Bapprox_eq (The permutation-free `B̃₀ = D̃' J̃' C̃`)

The following holds (Lean statement, in the scope of module Solutions.KFMB0):

theorem NLogN.KFM.Bapprox_eq (p s t α : ℕ) (V : ℕ → ℤ × ℤ) :
    IntMul.HvdH.Bapprox p s t α V =
      IntMul.HvdH.psInvPerm s t (B0approx p s t α (IntMul.HvdH.ptPerm s t V))

Context. The lemma belongs to the part The permutation-free B̃₀ = D̃' J̃' C̃: B0approx is the base Bapprox (HvdH Proposition 4.7(ii)) without its two data permutations P̃_t and P̃_s⁻¹: Bapprox = P̃_s⁻¹ ∘ B0approx ∘ P̃_t (Bapprox_eq). Lemma 19 (κ, §8.1) keeps both permutations in the transform, so the machine applies B0approx (see KFMR19, KFMBap0).

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.