rotP computes the Lemma 10 controlled shift, given the offset table offs Λ c A p (p < vol A) on stack 14

rotP computes the Lemma 10 controlled shift, given the offset table offs Λ c A p (p < vol A) on stack 14.

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

theorem NLogN.KIL.rotP_shift_spec [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) (t : IState HS) (xs : List ℤ) (Bb : ℕ)
    (R L1 L2 L4 L14 : 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)
    (h0 : t.G 0 = ints xs ++ R) (h1 : t.G 1 = .int (AModel.vol A) :: L1) (h2 : t.G 2 = .int Q :: L2)
    (h4 : t.G 4 = .int (AModel.vol C) :: L4)
    (h14 : t.G 14 = ints ((List.range (AModel.vol A)).map fun p => (offs Λ c A p : ℤ)) ++ L14) :
    rotP.ok t ∧ rotP.run t = (t.upd 0 (ints (shiftArr Λ i 1 c 0 xs) ++ R)).upd 14 L14 ∧
      rotP.cost t ≤ 200000 * (AModel.vol A * (Q * AModel.vol C) + 1) * (Bb + 2) + 10 ^ 8 * (Bb + 2) ^ 2

Context. The lemma belongs to the part The controlled shift of Lemma 10 as a block rotation: For a layout Λ on sh = A ++ Q :: C with the target field i at position |A|, unit coefficient u = 1 and controls only before the target (c j = 0 for j ≥ i), the controlled shift shiftArr Λ i 1 c is the block rotation pull [vol A, Q, vol C] [vol A, Q, vol C] (rotAddr Q (offs Λ c A)) where block p (the prefix digits) is rotated by offs Λ c A p = (Σ_j c_j · digit_j(decode A p)).val.

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.