Lemma 10 shift with any fixed unit coefficient u = ±k·d⁻¹ (k, d ≥ 1 prime to Q, reduced below Q), IL program

Lemma 10 shift with any fixed unit coefficient u = ±k·d⁻¹ (k, d ≥ 1 prime to Q, reduced below Q), IL program.

The following holds (Lean statement, in the scope of module Solutions.KIL3Unit):

theorem NLogN.KIL.shiftArr_il_unit [NeZero Q] (i : Fin n) (c : Fin n → ZMod Q)
    (hc : ∀ j, i ≤ j → c j = 0) (A C : Shape) (hsh : sh = A ++ Q :: C)
    (hA : (Λ.pos i : ℕ) = A.length) (k d : ℕ) (neg : Bool) (hk : 1 ≤ k) (hkQ : k ≤ Q)
    (hd : 1 ≤ d) (hdQ : d ≤ Q) (hQk : Nat.Coprime Q k) (hQd : Nat.Coprime Q d)
    (t : IState HS) (xs : List ℤ) (Bb : ℕ) (R : List Item)
    (hlen : xs.length = AModel.vol sh) (hb : Bnd Bb xs) (hB1 : 1 ≤ AModel.vol C) (hQ1 : 1 ≤ Q)
    (hP1 : 1 ≤ AModel.vol A) (hV : zsz ((AModel.vol A * (Q * AModel.vol C) : ℕ) : ℤ) ≤ Bb)
    (hQ2 : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb) (h0 : t.G 0 = ints xs ++ R) :
    Sp (shFrac Q k d (AModel.vol A) (AModel.vol C) neg (A.zip (csOf Λ c A))) t
      (t.upd 0 (ints (shiftArr Λ i (fracU Q k d neg) c 0 xs) ++ R))
      (cShiftF k d A.length (AModel.vol A * (Q * AModel.vol C)) Bb)

Context. The lemma belongs to the part KIL3: the Lemma 10 controlled shift with any fixed unit u = ±n·d⁻¹: For fixed naturals n, d ≥ 1 prime to Q, shiftArr Λ i (±n·d⁻¹) c is (rotation / reflection) ∘ (y ↦ n·y) ∘ (y ↦ d⁻¹·y) on [vol A, Q, vol C] (shiftArr_frac_eq, shiftArr_negfrac_eq); the two multiplicative permutations are the pass programs divPg (KIL3Gather) and mulPg (KIL3Scatter), skipped when the factor is 1.

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.