{
  "V": "Q₁₁(χ₈)(1)",
  "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": {
    "frobenius_values": {
      "3": {
        "frobenius_value": -3,
        "equals_one": false
      },
      "5": {
        "frobenius_value": -5,
        "equals_one": false
      },
      "7": {
        "frobenius_value": 7,
        "equals_one": false
      },
      "13": {
        "frobenius_value": -13,
        "equals_one": false
      },
      "17": {
        "frobenius_value": 17,
        "equals_one": false
      },
      "19": {
        "frobenius_value": -19,
        "equals_one": false
      },
      "23": {
        "frobenius_value": 23,
        "equals_one": false
      },
      "29": {
        "frobenius_value": -29,
        "equals_one": false
      },
      "31": {
        "frobenius_value": 31,
        "equals_one": false
      },
      "37": {
        "frobenius_value": -37,
        "equals_one": false
      },
      "41": {
        "frobenius_value": 41,
        "equals_one": false
      },
      "43": {
        "frobenius_value": -43,
        "equals_one": false
      }
    },
    "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,
  "conclusion": "H¹_f(Q,V) = H¹(G_{Q,{2,11,∞}}, V): a class unramified outside 2·11 is automatically in H¹_f. The extension is such a class by construction, so STEP 2 IS SETTLED.",
  "global_euler_characteristic": {
    "chi": -1,
    "H0": 0,
    "relation": "dim H¹ = 1 + dim H²",
    "with_units_dim_H1_f_equals_1": "forces H² = 0 — consistent"
  },
  "cited": [
    "local Euler characteristic formulas",
    "local Tate duality",
    "Bloch–Kato: dim H¹_f(Q_p,V) = dim H⁰ + dim D_dR/Fil⁰",
    "global Euler characteristic formula"
  ],
  "agrees": true
}