "C4": "RUN-094's chain: Thm A6.1 -> section A6.2 -> Cor A6.3(i) -> Thm A2.1 case (3), with RUN-092's dim H^1_f = 1 and RUN-093's vacuous local conditions",
    "C5": "no decomposition of D = C_arith (x) C_aux^{-1} into factors exists, so 'every defect factor' has no referent; and the criterion's mood is ENUMERATION, which no theorem label may discharge",
    "C6": "neither alternative is in hand: no theorem here supplies canonical period-de Rham cancellation, and no explicit residual pairing scalar eta_pdR has been exhibited",
    "C7": "neither alternative is in hand: no auxiliary-to-arithmetic determinant comparison map exists, and no explicit kappa_det has been exhibited",
    "C8": "the absence claim cannot be made for the target: 389.a1's prime-to-11 saturation of its rational modular-symbol line is unmeasured (RUN-090), so it is not known that no unit ambiguity is being attributed to a length there",
    "C9": "the subject does not exist: there are no actual comparison objects, hence no transport graph and no edges, so 'no residual holonomy' would be vacuous rather than true",
    "C10": "neither alternative is in hand: no canonical section, and q_fin = 1 is a number, not an anchor (RUN-090's ACCEPT_NUMERICAL_ANCHOR = False)"
  },
  "criterion_sentences": {
    "C1": "Target and calibrator use the same exact late character.",
    "C2": "No partner arithmetic is hidden in the common line.",
    "C3": "No analytic coordinate weight is hidden in exact normalization.",
    "C4": "The auxiliary leading tensor or equivalent q-blind source exists.",
    "C5": "Every defect factor is tagged E, L, G, or P.",
    "C6": "Either canonical cancellation or one explicit residual pairing scalar.",
    "C7": "Either canonical cohomological equality or explicit $\\kappa_{\\det}$.",
    "C8": "No exact unit ambiguity is attributed to a length / index.",
    "C9": "No residual holonomy.",
    "C10": "Canonical section or one exact anchor."
  },
  "criterion_moods": {
    "C1": "EQUALITY",
    "C2": "ABSENCE",
    "C3": "ABSENCE",
    "C4": "EXISTENCE",
    "C5": "ENUMERATION",
    "C6": "EXISTENCE",
    "C7": "EXISTENCE",
    "C8": "ABSENCE",
    "C9": "ABSENCE",
    "C10": "EXISTENCE"
  },
  "subitems": {
    "C2": {
      "parent_sentence": "No partner arithmetic is hidden in the common line.",
      "parent_verdict": "PASS",
      "children": {
        "C2-STRUCTURE": {
          "asks": "is partner arithmetic kept out of the common line",
          "in_parent_sentence": true,
          "verdict": "PASS"
        },
        "C2-VALUE": {
          "asks": "is the numerical value of the partner factor Pi_E/Pi_0 known",
          "in_parent_sentence": false,
          "verdict": "OPEN"
        }
      },
      "mapping_back": "the parent takes the verdict of the child its own sentence asks about; a child outside the sentence is carried in the outside-certificate ledger and cannot change the parent either way",
      "consistent": true
    },
    "C4": {
      "parent_sentence": "The auxiliary leading tensor or equivalent q-blind source exists.",
      "parent_verdict": "PASS",
      "children": {
        "C4-LINE": {
          "asks": "does the auxiliary leading tensor / q-blind source exist",
          "in_parent_sentence": true,
          "verdict": "PASS"
        },
        "C4-EXACT-NORM": {
          "asks": "is the comparison scalar c_ES/Kum fixed",
          "in_parent_sentence": false,
          "verdict": "OPEN"
        }
      },
      "mapping_back": "the parent takes the verdict of the child its own sentence asks about; a child outside the sentence is carried in the outside-certificate ledger and cannot change the parent either way",
      "consistent": true
    }
  },
  "outside_certificate": {
    "BF-COORD": "no computed representative of kappa_d(f,g), hence none of B_{E,2} or r_{0,1}",
    "C2-VALUE": "Pi_E/Pi_0, the Petersson/period ratio, has no known value",
    "C4-EXACT-NORM": "c_ES/Kum, the scalar between [Theta_aux] and kappa(epsilon), is not fixed",
    "C_PI-TRIVIALISATION": "b_f^+ is non-canonical (LR section A8.3), so C_pi has no numerical value until one trivialisation is fixed; anything built from C_aux inherits that ambiguity"
  },
  "rules": {
    "unmeasured_to_pass": false,
    "closure_requires_all": true,
    "vacuous_absence_is_pass": false,
    "admissible_without_object": [
      "COMPUTED_HERE",
      "CITED_THEOREM",
      "DERIVED_FROM_VERIFIED_INPUTS"
    ],
    "admissible_with_object": [
      "COMPUTED_HERE"
    ]
  },
  "inherited_hypotheses": {
    "389.a1": {
      "ordinary at p (a_p not divisible by p)": true,
      "unit root mod p": 7,
      "prime-to-p level (p does not divide N)": true,
      "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true,
      "p-distinguished (the two roots differ mod p)": true,
      "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true,
      "non-critical slope (ordinary means slope 0 < r_1 + 1)": true,
      "Hida (ordinary), as section B2 restricts to": true,
      "newform of its level, as section B2 assumes": true,
