SwapNeed → PhiNeed

SwapNeed → PhiNeed.

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

theorem NLogN.KFUU.phiNeed_of_swapNeed (hsw : KFM.SwapNeed) : KFM.PhiNeed

Context. The lemma belongs to the part KFUU §10: SwapNeed → PhiNeed: The table is KFM's merged table mTable ext₁ ∅ ∅ own (KFM.mExt): the SwapNeed extensions relabelled (IRuns.relabel via KFM.mTable_g1), then own = [phiFU, phiBU]. The swap entry of the reversals is the relabelled SwapNeed entry. Costs: cFwd, cBwd ≤ natCost + (d+1)·cSwV and natCost_le, swq_le absorb everything into (C₁ + 781 K) · cPhiU.

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.