{
  "attack": "C4 step 2 — the transverse extension, its Selmer line, and the local conditions",
  "C4_STATUS": {
    "target_selmer_line": {
      "group": "H^1_f(Q, Q_11(chi8)(1))",
      "dimension": 1,
      "generator": "kappa(epsilon), epsilon = 1 + sqrt2"
    },
    "transverse_extension": {
      "constructed": false,
      "extension_group": "Ext¹(1, ψ) = H^1(Q, Q_11(chi_8)(1))",
      "cocycle": "c(gh) = ψ(g)c(h) + c(g)",
      "which_off_diagonal_entry": "the upper-right entry c(g)",
      "basis_frame": "e₁ spans the sub; e₂ is any lift of the quotient — the class is independent of that lift",
      "ordering_convention": "0 → ψ → V_Eis → 1 → 0",
      "local_conditions": {
        "at_11": {
          "chi8_unramified_at_11": true,
          "chi8_at_11": -1,
          "H0": 0,
          "H2": 0,
          "H1": 1,
          "H1_f": 1,
          "vacuous": true,
          "why": "H¹ = H¹_f, both of dimension 1, so every class is automatically crystalline at 11"
        },
        "at_2": {
          "chi8_ramified_at_2": true,
          "H0": 0,
          "H2": 0,
          "H1": 0,
          "vacuous": true,
          "why": "the whole local H¹ vanishes"
        },
        "at_infinity": {
          "chi8_even": true,
          "conjugation_acts_by": -1,
          "V_c": 0,
          "vacuous": true,
          "why": "11 is odd, so H¹(Gal(C/R), V) = 0"
        },
        "away_from_2_11": {
          "vacuous": true,
          "why": "(χ₈χ_cyc)(Frob_ℓ) = χ₈(ℓ)ℓ is never 1, so H⁰ = 0 and H¹(Q_ℓ,V) = 0"
        },
        "membership_automatic": true
      },
      "nonzero_witness": null
    },
    "comparison": {
      "relation": "Theta_aux = c_ES/Kum * kappa(epsilon)",
      "c_ES_Kum": "UNMEASURED",
      "why": "the round's instruction is class-level identification first; and the value log₁₁(ε)/(11√2) ≡ 5 is explicitly not allowed to suggest a scalar",
      "how_many_scalars": "ONE, if the Eichler–Shimura frame's own normalisation is folded in: of the four frames only ES is non-canonical (RUN-092), so its scalar and c_ES/Kum are not independent — fixing the ES frame fixes the comparison. Two become one exactly when the ES frame is pinned to the canonical quotient frame, which RUN-091 showed is canonical. That folding is a statement about the ES normalisation and is itself UNMEASURED."
    },
    "verdict": "UNMEASURED",
    "split": {
      "C4-membership": "CLOSED — automatic, all four local conditions computed vacuous",
      "C4-nonvanishing": "OPEN — no witness for the transverse class",
      "C4-exact-scalar": "OPEN — not attempted, by instruction"
    }
  },
  "minimal_map_still_missing_for_C1": "step 3 — the Rankin–Eisenstein/Beilinson–Flach class for the pair over the slope-1 Coleman family. Everything downstream of it is either built here or standard once it exists.",
  "c1_chain": {
    "question": "what is the minimal map still missing to carry the auxiliary object to 389.a1 and 19.a1?",
    "finding": "the auxiliary transverse extension does NOT transport to the partners by itself. It is a global object attached to g alone; B̂_i needs the Euler system class of the PAIR (E_i, g). So C4's object and C1's object sit on different paths and meet only at the normalisation.",
    "shortest_actual_chain": [
      "1. the auxiliary family g = E_k(1,χ₈) in the weight direction — BUILT (RUN-091)",
      "2. the partner newform f_i (389.a1 / 19.a1), ordinary at 11 — BUILT (RUN-091)",
      "3. the Rankin–Eisenstein / Beilinson–Flach class for the PAIR (f_i, g) over the family — MISSING",
      "4. its image under the Perrin-Riou / Coleman map: R_i(X,t) = c_f(X)·U_i(X,t) — follows from 3",
      "5. the X-leading term — standard once 4 exists",
      "6. projection through the canonical one-dimensional Eisenstein quotient — BUILT (RUN-091)",
      "7. B̂_i"
    ],
    "the_one_missing_map": "step 3 — the Rankin–Eisenstein/Beilinson–Flach class for the pair over the slope-1 Coleman family. Everything downstream of it is either built here or standard once it exists.",
    "which_of_the_two_routes": "neither of the two routes as stated. The chain does NOT run 'auxiliary generalised eigenspace → ES comparison → partner BF class', because the auxiliary eigenspace is partner-blind; and it does not need 'pseudocharacter → Galois rep → local quotient → ES → BF' either, since the pseudocharacter and the canonical quotient are already built. The BF class for the pair is the single input, and it is upstream of the ES comparison, not downstream of it.",
    "consequence_for_sequencing": "closing C4 does NOT unblock C1: they need different objects. C4 needs the θ-critical extension; C1 needs the pair's BF class. Either can be attacked first."
  },
  "obstruction_computed_this_round": {
    "B_2_chi8": "2",
    "L_minus_1_chi8": "-1",
    "v_11": 0,
    "no_classical_mod_11_eisenstein_congruence": true,
    "consequence": "the transverse class cannot be produced by a cuspidal congruence mod 11; if it is nonzero it must come from the θ-critical overconvergent direction",
    "label": "COMPUTED_HERE"
  },
  "claim_labels_legend": [
    "COMPUTED_HERE",
    "CITED_THEOREM",
    "DERIVED_FROM_VERIFIED_INPUTS",
    "UNMEASURED",
    "FAILED"
  ],
  "status": {
    "C1": "UNMEASURED — auxiliary side built; the pair's BF class is the one missing map",
    "C2": "UNMEASURED — valuation improved, exact unit unresolved",
    "C3": "PASS",
    "C4": "UNMEASURED — membership CLOSED, nonvanishing OPEN, scalar OPEN",
    "C5-C7": "BLOCKED BY C4",
    "C8": "PARTIAL",
    "C9-C10": "NOT YET READY",
    "BSD": "OPEN"
  }
}