{
  "quantity": "Π_E/Π_0, the adjoint Petersson/period ratio, with D_i = Π_i·E_i (Attack 09 (14)–(15))",
  "cited_theorem": "Hida: the adjoint p-adic L-value L_p(Ad f)(0) generates the congruence ideal of f, so v₁₁(Π_i) = v₁₁(η_{f_i}). CITED_THEOREM — it pins the ideal, not the value.",
  "congruence_389": {
    "level": 389,
    "dual_manin_dimension": 65,
    "genus": 32,
    "hecke_primes": [
      2,
      3,
      5
    ],
    "a_q_mod_11": {
      "2": 9,
      "3": 9,
      "5": 8
    },
    "per_prime": {
      "2": {
        "a_q": 9,
        "dim_ker": 2,
        "dim_ker_power": 2
      },
      "3": {
        "a_q": 9,
        "dim_ker": 2,
        "dim_ker_power": 2
      },
      "5": {
        "a_q": 8,
        "dim_ker": 2,
        "dim_ker_power": 2
      }
    },
    "nilpotency_power": 8,
    "ordinary_eigenspace_dim": 2,
    "generalised_eigenspace_dim": 2,
    "equal": true,
    "eigensystem_is_eisenstein": false,
    "congruence_free": true,
    "v_11_eta": 0,
    "agrees": true
  },
  "congruence_19": {
    "level": 19,
    "dual_manin_dimension": 3,
    "cusp_dimension": 2,
    "genus": 1,
    "dim_S2": 1,
    "reason": "dim S₂(Γ₀(19)) = 1: a single newform, so no cusp form can be congruent to it",
    "v_11_eta": 0,
    "agrees": true
  },
  "v_11_Pi_E": 0,
  "v_11_Pi_0": 0,
  "v_11_ratio": 0,
  "conclusion": "Π_E/Π_0 is an 11-adic UNIT",
  "consequence": "A_0(0)/A_E(0) ≡ 6·(Π_E/Π_0) (mod 11) is a nonzero unit — the partner ratio neither vanishes nor blows up mod 11, which is what PC-001's non-vanishing needs",
  "residue_mod_11": null,
  "residue_label": "UNMEASURED",
  "why_residue_unmeasured": "Hida's theorem identifies the congruence IDEAL; recovering the residue needs the actual Petersson norms against a stated period — the normalisation PC-001 §3 declines to fix ('其完整 period 比值 D_0 沒有在本輪算出'), plus the Manin constants and the optimality of 19.a1 = Cremona 19a2 (NOT the X₀-optimal curve of its class)",
  "valuation_label": "COMPUTED_HERE",
  "agrees": true
}