The opposite twist of round h (J_{-E}), uniform

The opposite twist of round h (J_{-E}), uniform.

The following holds (Lean statement, in the scope of module Solutions.KFV19Layer):

theorem NLogN.KFV.layerU_twO (hK : 0 < K) (hℓ : 1 ≤ ℓ) {h U W p : ℕ} (hh : h < ℓ)
    (hDU : D ≤ U) (hnU : (tgtL D ℓ K long).length ≤ U) (h2l : 3 * 2 ^ (ℓ + 1) ≤ U)
    (h2n : 2 ^ (tgtL D ℓ K long).length + 1 ≤ U) (hmU : D * h ≤ U) :
    LayerU (r := 2 ^ ℓ) I iprog (RCtx (pLP D ℓ K long) (flOf D long) h) (twistU true)
      (twLc (2 ^ ℓ) (binSh (tgtL D ℓ K long).length)
        (fun j => -roundEx (paperLayout D ℓ K long hK hℓ) ℓ h (partAxes (al D ℓ long) h) j))
      (2 ^ (tgtL D ℓ K long).length) p W
      (cTwistU (tgtL D ℓ K long).length ℓ D h
        (genG (pLP D ℓ K long) (flOf D long) true h).length W (zsz (U : ℤ)))

Context. The lemma belongs to the part KFV §23: uniform layers in a context: LayerU Ctx code T N p W cost: on any state encK s (o0 (gflat zs ++ l)) (only stack 0 managed) whose base s satisfies Ctx, code replaces the array by one representing T xs. The round's context (layout registers, flags) lives in the base, so a layer may call procedures that need every other managed stack empty (Proposition 14).

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.