Controlled shift of one field (block rotation)

Controlled shift of one field (block rotation). With the array xs (shape [P, Q, B]) on stack 0, P, Q, B on stacks 1, 2, 4 and the offsets a_0, …, a_{P-1} (a_p ≤ Q) on stack 14, rotP replaces xs by pull [P, Q, B] [P, Q, B] (rotAddr Q as) 0 xs (record y of block p moves to (y + a_p) mod Q), pops the offsets and restores every other stack; cost O((V + 1)(Bb + 2) + (Bb + 2)^2).

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

theorem NLogN.KIL.rotP_spec (t : IState HS) (xs : List ℤ) (as : ℕ → ℕ) (P Q B Bb : ℕ)
    (R L1 L2 L4 L14 : List Item) (has : ∀ p, as p ≤ Q)
    (hlen : xs.length = P * Q * B) (hb : Bnd Bb xs) (hB1 : 1 ≤ B) (hQ1 : 1 ≤ Q) (hP1 : 1 ≤ P)
    (hV : zsz ((P * (Q * B) : ℕ) : ℤ) ≤ Bb)
    (h0 : t.G 0 = ints xs ++ R) (h1 : t.G 1 = .int P :: L1) (h2 : t.G 2 = .int Q :: L2)
    (h4 : t.G 4 = .int B :: L4)
    (h14 : t.G 14 = ints ((List.range P).map fun p => (as p : ℤ)) ++ L14) :
    rotP.ok t ∧ rotP.run t =
        (t.upd 0 (ints (AModel.pull [P, Q, B] [P, Q, B] (rotAddr Q as) 0 xs) ++ R)).upd 14 L14 ∧
      rotP.cost t ≤ 200000 * (P * (Q * B) + 1) * (Bb + 2) + 10 ^ 8 * (Bb + 2) ^ 2

Context. The lemma belongs to the part Primitive (d): controlled shift as a rotation of blocks: rotP reads an array on [P, Q, B] (stack 0) and a list of offsets a_p < Q (stack 14, consumed) and rotates the Q records of block p cyclically by a_p: record y of block p moves to (y + a_p) mod Q. Cost O(V + P · Q) item operations.

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.