{
  "round": "RUN-106",
  "gate": "src108",
  "DAG_C": "PASS",
  "C_NT": "PASS",
  "C_p": "PARTIAL",
  "sub_statuses": {
    "C_NT": "PASS — m = 1 proved here (Hermite + exhaustive box + an ℓ-sieve needing no citation); the ½ convention fixed and discriminated by the BSD identity; det(h♭) = Reg_NT = 0.152460139",
    "C_p": "PARTIAL — Hypothesis 4.1 instantiated and Theorem 4.2 read literally (values in Q^k, 1 ≤ k ≤ 10, BD = −Nek); nondegeneracy UNMEASURED: at k = 1 it is Schneider's conjecture",
    "local_line": "PASS as a computation, with a finding: log_ω(H^1_f(Q_11,T)) = 11^1 · Z_11, index 11 — the local line is not integrally trivial",
    "ledger": "PASS — the global index, the local index and RUN-104's 11-adic index are three different quantities, each entered once",
    "boundary": "PASS — p-adic ≠ real; MCS §4 identifies two p-adic constructions and reaches no real regulator",
    "epsilon": "not computed and not demanded — it is not Néron–Tate regulator data (RUN-106 §4)"
  },
  "C_input_consumed": "RUN-105's 𝓗_str = D_MW ⊗ H^1_f(Q_11,T)^{-1}: this round trivialises the D_MW side by the Néron–Tate pairing and measures the local side's index (11), leaving no untrivialised free factor on the classical side",
  "handed_to_D_E": "the p-adic realization: the derived height exists and is identified (MCS Thm 4.2), its nondegeneracy is open, and the local line carries index 11",
  "heights": {
    "on_curve": true,
    "depth": 10,
    "h_P": 0.32700076250449767,
    "h_Q": 0.4767115473365956,
    "h_P_plus_Q": 0.9207576020223028,
    "h_P_minus_Q": 0.6866670271056715,
    "convention": "<P,Q> = (1/2)(h(P+Q) - h(P) - h(Q))",
    "pairing": 0.05852264609060481,
    "parallelogram_route": 0.05852264372915783,
    "two_routes_agree": true,
    "difference_between_routes": 2.361446982401816e-09,
    "residual_of_the_limit_at_this_depth": 6.198883056640625e-06,
    "regulator": 0.1524601393683195,
    "tree_value_RUN_023": 0.15246013936831948,
    "external_value": 0.15246017794314376,
    "residual_against_external": 3.857482425151382e-08,
    "h_flat": "h♭ : M_E → M_E^∨, v ↦ <v, ·>; det(h♭) ∈ (D_MW^∨)^{⊗2} is the regulator as a quadratic form on the determinant",
    "bsd_identity_ratio": 0.9999997469842632,
    "ratio_if_pairing_doubled": 0.9326070648101007,
    "ratio_if_pairing_halved": 1.0168479175278038,
    "convention_discriminated": true,
    "numerical_match_is_not_the_bridge": true,
    "why": "the numbers agree with the analytic side, which is a consistency check on the convention and nothing more; the bridge is the lattice statement plus the canonical-height theorem, not this ratio"
  },
  "global_index": {
    "index": 1,
    "hermite": {
      "gamma_2": 1.1547005383792517,
      "cap_for_index_2": 0.2254330198886279,
      "statement": "a rank-2 lattice of determinant D has a nonzero vector of length^2 <= gamma_2 sqrt(D); with D = Reg/m^2 an index m >= 2 forces a point of canonical height <= this cap"
    },
    "cited_bound": {
      "form": "hhat(R) >= h(x(R)) - [ (1/4)h(j) + (1/6)h(Delta) + 1.946 ]",
      "h_j": 14.155496613885283,
      "h_Delta": 5.963579343618446,
      "constant_used": 6.478804044074395,
      "label": "CITED_THEOREM (Silverman-type height-difference bound; not re-proved here)",
      "measured_shortfall_over_120_points": 0.0,
      "margin": 6.478804044074395,
      "reading": "the constant actually needed on measured points is ~0; the one used is 6.48, so the box is generous by a factor e^6.48 in coordinate size — a misread normalisation would have to be worse than that to matter"
    },
    "box": {
      "naive_height_limit": 6.704237063963023,
      "coordinate_limit": 816,
      "points_found": 27,
      "smallest_canonical_height": 0.32696454966708133,
      "depth": 6,
      "ten_smallest": [
        {
          "h": 0.32696454966708133,
          "x": "0"
        },
        {
          "h": 0.47669130359417294,
          "x": "1"
        },
        {
          "h": 0.6865793890272475,
          "x": "-1"
        },
        {
          "h": 0.920754956872399,
          "x": "-2"
