gsw_iruns (KFB: swapping two address bits of a Gaussian array (KFC field moves))

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

theorem NLogN.KFB.gsw_iruns {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    (s : IState HS) (wb W K P Q : ℕ) (hW : 1 ≤ W) (hPQ : P < Q) (hQK : Q < K)
    (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ) (hl : zs.length = W * 2 ^ K)
    (h0 : f 0 = gflat zs ++ l) (hb : NLogN.KFB.GB wb zs) :
    IRuns I iprog (gswP W K P Q) (encK s f)
      (encK s (Function.update f 0 (gflat (gswL P Q zs) ++ l))) (cGsw W K P Q wb)

Context. The lemma belongs to the part KFB: swapping two address bits of a Gaussian array (KFC field moves): With q = 2, KFD's fine field interchange swapEF (one fwdF and one bwdF, linear cost) swaps two binary fields of the integer array gflat zs. Viewing the Gaussian array of W · 2^K entries as the shape [W 2^{K-Q-1}, 2, 2^{Q-P-1}, 2, 2^{P+1}] of integers, the two binary fields are the index bits Q > P.

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_Defs18.