SwapNeed: uniform Lemma 12 on Gaussian arrays, in the merged table kTable with one extra procedure (swGU…

SwapNeed: uniform Lemma 12 on Gaussian arrays, in the merged table kTable with one extra procedure (swGU q).ren id (gL 1) (entry 7) and the descriptor on dS = 7.

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

theorem NLogN.KFW.swapNeed : KFM.SwapNeed

Context. The lemma belongs to the part KFW: uniform Lemma 12 on Gaussian arrays (SwapNeed): swGU q = descP ; swWU q ; popP: read the descriptor descSw w u p q sh on dS into the registers of swWU (descP), run the register-driven integer swap on the shape sh ++ [2], and pop the registers again. swGU_spec, and swapNeed : KFM.SwapNeed with the table kTable (fun _ => (swGU q).ren id (gL 1)) (one fixed prime q).

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs22.