carW_spec (KFT §3: linear cost bounds, the `bin_{2n}` write, and carries-then-write)

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

theorem NLogN.KFT.carW_spec (b B : ℕ) (hb : 1 ≤ b) (zs : List GI) (hre : ∀ z ∈ zs, 0 ≤ z.1)
    (hB : NLogN.KFB.GB B zs) {W : Fin HS} (hW0 : W ≠ 0) (s : IState HS) (R0 LW : List Item) (m : ℕ)
    (hv : carOut b zs < 2 ^ m) (h0 : s.G 0 = ints (gflat zs) ++ R0)
    (hW : s.G W = .int (2 ^ m) :: LW) :
    IRuns I iprog (carW b zs.length W) s
      { (s.upd 0 R0).upd W LW with out := s.out ++ bin m (carOut b zs) }
      (cCar b B zs.length + outRCost m)

Context. The lemma belongs to the part KFT §3: linear cost bounds, the bin_{2n} write, and carries-then-write.

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.