The read-out

The read-out.

The following holds (Lean statement, in the scope of module Solutions.KFMBig9):

theorem NLogN.KFM.fin_phase (C : BigCtx ix Isym extM n D) (t : IState HS) (hR : BaseOK t n) (x y : ℕ)
    (hx : x < 2 ^ n) (hy : y < 2 ^ n) (zsZ : List GI)
    (Z : ((i : Fin (D + 1)) → ZMod (sBD n D i)) → ℂ)
    (h0 : t.G 0 = ints (gflat zsZ)) (hZ : RepPad (pP n) (sBD n D) (tBD n D) zsZ Z)
    (hg : NLogN.KFB.GB (NLogN.KFM.wU n D) zsZ) (hcl : Close C x y Z) (hout : t.out = []) :
    ∃ s' c, IRuns Isym (kTable extM) (finP ix) t s' c ∧ s'.out = bin (2 * n) (x * y) ∧
      (c : ℝ) ≤ kFin C

Context. The lemma belongs to the part KFM Big9: the read-out: After the rounding passes (finA), Φ⁻¹; carries; write outputs bin(2n, x·y).

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.