fwd18_ok (KFV §51: the two layer programs of the Proposition 18 core)

The following holds (Lean statement, in the scope of module Solutions.KFV51Fwd):

theorem NLogN.KFV.fwd18_ok (hK : 0 < K) (hℓ : 2 ≤ ℓ) (hq1 : 1 ≤ qq ℓ K) {e14 eSw : Fin Pp} {Kc : ℝ}
    (HSw : SwOK I iprog eSw dW18 Kc) (hKc : 0 ≤ Kc) {p W Z B cCh U : ℕ}
    (HC : ChunkU D ℓ K long I iprog e14 hK (by omega) p W Z cCh) (hW : p + 1 ≤ W)
    (hB : (pLP D ℓ K long).n + ℓ + 5 ≤ B) (hNB : D * qq ℓ K + D + 1 ≤ B) (hWB : W ≤ B)
    (hbL : bL D ℓ K long ≤ B) (hU : TwB D ℓ K long U) :
    LayerU (r := 2 ^ ℓ) I iprog (Ctx18 D ℓ K long p W Z) (fwd18 e14 eSw)
      (procQc p (paperLayout D ℓ K long hK (by omega)) ℓ (partAxes (al D ℓ long))
          (paperSel D ℓ K long hK (by omega)) ℓ ∘
        pull (binSh (pLP D ℓ K long).n) (binSh (pLP D ℓ K long).n)
          (relabel (initL D ℓ long) (tgtL D ℓ K long)) 0)
      (2 ^ (pLP D ℓ K long).n) p W
      (cLayIn D ℓ K long Kc (pLP D ℓ K long).n W B + NLogN.KFV.cFwd D ℓ K long W B cCh U)

Context. The lemma belongs to the part KFV §51: the two layer programs of the Proposition 18 core: Both on the context Ctx18 (layout registers and a round frame), with the chunk rounds given by ChunkU (from the uniform Proposition 14) and the exchanges by SwOK (descriptor on stack 7).

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.