Eval Consistency of rel Out star

theorem evalConsistency_of_relOut_star (hq5 : q % 8 = 5) {b ω γ : ℕ} (hκ : (2 * ω) ^ 2 < q)    (stmt : QuadEvalStatement 𝓜(q, α) innerRows (2 ^ m) messageDigits outerRows (2 ^ r) innerDigits      dRows)    (v : CarrierCom 𝓜(q, α) dRows)    (fam : Fin (2 ^ r + 1) → (Fin (2 ^ r) → ShortChallenge 𝓜(q, α) ω))    (resp : Fin (2 ^ r + 1) →      QuadEvalResponse 𝓜(q, α) innerRows (2 ^ m) messageDigits (2 ^ r) innerDigits zDigits)    (hrel : ∀ j, ((stmt, v, fam j), resp j) ∈ relOut 𝓜(q, α) (b : ZMod q) ω γ)    (hstar : ∃ e, StarAt fam e)    (hw : ∀ j, (resp j).carrierDec = (resp (central fam)).carrierDec) :    evalConsistency 𝓜(q, α) (b : ZMod q) stmt.avec stmt.bvec stmt.y      (extractedOpening 𝓜(q, α) (b : ZMod q) fam resp) := by  refine evalConsistency_of_star 𝓜(q, α) (b : ZMod q) stmt.avec stmt.bvec stmt.y    (extractedOpening 𝓜(q, α) (b : ZMod q) fam resp)    (gadgetMatrix 𝓜(q, α) (b : ZMod q) (2 ^ r) messageDigits *ᵥ (resp (central fam)).carrierDec)    ?_ ?_  · -- c3: verbatim c3 row of the central branch.    obtain ⟨-, -, hc3, -, -, -, -, -⟩ := hrel (central fam)    exact hc3  · -- c4 j: the coordinate-isolated, unit-divided c4-subtract chain.    intro j    obtain ⟨-, -, -, hc4s, -, -, -, -⟩ := hrel (sib fam j)    obtain ⟨-, -, -, hc4e, -, -, -, -⟩ := hrel (central fam)    rw [hw (sib fam j)] at hc4s    have hunit := slack_isUnit hq5 hκ fam hstar j    have hcoord : CoordEq j (fun i => (fam (sib fam j) i).val)        (fun i => (fam (central fam) i).val) :=      ShortChallenge.coordEq_val 𝓜(q, α) (coordEq_symm (sib_coordEq fam hstar j))    -- Subtract-and-isolate: the two branches' c4 rows, sharing `ŵ`, give `c̄ⱼ · wⱼ = aᵀ G Δzⱼ`.    have hchain : dot stmt.avec (gadgetMatrix 𝓜(q, α) (b : ZMod q) (2 ^ m) messageDigits *ᵥ          ((Hachi.jMatrix 𝓜(q, α) (b : ZMod q) ((2 ^ m) * messageDigits) zDigits *ᵥ              (resp (sib fam j)).zDec)           - (Hachi.jMatrix 𝓜(q, α) (b : ZMod q) ((2 ^ m) * messageDigits) zDigits *ᵥ              (resp (central fam)).zDec)))        = ((fam (sib fam j) j).val - (fam (central fam) j).val) *            (gadgetMatrix 𝓜(q, α) (b : ZMod q) (2 ^ r) messageDigits *ᵥ              (resp (central fam)).carrierDec) j := by      rw [matVecMul_sub, dot_sub, ← hc4s, ← hc4e, ← Hachi.tensorG1_sub_challenge,        Hachi.tensorG1_coord_diff 𝓜(q, α) (b : ZMod q) messageDigits hcoord]    -- Divide by `c̄ⱼ`: the extracted message `sⱼ := c̄ⱼ⁻¹ •ᵥ Δzⱼ` recovers `wⱼ` under `aᵀ G`.    change (gadgetMatrix 𝓜(q, α) (b : ZMod q) (2 ^ r) messageDigits *ᵥ          (resp (central fam)).carrierDec) j        = dot stmt.avec (gadgetMatrix 𝓜(q, α) (b : ZMod q) (2 ^ m) messageDigits *ᵥ            (Ring.inverse ((fam (sib fam j) j).val - (fam (central fam) j).val) •ᵥ              ((Hachi.jMatrix 𝓜(q, α) (b : ZMod q) ((2 ^ m) * messageDigits) zDigits *ᵥ                  (resp (sib fam j)).zDec)               - (Hachi.jMatrix 𝓜(q, α) (b : ZMod q) ((2 ^ m) * messageDigits) zDigits *ᵥ                  (resp (central fam)).zDec))))    rw [matVecMul_scalarVecMul, dot_scalarVecMul, hchain, ← mul_assoc,      Ring.inverse_mul_cancel _ hunit, one_mul]