{
  "round": "RUN-097",
  "gate": "src99",
  "R0": "PASS",
  "why": "every arrow actual, every junction typed and compatible, every scalar identity lifted by a named theorem, both endpoints realised, the partner slot separate",
  "C5_rerun_against_the_actual_chain": {
    "verdict": "PASS",
    "survivors_delta_factors": {
      "G": [
        "q_f",
        "Gauss(chi_8)"
      ],
      "L": [
        "plus_index_2 (19.a1)",
        "index (389.a1, plus)",
        "index (389.a1, minus)"
      ],
      "E": [
        "eta_pdR",
        "kappa_det",
        "id(LR class, PC-001 projected class)",
        "id(Kato class, modular symbol)"
      ]
    },
    "cancel_in_delta": [
      "C_pi",
      "c_lead",
      "j",
      "log11_gamma",
      "S5",
      "lambda_i(0)",
      "E_adj_i / P-part",
      "E_adj_i / L-part",
      "Pi_i",
      "kappa_E",
      "ell_0"
    ],
    "post_chain": [
      "16 = e11/theta",
      "416 = 16*26"
    ],
    "reading": "the composite's scalar keeps exactly the frames (G: q_f = G_b measured against omega_f, Gauss), the lattice indices (L), and the exact comparisons (E: eta_pdR, kappa_det, the two identifications); every partner factor and C_d and the cyclotomic log cancel between Psi_1^{-1} and Psi_3 Psi_4 - which is Round 030 (16) computed: delta = a_D q_D eta_D with the P factors removed, and (15.2) delta = c_lead/c_aux"
  },
  "typed_dictionary": {
    "C_aux": {
      "object": "D_cris(F+ D(f)^*(-1-k)), framed by the unique omega_f : -> X^{-n} O(V_1)",
      "type": {
        "category": "crystalline (D_cris of a (phi,Gamma)-submodule)",
        "ring": "O(V_1)",
        "rank": 1,
        "character": "psi^{-1} eps^{-1-r} side of the triangulation",
        "tate_twist": -1,
        "orientation": "line"
      },
      "q_blind": true,
      "why_q_blind": "omega_f is canonical (C1.10 unique; C1.11 determined by f); b_f+ is a second, non-canonical framing of the same line and q_f = b_f+(0) is its value",
      "weight_factor": {
        "round030": "Sym^m(T_X^*)",
        "LR": "X^{-n} O(V_1) (C1.10)",
        "m_equals_n": true
      },
      "frame": "omega_f (canonical); b_f+ (non-canonical, gauge u)",
      "source": "LR A8.3, C1.10, C1.11"
    },
    "C_arith": {
      "object": "the line spanned by L_p(g x psi) in H_Gamma, constructed from modular symbols (Mazur-Tate-Teitelbaum)",
      "type": {
        "category": "H_Gamma (Iwasawa algebra / distributions)",
        "ring": "H_Gamma",
        "rank": 1,
        "character": "g x psi",
        "tate_twist": 0,
        "orientation": "line"
      },
      "q_blind": true,
      "frame": "the modular-symbol period frame Omega^+- (G) and the lattice index (L)",
      "source": "Kato [Kat04] + MTT; in-tree: src83's series (RUN-081), read back from RUN-090's log",
      "realised_in_tree": {
        "series_first_eleven_mod_11": {
          "L_E": [
            0,
            0,
            2,
            2,
            2,
            0,
            5,
            6,
            10,
            7,
            0
          ],
          "lambda_E": [
            5,
            9,
            10,
            6,
            6,
            2,
            0,
            8,
            8,
            9,
            10
          ],
          "L_0": [
            9,
            3,
            10,
            3,
            1,
            6,
            9,
            6,
            2,
            0,
            6
          ],
          "lambda_0": [
            4,
            3,
            10,
            9,
            4,
            10,
            2,
            7,
            3,
            8,
            3
          ]
        },
        "order_at_t_equals_0": {
          "L_E": 2,
          "lambda_E": 0,
          "L_0": 0,
          "lambda_0": 0
        },
        "target_ordinary_vanishes_to_order_2_(rank_2)": true,
        "twisted_series_nonzero_at_0": true,
        "nonzero_as_elements_of_H_Gamma": true,
        "source": "RUN-081 (src83) reproduced PC-002 coefficient for coefficient; RUN-090 (src92) re-read them in the gamma = 12 coordinate",
        "agrees": true
      }
    },
    "Q": {
      "object": "the quotient-transport line: the value q_f = b_f+(0) = c_r eta_f of the non-canonical framing",
      "type": {
        "category": "scalar over the coefficient field",
        "ring": "L",
        "rank": 1,
        "character": "-",
        "tate_twist": 0,
        "orientation": "line"
      },
      "q_blind": false,
      "frame": "b_f+ (gauge u)",
      "source": "LR A8.3 bad-fibre relation; PC-001 section 4; RUN-096 gauge law (1,0)"
    },
    "N_pi": {
      "object": "the late projected line: the line of B_hat_i = C_pi A_i j z_i",
      "type": {
        "category": "H_Gamma",
        "ring": "H_Gamma",
        "rank": 1,
        "character": "g x psi",
        "tate_twist": 0,
        "orientation": "line"
      },
      "q_blind": true,
      "frame": "-",
      "source": "PC-001 (1) = LR C1.13 composed with the projection; RUN-096: B_hat_i gauge-invariant"
    },
    "extension_class_separately": {
      "object": "the extension class [Theta_aux] of 0 -> psi -> V(f_beta) -> tau eps^{-1-r} -> 0 (RUN-092..094)",
      "type": {
        "category": "etale (Galois cohomology H^1_f)",
        "ring": "Q_11",
        "rank": 1,
        "character": "chi_8",
        "tate_twist": 1,
        "orientation": "line"
      },
      "q_blind": true,
      "frame": "Kummer (canonical) / ES (non-canonical)",
      "source": "LR Thm A2.1 case (3), Cor A6.3(i)",
      "relation_to_C_aux": "the vertical (global-reducibility) line, transverse to the horizontal (triangulation) line F+ inside D(f_beta)^* with cokernels t^{r+1} (LR, diagram before A8.1); the transversality IS non-criticality (A4.5). Not the same line; the scalar between them is C4-EXACT-NORM."
