The uniform twiddle pass (twC_spec with N, 2r, 2r on qX, qB, qZ)

The uniform twiddle pass (twC_spec with N, 2r, 2r on qX, qB, qZ).

The following holds (Lean statement, in the scope of module Solutions.KFV15Tw):

theorem NLogN.KFV.twCU_runs (N r w : ℕ) (hr : 0 < r) (sv : ℕ → ℕ) (hsv : ∀ p, sv p ≤ 2 * r)
    (zs : List GI) (hzs : zs.length = N * r) (hb : NLogN.KFB.GB w zs) (t : IState HS) (R L14 : List Item)
    (h0 : t.G 0 = ints (gflat zs) ++ R)
    (h14 : t.G 14 = ints ((List.range N).map fun p => (sv p : ℤ)) ++ L14)
    {L1 L2 L3 : List Item} (h1 : t.G NLogN.KFUU.qX = .int (N : ℤ) :: L1)
    (h2 : t.G NLogN.KFUU.qB = .int ((2 * r : ℕ) : ℤ) :: L2) (h3 : t.G NLogN.KFUU.qZ = .int ((2 * r : ℕ) : ℤ) :: L3) :
    IRuns I iprog twCU t ((t.upd 0 (ints (gflat (twZ r N sv zs)) ++ R)).upd 14 L14)
      (2000 * cTw N r w)

Context. The lemma belongs to the part KFV §19: the uniform twiddle pass twCU: KFV.twC N r = dblI N (2r) ; rotC N (2r) ; unpadI N (2r) (4r) carries the literals N and 2r. twCU reads them from the parameter stacks qX = N, qB = 2r, qZ = 2r (KFUU.π3): dblIU (register-counted doubling), rotCU (const 1 N, const 2 Q become dups), and KFUU's unpadIU. twCU_runs: twC_spec transported by KFV.ISim.runs.

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.