      "weight 2, trivial nebentypus": true,
      "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true,
      "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true,
      "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true
    },
    "19.a1": {
      "ordinary at p (a_p not divisible by p)": true,
      "unit root mod p": 3,
      "prime-to-p level (p does not divide N)": true,
      "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true,
      "p-distinguished (the two roots differ mod p)": true,
      "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true,
      "non-critical slope (ordinary means slope 0 < r_1 + 1)": true,
      "Hida (ordinary), as section B2 restricts to": true,
      "newform of its level, as section B2 assumes": true,
      "weight 2, trivial nebentypus": true,
      "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true,
      "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true,
      "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true
    }
  },
  "smoothing": {
    "formula": "C_d(f, g, j) = d^2 - d^(j - k1 - k2) eps_f^{-1}(d) eps_g^{-1}(d)   [LR equation (8)]",
    "d": 5,
    "eps_f": "chi_8",
    "eps_f_inverse_at_d": -1,
    "k1": 0,
    "per_partner": {
      "389.a1": {
        "eps_g_inverse_at_d": 1,
        "k2": 0,
        "C_d_at_j": {
          "0": 26,
          "1": 30,
          "2": 50
        }
      },
      "19.a1": {
        "eps_g_inverse_at_d": 1,
        "k2": 0,
        "C_d_at_j": {
          "0": 26,
          "1": 30,
          "2": 50
        }
      }
    },
    "partner_entry_pairs_eps_g_and_k2": {
      "389.a1": [
        1,
        0
      ],
      "19.a1": [
        1,
        0
      ]
    },
    "partner_entries_agree": true,
    "C_d_common_for_our_pair": true,
    "pc001_S5_at_0": 26,
    "j_values_matching_pc001": [
      0
    ],
    "j_pinned_to_k1_plus_k2": true,
    "derived_not_asserted": "RUN-090 recorded S5_partner_independent as a constant; here the commonality is derived from (eps_g(d), k2) agreeing, which a mutation can break",
    "agrees": true
  },
  "late_factor": {
    "C_pi": "c_lead / q_f",
    "constituents": {
      "c_lead": {
        "source": "LR Thm C1.9: c_f(k) is a meromorphic function on V_1 alone",
        "side": "auxiliary",
        "canonical": true
      },
      "q_f": {
        "source": "PC-001 section 4 = LR section A8.3: b_f^+(0) = c_r eta_f, a trivialisation of D_cris(F^+ D(f)^*(-1-k)), a module built from f alone",
        "side": "auxiliary",
        "canonical": false
      }
    },
    "sides": {
      "c_lead": "auxiliary",
      "q_f": "auxiliary"
    },
    "every_constituent_auxiliary_only": true,
    "partner_enters_only_through": [
      "eta_g"
    ],
    "non_canonical_constituents": [
      "q_f"
    ],
    "nonvanishing_source": "C1.13",
    "c19_nonvanishing_excepts_k_equals_r": true,
    "nonvanishing_note": "our point is k = r, which is exactly the weight C1.9's non-vanishing clause excepts; the nonvanishing comes from C1.13's c*_f(r) in L^x instead",
    "pc001_text": {
      "writes_C_pi_as_c_lead_over_q_f": true,
      "refuses_to_assume_q_f_is_1": true,
      "calls_the_common_frame_a_necessary_assumption": true,
      "says_the_factor_cancels_in_(2)": true
    },
    "conclusion": "every constituent of C_pi is auxiliary-only, so no property of either partner can obstruct fixing one frame: the common frame PC-001 calls a necessary assumption is a free, partner-independent normalisation. b_f^+ is non-canonical (section A8.3), so C_pi has no numerical value until one trivialisation is fixed, and anything built from C_aux inherits that ambiguity.",
    "common_frame_has_a_partner_dependent_obstruction": false,
    "agrees": true
  },
  "claim_labels_legend": [
    "COMPUTED_HERE",
    "CITED_THEOREM",
    "DERIVED_FROM_VERIFIED_INPUTS",
    "UNMEASURED",
    "FAILED"
  ],
  "labels": {
    "COMPUTED_HERE": [
      "the inherited hypotheses on both curves, semistability from the Weierstrass invariants included",
      "LR equation (8) instantiated and j pinned by PC-001's 26",
      "the criterion sentences parsed from Round 030 and their moods",
      "the provenance of every constituent of C_pi"
    ],
    "CITED_THEOREM": [
      "LR Assumption B1.1, section B2, section A8.3, Thm A2.1/A6.1/A6.3(i), Thm C1.4, C1.9, C1.10, Remark C1.11, Thm C1.13",
      "Mazur, rational isogenies of prime degree"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-090's C3 and its lattice ledger",
      "RUN-092/093's Selmer line",
      "RUN-094's C4 chain and the C1.13 dictionary"
    ],
    "UNMEASURED": [
      "BF-COORD",
      "C2-VALUE",
      "C4-EXACT-NORM",
      "C_PI-TRIVIALISATION",
      "C5",
      "C6",
      "C7",
      "C8",
      "C9",
      "C10"
    ],
    "FAILED": []
  },
  "status": {
    "C1": "PASS",
    "C2": "PASS",
    "C3": "PASS",
    "C4": "PASS",
    "C5": "UNMEASURED - first blocker in the certificate's own numbering",
    "C6-C10": "UNMEASURED",
    "BSD": "OPEN"
  }
}