The B̃₀ pass as a list pipeline (the list maps (List.range s).map (B0P …), which is KFMBap0.b0L by…

The B̃₀ pass as a list pipeline (the list maps (List.range s).map (B0P …), which is KFMBap0.b0L by b0L_eq_B0P) is tensorMapZ of the B0P maps, read on the output box.

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

theorem NLogN.KFM.b0_tensorGo (p α K : ℕ) (ts ss : List ℕ) (ht : ts.length = K) (hs : ss.length = K)
    (hpos : ∀ i < K, 0 < ts.getD i 1) (g : (Fin K → ℕ) → ℤ × ℤ) :
    Ref.tensorGo ((List.range K).map fun i => (ss.getD i 1, fun X : List (ℤ × ℤ) =>
        (List.range (ss.getD i 1)).map (B0P p (ss.getD i 1) (ts.getD i 1) α fun j =>
          X.getD j (0, 0)))) ts
      (Ref.listOf ts fun j => g fun i => j i) =
    Ref.listOf ss fun j =>
      tensorMapZ (fun i : Fin K => B0P p (ss.getD i 1) (ts.getD i 1) α) g fun i => j i

Context. The lemma belongs to the part Item 12 (math side, part 2): line maps applied axis by axis: The machine applies the per-axis fixed-point maps (Sapprox, B0P) slice by slice. That is the base tensorMapZ (via tensorGo_eq, KFMBaseTGo). This file identifies the exact counterpart, the base tensorMap of the lifted line maps liftL (L i), with the explicit tensor witness tensCLM (kerL L) of KFMR19. Then base fp_tensor gives the error of the whole pass.

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.