The B̃₀ axis step

The B̃₀ axis step.

The following holds (Lean statement, in the scope of module Solutions.KFMBap0Ax):

theorem NLogN.AX.bap0AxP_all (M : MulSpec) (p s t α m kb Bi Bo mm Φ Bacc Q Bq : ℕ) (rest : List ℕ)
    (hM : MulBnd M mm Φ) (H : BapHyp p s t α kb Bi Bo mm Bacc Q) (HT : TabHypB p s t α Bq)
    (hα : 1 ≤ α) (HQ : AxQ t s m kb Bi Bo Q) (hmr : m = rest.prod)
    (st : IState HS) (L : List (ℤ × ℤ)) (Rest : List Item) (hR : BapRegs p s t α m kb st)
    (hD : st.G NLogN.GL.gD = gints L ++ Rest) (hL : L.length = t * m) (hLB : NLogN.GL.GB Bi L) :
    bap0AxP.run st = st.upd NLogN.GL.gD (gints (tensorGo [(s, b0L p s t α)] (t :: rest) L) ++ Rest) ∧
      bap0AxP.ok M st ∧ bap0AxP.cost M st ≤ bap0AxCost p s t α m kb Bi mm Φ Bacc Q Bq ∧
      (tensorGo [(s, b0L p s t α)] (t :: rest) L).length = m * s ∧
      NLogN.GL.GB Bo (tensorGo [(s, b0L p s t α)] (t :: rest) L)

Context. The lemma belongs to the part Item 12 (machine side, part 2): the B̃₀ axis program bap0AxP: nlogn's bapAxP (HAx8) with the slice program bapSlP replaced by the permutation-free bap0SlP (KFMBap0). The table building is unchanged; the two key tables (ptKR, psKR) are still built and removed but no longer read. That costs O((t+s)·Bq²) per axis call. On gD = gints L ++ R (L of dims t :: rest) it computes gints (tensorGo [(s, b0L p s t α)] (t :: rest) L) ++ R. The proof is nlogn's bapAxP_all with the slice lemmas renamed. The declarations live in NLogN.AX (as in HAx8) so that names resolve exactly as there.

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.