Base state of rotP (stack form, as KIL's rotP_spec)

Base state of rotP (stack form, as KIL's rotP_spec).

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

theorem NLogN.KFD.rotP_fspec (t : IState HS) (xs : List ℤ) (as : ℕ → ℕ) (P Q B w Bb : ℕ)
    (R L1 L2 L4 L14 : List Item) (has : ∀ p, as p ≤ Q)
    (hlen : xs.length = P * Q * B) (hb : Bnd w 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) :
    FSp rotP t ((t.upd 0 (ints (AModel.pull [P, Q, B] [P, Q, B] (rotAddr Q as) 0 xs) ++ R)).upd 14 L14)
      (10 ^ 6 * (P * (Q * B) + 1) * (w + 2) + 10 ^ 8 * (Bb + 2) ^ 2)

Context. The lemma belongs to the part KFD: the fine block rotation rotP: Fine specification of KILRot's rotP (program unchanged). Every record move is a fine mvK (O(B·(w+2)) for a record of B entries of size ≤ w), every per-block number (Q, a_p, Q - a_p) is bounded by Q itself (zsz Q ≤ Q), so a block of Q records costs O(Q·B·(w+2)); one multiplication P·Q·B per call costs O((Bb+2)^2).

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.