    }
  },
  "chain": [
    {
      "name": "Psi_1",
      "from": "L1 (BF family line)",
      "to": "L0 (C_aux) (x) partner line (x) H_Gamma",
      "theorem": "Perrin-Riou regulator L^PR (LZ16) on loc_p; image computed by LR C1.9",
      "line_level": "class-level (a map of modules)",
      "partner_dependence": "ONLY through the separate tensor factor D_cris(F- D(g)^*) and eta_g"
    },
    {
      "name": "Psi_2",
      "from": "L1 (BF family line)",
      "to": "L2 (BF leading line)",
      "theorem": "C1.6 (tau-quotient image zero) + C1.7 (unique lift to H^1(Q,V(f,g)^*), image in the psi-quotient) + leading term X^n kappa-hat_d mod X",
      "line_level": "class-level (quotient maps of Galois modules)",
      "partner_dependence": "the target module is V(g)^*(psi^{-1}) (x) H_Gamma"
    },
    {
      "name": "Psi_3",
      "from": "L2 (BF leading line)",
      "to": "L3 (Kato line)",
      "theorem": "C1.13: kappa-hat*_d(f_beta,g) = [C C_d log L_p(g x tau)/L_p(Ad g)] kappa(g x psi); C1.1's proof: the module is free of rank 1 over H_Gamma",
      "line_level": "scalar-first, lifted to classes by Prop C1.1",
      "partner_dependence": "L_p(g x tau), L_p(Ad g), kappa(g x psi)"
    },
    {
      "name": "Psi_4",
      "from": "L3 (Kato line)",
      "to": "L4 (C_arith)",
      "theorem": "Kato [Kat04]: localisation at p, Perrin-Riou regulator for F- V(g)^*, lands on L_p(g x psi); and Kato's L_p equals the Mazur-Tate-Teitelbaum L_p built from modular symbols, up to the period frame",
      "line_level": "class-level; the equality of Kato's and MTT's L_p is an equality of elements of H_Gamma",
      "partner_dependence": "entirely partner-side"
    }
  ],
  "junctions": {
    "L1 shared by Psi_1 and Psi_2": {
      "compatible": true,
      "problems": []
    },
    "L2: Psi_2 -> Psi_3": {
      "compatible": true,
      "problems": []
    },
    "L3: Psi_3 -> Psi_4": {
      "compatible": true,
      "problems": []
    }
  },
  "composite": "Psi = Psi_4 . Psi_3 . Psi_2 . Psi_1^{-1} : C_aux (x) (partner line) -> C_arith, over H_Gamma-lines",
  "positive_control_accepted": true,
  "sharp_question_1": "LR proves equal regulator images, then equality of CLASSES by Prop C1.1 (injectivity): line-level",
  "sharp_question_2": "id(Kato ~ modular symbol) is Kato's theorem that his p-adic L-function is the modular-symbol one up to the stated period frame - a theorem-induced identification of generators of the same line, with the frame named as G; not bookkeeping",
  "refinement_of_RUN_096": "C1.10: Attack 09's Eichler-Shimura frame IS omega_f, which is canonical; so RUN-096's v is not free and G_b = u/v is the b_f+ -> omega_f ratio, exactly the b_f+-dependence of c_f(k) that C_pi = c_lead/q_f removes",
  "not_claimed": [
    "the values of eta_pdR, kappa_det, G_b, Pi_i (C5/C6/C7)",
    "any leading-coefficient extraction in t (PC-001 (2))"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "the typed dictionary and the junction type checks",
      "the in-tree realisation of C_arith (the four series, orders at t = 0)"
    ],
    "CITED_THEOREM": [
      "LR Prop C1.1, Thm C1.4, Prop C1.6, C1.7, Prop C1.8, Thm C1.10, Remark C1.11, Thm C1.13 and its proof, sections A8.1-A8.3, A4.5",
      "Kato [Kat04]; Perrin-Riou / LZ16 regulator"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-094/095's hypotheses",
      "RUN-096's gauge law and C8 indices",
      "RUN-081/090's series"
    ],
    "UNMEASURED": [
      "the values of eta_pdR, kappa_det, G_b, Pi_i (C5/C6/C7)",
      "any leading-coefficient extraction in t (PC-001 (2))"
    ],
    "FAILED": []
  },
  "status": {
    "C1-C4": "PASS",
    "R0": "PASS",
    "C5": "PASS",
    "C6": "UNMEASURED",
    "C7": "UNMEASURED",
    "C8": "PASS",
    "C9-C10": "UNMEASURED",
    "BSD": "OPEN"
  }
}