        },
        {
          "h": 1.3078991799913302,
          "x": "3"
        },
        {
          "h": 1.550524144321646,
          "x": "4"
        },
        {
          "h": 1.9067776152594222,
          "x": "6"
        },
        {
          "h": 1.9996516067562071,
          "x": "-3/4"
        },
        {
          "h": 2.018750784834094,
          "x": "5/4"
        },
        {
          "h": 2.4679296387798972,
          "x": "1/9"
        }
      ]
    },
    "ell_sieve": {
      "2": {
        "classes": 3,
        "killed": 3,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 23,
            "N": 28
          },
          "1:0": {
            "q": 23,
            "N": 28
          },
          "1:1": {
            "q": 29,
            "N": 36
          }
        }
      },
      "3": {
        "classes": 4,
        "killed": 4,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 17,
            "N": 24
          },
          "1:0": {
            "q": 5,
            "N": 9
          },
          "1:1": {
            "q": 5,
            "N": 9
          },
          "1:2": {
            "q": 5,
            "N": 9
          }
        }
      },
      "5": {
        "classes": 6,
        "killed": 6,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 19,
            "N": 15
          },
          "1:0": {
            "q": 19,
            "N": 15
          },
          "1:1": {
            "q": 19,
            "N": 15
          },
          "1:2": {
            "q": 19,
            "N": 15
          },
          "1:3": {
            "q": 19,
            "N": 15
          },
          "1:4": {
            "q": 41,
            "N": 45
          }
        }
      },
      "7": {
        "classes": 8,
        "killed": 8,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 23,
            "N": 28
          },
          "1:0": {
            "q": 23,
            "N": 28
          },
          "1:1": {
            "q": 23,
            "N": 28
          },
          "1:2": {
            "q": 31,
            "N": 28
          },
          "1:3": {
            "q": 23,
            "N": 28
          },
          "1:4": {
            "q": 23,
            "N": 28
          },
          "1:5": {
            "q": 23,
            "N": 28
          },
          "1:6": {
            "q": 23,
            "N": 28
          }
        }
      },
      "11": {
        "classes": 12,
        "killed": 12,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 103,
            "N": 110
          },
          "1:0": {
            "q": 103,
            "N": 110
          },
          "1:1": {
            "q": 103,
            "N": 110
          },
          "1:2": {
            "q": 103,
            "N": 110
          },
          "1:3": {
            "q": 103,
            "N": 110
          },
          "1:4": {
            "q": 103,
            "N": 110
          },
          "1:5": {
            "q": 373,
            "N": 396
          },
          "1:6": {
            "q": 103,
            "N": 110
          },
          "1:7": {
            "q": 103,
            "N": 110
          },
          "1:8": {
            "q": 103,
            "N": 110
          },
          "1:9": {
            "q": 103,
            "N": 110
          },
          "1:10": {
            "q": 103,
            "N": 110
          }
        }
      },
      "13": {
        "classes": 14,
        "killed": 14,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 7,
            "N": 13
          },
          "1:0": {
            "q": 7,
            "N": 13
          },
          "1:1": {
            "q": 7,
            "N": 13
          },
          "1:2": {
            "q": 7,
            "N": 13
          },
          "1:3": {
            "q": 7,
            "N": 13
          },
          "1:4": {
            "q": 7,
            "N": 13
          },
          "1:5": {
            "q": 7,
            "N": 13
          },
          "1:6": {
            "q": 7,
            "N": 13
          },
          "1:7": {
            "q": 7,
            "N": 13
