rotF_spec (KFC: the self-contained fine-cost block rotation `rotF`)

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

theorem NLogN.KFC.rotF_spec (t : IState HS) (xs : List ℤ) (as : ℕ → ℕ) (P Q B w : ℕ)
    (R L1 L2 L4 L14 : List Item) (has : ∀ p, as p ≤ Q)
    (hlen : xs.length = P * Q * B) (hb : Bnd w xs)
    (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) :
    IRuns I iprog rotF t
      ((t.upd 0 (ints (AModel.pull [P, Q, B] [P, Q, B] (KIL.rotAddr Q as) 0 xs) ++ R)).upd 14 L14)
      (NLogN.KFC.cRotF P Q B w)

Context. The lemma belongs to the part KFC: the self-contained fine-cost block rotation rotF: rotF_spec: the same result as KIL.rotP_spec ((t.upd 0 (ints (pull [P,Q,B] [P,Q,B] (rotAddr Q as) 0 xs) ++ R)).upd 14 L14), with cost cRotF P Q B w ≤ 68·kap w·(P·Q·B) (cRotF_le), w a bound on the bit size of the entries.

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.