The fine uniform scatter

The fine uniform scatter.

The following holds (Lean statement, in the scope of module Solutions.KFDScat):

theorem NLogN.KFD.mulPgU_fspec (t : IState HS) (xs : List ℤ) (P Q c B w b bL Bb : ℕ) (R : List Item)
    (hlen : xs.length = P * Q * B) (hb : Bnd w xs) (hc2 : 2 ≤ c)
    (hQc : Nat.Coprime Q c) (zc : zsz ((2 * c : ℕ) : ℤ) ≤ b) (zB : zsz (B : ℤ) ≤ bL)
    (zc' : zsz (c : ℤ) ≤ bL)
    (zQ : zsz (Q : ℤ) ≤ bL) (zQB : zsz ((Q * B : ℕ) : ℤ) ≤ bL) (zP : zsz (P : ℤ) ≤ Bb)
    (zQg : zsz (Q : ℤ) ≤ Bb) (zcg : zsz ((2 * c : ℕ) : ℤ) ≤ Bb)
    (zV : zsz ((P * (Q * B) : ℕ) : ℤ) ≤ Bb) (h0 : t.G 0 = ints xs ++ R)
    {LQ LB LA LV LQB : List Item}
    (hQs : t.G sQQ = .int Q :: LQ) (hBs : t.G sPC = .int B :: LB)
    (hAs : t.G sPA = .int P :: LA) (hVs : t.G sVV = .int ((P * (Q * B) : ℕ) : ℤ) :: LV)
    (hQBs : t.G sQB = .int ((Q * B : ℕ) : ℤ) :: LQB) :
    FSp (mulPgU c) t (t.upd 0 (ints (AModel.pull [P, Q, B] [P, Q, B] (mulAddr Q c) 0 xs) ++ R))
      (cMulPgUF P Q c B w b bL Bb)

Context. The lemma belongs to the part KFD: the fine scatter mulPgU (multiplication of the target field by c): Fine specifications of KIL3Scatter / KIL4Mul (programs unchanged), with the bounds w, b, bL, Bb of KFDPass.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.