vol_ctB (KFUU §2f: the layouts of phase 2 (swaps of the main fields))

The following holds (Lean statement, in the scope of module Solutions.KFUURev6):

theorem NLogN.KFUU.vol_ctB (u cz : ℕ) (bs : List Bool) (hcz : 1 ≤ cz) :
    (2 ^ u) ^ bs.length * vol (ctB u cz bs) = vol (shP u bs)

Context. The lemma belongs to the part KFUU §2f: the layouts of phase 2 (swaps of the main fields): After phase 1 the layout is lay1 bg n 0 = mains ++ bitsR ++ [2n], of field sizes 2^u (n times) ++ ctB ++ [cz]. Swap k exchanges the main fields k and n-1-k; on the data it is the 5-field swap [U^k, U, U^{n-2-2k}, U, U^k·vol ctB·cz], U = 2^u.

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.