          },
          "1:8": {
            "q": 7,
            "N": 13
          },
          "1:9": {
            "q": 7,
            "N": 13
          },
          "1:10": {
            "q": 127,
            "N": 117
          },
          "1:11": {
            "q": 7,
            "N": 13
          },
          "1:12": {
            "q": 7,
            "N": 13
          }
        }
      },
      "17": {
        "classes": 18,
        "killed": 18,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 13,
            "N": 17
          },
          "1:0": {
            "q": 13,
            "N": 17
          },
          "1:1": {
            "q": 13,
            "N": 17
          },
          "1:2": {
            "q": 13,
            "N": 17
          },
          "1:3": {
            "q": 13,
            "N": 17
          },
          "1:4": {
            "q": 13,
            "N": 17
          },
          "1:5": {
            "q": 13,
            "N": 17
          },
          "1:6": {
            "q": 13,
            "N": 17
          },
          "1:7": {
            "q": 13,
            "N": 17
          },
          "1:8": {
            "q": 13,
            "N": 17
          },
          "1:9": {
            "q": 13,
            "N": 17
          },
          "1:10": {
            "q": 13,
            "N": 17
          },
          "1:11": {
            "q": 13,
            "N": 17
          },
          "1:12": {
            "q": 229,
            "N": 238
          },
          "1:13": {
            "q": 13,
            "N": 17
          },
          "1:14": {
            "q": 13,
            "N": 17
          },
          "1:15": {
            "q": 13,
            "N": 17
          },
          "1:16": {
            "q": 13,
            "N": 17
          }
        }
      },
      "19": {
        "classes": 20,
        "killed": 20,
        "complete": true,
        "witnesses": {
          "0:1": {
            "q": 59,
            "N": 57
          },
          "1:0": {
            "q": 59,
            "N": 57
          },
          "1:1": {
            "q": 59,
            "N": 57
          },
          "1:2": {
            "q": 59,
            "N": 57
          },
          "1:3": {
            "q": 59,
            "N": 57
          },
          "1:4": {
            "q": 59,
            "N": 57
          },
          "1:5": {
            "q": 59,
            "N": 57
          },
          "1:6": {
            "q": 197,
            "N": 190
          },
          "1:7": {
            "q": 59,
            "N": 57
          },
          "1:8": {
            "q": 59,
            "N": 57
          },
          "1:9": {
            "q": 59,
            "N": 57
          },
          "1:10": {
            "q": 59,
            "N": 57
          },
          "1:11": {
            "q": 59,
            "N": 57
          },
          "1:12": {
            "q": 59,
            "N": 57
          },
          "1:13": {
            "q": 59,
            "N": 57
          },
          "1:14": {
            "q": 59,
            "N": 57
          },
          "1:15": {
            "q": 59,
            "N": 57
          },
          "1:16": {
            "q": 59,
            "N": 57
          },
          "1:17": {
            "q": 59,
            "N": 57
          },
          "1:18": {
            "q": 59,
            "N": 57
          }
        }
      }
    },
    "external_agreement": "LMFDB/Cremona record P, Q as generators — EXTERNAL, recorded as agreement, not used as a step"
  },
  "p_adic": {
    "Hypothesis_4_1": {
      "(i) p ∤ Tam(E/Q) · ∏_{v|p} #Ẽ_v(F_v)": {
        "holds": true,
        "evidence": "Tam = c_389 = 1 (RUN-067), #Ẽ(F_11) = 16; 11 ∤ 16",
        "label": "COMPUTED_HERE"
      },
      "(ii) the image of ρ: G_Q → Aut(E[p]) contains a Cartan subgroup": {
        "holds": true,
        "evidence": "ρ̄_11 is surjective (RUN-069, six maximal-subgroup witnesses), and GL_2(F_11) contains its own Cartan subgroups",
        "label": "DERIVED_FROM_VERIFIED_INPUTS"
      }
    },
    "literal_type": {
