bapP0 computes b0L

bapP0 computes b0L.

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

theorem NLogN.KFM.bapP0_run (p s t α kb : ℕ) (hs : 0 < s) (hst : s ≤ t) (V : List (ℤ × ℤ))
    (hV : V.length = t) (s0 : IState HS) (RU : List Item) (hU : s0.G CU = gints V ++ RU)
    (hP : BParams s0 p s t α kb) :
    bapP0.run s0 = s0.upd CU (gints (b0L p s t α V) ++ RU)

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.