The uniform level-0 code computes the digit interchange of P :: L :: swShape A B Cs q for every descriptor…

The uniform level-0 code computes the digit interchange of P :: L :: swShape A B Cs q for every descriptor at level 0.

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

theorem NLogN.KFD.lev0F_runs {q m P L : ℕ} {A B Cs : Shape} (t : IState HS) (xs : List ℤ) (R : List Item)
    (Bb w : ℕ) (hD : DescIn t q m 0 P L A B Cs) (h0 : t.G 0 = ints xs ++ R)
    (hpos : ∀ a ∈ P :: L :: swShape A B Cs q, 1 ≤ a)
    (hlen : xs.length = vol (P :: L :: swShape A B Cs q)) (hb : Bnd w xs)
    (hV : zsz ((vol (P :: L :: swShape A B Cs q) : ℕ) : ℤ) ≤ Bb) :
    IRuns I iprog lev0F t
      (t.upd 0 (ints (AModel.pull (P :: L :: swShape A B Cs q) (P :: L :: swShape A B Cs q)
        (swapPos (A.length + 2) (A.length + B.length + 3)) 0 xs) ++ R))
      (cLev0F q (vol (P :: L :: swShape A B Cs q)) Bb w)

Context. The lemma belongs to the part KFD: the fine uniform Prop. 11 procedure.

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.