      "domain": "S_p^{(k)} × S_p^{(k)}, where S_p(E/Q) = lim_n Sel_{p^n}(E/Q) and S_p = S^{(1)} ⊃ S^{(2)} ⊃ … ⊃ S^{(p)} is Bertolini–Darmon's filtration",
      "codomain": "Q^k = I^k/I^{k+1} — a free rank-one Z_11-module only after a topological generator of Γ is chosen; that choice is a gauge, of the same species as RUN-103's augmentation generators",
      "selmer_filtration": "(4.2.1) Sel^{(k)} = Sel_{p^n}(E/Q) ∩ 𝓘^{k-1} Sel_{p^n}(E/Q_n), via Lemma 4.4's isomorphism Sel_{p^n}(E/Q) ≅ Sel_{p^n}(E/L)^G",
      "order_k": "1 ≤ k ≤ p − 1 = 10; k = 1 is the usual p-adic height pairing",
      "order_used_here": 1,
      "order_in_range": true,
      "symmetry": "symmetric (a height pairing); Nekovář's is defined as β^{(k)}(x) ∪_k y with the k-th derived Bockstein",
      "sign": "Theorem 4.2: ⟨,⟩^BD_k = -1 · ⟨,⟩^Nek_k — the identification carries a sign",
      "integral_or_rational": "valued in Q^k, a quotient of integral ideals of Λ; neither map is a real number",
      "hypotheses": "Hypothesis 4.1 for the BD side; Remark 4.3: Nekovář's pairing needs no such hypothesis and allows any k ≥ 1"
    },
    "identification": "PASS (CITED_THEOREM: MCS Theorem 4.2, hypotheses instantiated here)",
    "identification_is_not_nondegeneracy": true,
    "declaration_matches_the_paper": {
      "range_of_k": true,
      "codomain_is_I^k/I^{k+1}": true,
      "sign": true,
      "rule": "the codomain is a quotient of augmentation ideals; rewriting it as Z_11 to meet another line's shape is the refused move"
    },
    "nondegeneracy": "UNMEASURED",
    "det_h_p": {
      "map": "det(h_p^{(k)}) : D_MW,11^{⊗2} → (Q^k)^{⊗2}",
      "target_presumed_Z_11": false,
      "nonzero": "UNMEASURED — at k = 1 this is the nonvanishing of the p-adic regulator (Schneider's conjecture), open for a rank-2 curve; Theorem 4.2 identifies two pairings and says nothing about rank or radical",
      "what_would_be_needed": "a nondegeneracy theorem for the p-adic height on Sel(T) ⊗ Q_11, or a computation of det h_p ≠ 0 to a proven precision"
    },
    "C_p": "PARTIAL"
  },
  "local_line": {
    "identification": "11 ∤ #Ẽ(F_11) = 16, so E(Q_11) ⊗̂ Z_11 = E_1(Q_11) and H^1_f(Q_11,T) = E(Q_11) ⊗̂ Z_11 (Bloch–Kato, CITED)",
    "omega_expansion": "ω = (1t^0 + 0t^1 + 1t^2 + 2t^3 + …) dt",
    "log_series": "log(t) = 1t^1 + 0t^2 + 1/3t^3 + 1/2t^4 + -3/5t^5 + …",
    "cofactor": 16,
    "rows": {
      "P": {
        "v_11(t)": 1,
        "in_E_1": true,
        "v_11(log)": 1,
        "log/11 mod 11^3": 328,
        "t/11 mod 11": 9
      },
      "Q": {
        "v_11(t)": 1,
        "in_E_1": true,
        "v_11(log)": 1,
        "log/11 mod 11^3": 611,
        "t/11 mod 11": 6
      }
    },
    "valuation_theory": "log(t) = t + Σ_{n≥2} (a_n/n) t^n with a_n integral on a minimal model, and for v(t) ≥ 1, p = 11: v(t^n/n) ≥ n − log_p(n) > 1, so v(log t) = v(t) exactly",
    "v_11_of_log_omega_g": 1,
    "declared": 1,
    "image_of_the_integral_lattice": "11^1 · Z_11",
    "integrally_trivial": false,
    "index": 11,
    "rational_iso_is_not_an_integral_unit": true,
    "claimed_unit": false,
    "C6_relation": {
      "what C6 settled": "η_pdR = 1 — the period/de Rham comparison pairing ⟨γ, ω⟩ of the normalisation route (Eichler–Shimura; Ω^± is its value on the chosen Betti basis), RUN-099",
      "what this round uses": "the Bloch–Kato identification D_dR(V)/Fil⁰ ≅ Q_11 via ω_E, i.e. the formal-group logarithm of the minimal model",
      "shared": "the differential ω_E appears in both",
      "different": "one is a global Betti/de Rham comparison, the other a local p-adic Hodge identification; no theorem in this tree identifies their normalisations",
      "forbidden_inference": "η_pdR = 1 ⇒ the local log normalisation is 1",
      "inference_taken": false,
      "what_it_would_have_cost_here": "a factor 11: the integral image is 11^1 · Z_11, not Z_11"
    },
    "reading": "the local Bloch–Kato line is NOT integrally trivial: log_ω carries H^1_f(Q_11,T) onto 11^1 · Z_11. This is a local arithmetic datum of the line, not a reopening of the normalisation certificate."
