b0L_GB (Item 12 (machine side, part 1): the permutation-free `B̃₀` slice program)

The following holds (Lean statement, in the scope of module Solutions.KFMBap0):

theorem NLogN.KFM.b0L_GB (p s t α Bi Bo : ℕ) (hc : BD.cB1 p α (HvdH.jIter p s t α) * 2 ^ Bi ≤ 2 ^ Bo)
    (x : List (ℤ × ℤ)) (hx : NLogN.GL.GB Bi x) : NLogN.GL.GB Bo (b0L p s t α x)

Context. The lemma belongs to the part Item 12 (machine side, part 1): the permutation-free B̃₀ slice program.

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.