VS_eq (KFUU §9b: the uniform `Φ`)

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

theorem NLogN.KFUU.VS_eq (d : ℕ) (sF : ℕ → ℕ) : vol (NLogN.KFU.shR d (NLogN.KFU.sP d sF)) = Pv sF d * 2

Context. The lemma belongs to the part KFUU §9b: the uniform Φ: phiFU: read the descriptor (readD), build the shape lists (mkL), pad S_k → T_k, reverse (revU, the SwapNeed entry), unpad T_r → S_r, the d forward shifts (rotsFU), pad S_r → T_r, clear. phiFU_runs: on the managed state of a PhiNeed call it computes KFU.PhiD.

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.