toBitsI_spec (KFL §16: integers ↔ bits in IL, linear in the bit width)

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

theorem NLogN.KFL.toBitsI_spec (t : IState HS) (xs : List ℤ) (w : ℕ) (hxs : ∀ x ∈ xs, zsz x ≤ w)
    (R : List Item) (h0 : t.G 0 = ints xs ++ R) :
    IRuns I iprog (toBitsI xs.length w) t (t.upd 0 (ints (bitsL w xs) ++ R)) (cTo xs.length w)

Context. The lemma belongs to the part KFL §16: integers ↔ bits in IL, linear in the bit width: encW w z = sgn :: [bit 0, …, bit (w-1)] of |z| (zsz z ≤ w), bitsL w xs = xs.flatMap (encW w).

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.