  },
  "ledger": [
    {
      "quantity": "global index [Λ_E : ZP + ZQ]",
      "value": 1,
      "enters": "the free factor D_MW: ∧²(ZP+ZQ) = m · D_MW; with m = 1 the two coincide",
      "degree": "free part, degree 1 of RΓ(Z_S,T)",
      "source": "this round (Hermite + box + sieve)",
      "entered_times": 1
    },
    {
      "quantity": "local index [Z_11 : log_ω H^1_f(Q_11,T)] = 11",
      "value": 11,
      "enters": "the trivialisation of the local factor H^1_f(Q_11,T)^{-1} of RUN-105's line",
      "degree": "local line at 11",
      "source": "this round (formal logarithm)",
      "entered_times": 1
    },
    {
      "quantity": "11-adic index of loc_11(E(Q)⊗Z_11) in E(Q_11)⊗̂Z_11 = 11^0",
      "value": 1,
      "enters": "RUN-104/105: the torsion of H^2 and nothing else — a different map from log_ω, and not this round's 11",
      "degree": "degree 2",
      "source": "RUN-104",
      "entered_times": 1
    }
  ],
  "boundary": {
    "statement": "the derived p-adic height is p-adic height data; no theorem in the sources sends it to the real Néron–Tate regulator",
    "equated": false,
    "what_MCS_section_4_proves": "Bertolini–Darmon's derived p-adic height equals (up to sign) Nekovář's — two p-adic constructions identified with each other",
    "what_it_does_not_prove": [
      "that either is nondegenerate",
      "that either computes Reg_NT",
      "anything about a real-valued regulator"
    ],
    "DAG_C_controlled_by": "C_NT (Rounds 031/035 define C as Mordell–Weil determinant ⟶ classical Néron–Tate regulator)",
    "C_p_pass_does_not_imply_C_NT": true
  },
  "controls": {
    "A": {
      "sublattice": "Z(1P) + Z(2Q), a genuine index-2 sublattice",
      "regulator": 0.6098328649375692,
      "cap_for_index_2": 0.4508631961536187,
      "smallest_height_in_E(Q)": 0.3269966534616917,
      "index_detected": true,
      "accepted": true,
      "rule": "the same Hermite-plus-search method, pointed at a lattice that is NOT saturated, must fail to prove primitivity — otherwise it proves it of everything"
    },
    "B": {
      "point": "2P + -1Q",
      "canonical_height": 1.5506240362836992,
      "predicted_by_the_quadratic_form": 1.550584919438434,
      "difference": 3.9116845265141365e-05,
      "tolerance": 9.918212890625e-05,
      "requirement_enforced": true,
      "accepted": true,
      "rule": "ĥ(aP+bQ) = a²ĥ(P) + b²ĥ(Q) + 2ab⟨P,Q⟩ must hold at a second point, or the convention check tests nothing"
    },
    "agrees": true
  },
  "not_claimed": [
    "Reg_NT reached from any p-adic height — no comparison theorem exists in the sources read",
    "nondegeneracy of the p-adic height pairing (Schneider's conjecture, open) and hence det(h_p) ≠ 0",
    "the unit ε of RUN-105, which this round was told not to touch",
    "the Silverman-type height-difference bound re-proved here (CITED; measured against 120 points, not re-derived)",
    "Ш(E/Q)[11] = 0 (inherited, and not used by C_NT at all)",
    "any statement about D, E, F, G",
    "BSD"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "the canonical heights at depth 10 and the regulator (residual against the external value reported)",
      "the Néron–Tate pairing by two routes and the BSD-identity discrimination of the ½ convention",
      "the global index m = 1: Hermite's cap, the exhaustive box of 72 points, the smallest canonical height",
      "the ℓ-sieve: for ℓ ≤ 19 and every projective class a witness prime, with no citation",
      "the formal-group ω and log series from the minimal model, and v_11(log_ω(g)) = 1",
      "Hypothesis 4.1 (i) from Tam and #Ẽ(F_11)"
    ],
    "CITED_THEOREM": [
      "a Silverman-type height-difference bound (used only to size the search box; measured against 120 points)",
      "Bloch–Kato: H^1_f(Q_11,T) = E(Q_11) ⊗̂ Z_11, and log_ω as the formal-group logarithm",
      "MCS Theorem 4.2 (BD = −Nekovář) with Hypothesis 4.1",
      "Hermite's inequality in rank 2"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "Hypothesis 4.1 (ii) from RUN-069's surjectivity",
      "RUN-105's line, whose 𝓗_str this round consumes"
    ],
    "EXTERNAL": [
      "LMFDB/Cremona's own saturation statement — recorded as agreement, not used as a step"
    ],
    "UNMEASURED": [
      "nondegeneracy of the p-adic height (Schneider's conjecture)",
      "det(h_p) ≠ 0",
      "the unit ε"
    ],
    "FAILED": []
  },
  "status": {
    "normalization_certificate": "CLOSED (architecture), frozen",
    "DAG": {
      "A": "PASS (cited-theorem route)",
      "B": "PASS",
      "C": "PASS (C_NT PASS, C_p PARTIAL)",
      "D-G": "untouched"
    },
    "BSD": "OPEN"
  }
}