Uniform pointwise Gaussian product pass in the merged table, with FM products

Uniform pointwise Gaussian product pass in the merged table, with FM products.

The following holds (Lean statement, in the scope of module Solutions.KFMU1):

theorem NLogN.KFM.gmulPU_kTable {E : ℕ} (ext : Fin E → IStmt HS (7 + E)) {I : List Sym} (w Δ : ℕ)
    (s : IState HS) (zs ts : List GI) (R0 RW LR : List Item) (hlen : ts.length = zs.length)
    (h0 : s.G 0 = ints (gflat zs) ++ R0) (hW : s.G pW = ints (gflat ts) ++ RW)
    (hR : s.G pR = .int (2 ^ Δ) :: LR) (hz : NLogN.KFB.GB w zs) (ht : NLogN.KFB.GB w ts) (LX LX2 : List Item)
    (hX : s.G nX = .int (zs.length : ℤ) :: LX) (hX2 : s.G nX2 = .int ((2 * zs.length : ℕ) : ℤ) :: LX2) :
    IRuns I (kTable ext) (gmulPU (pMul E)) s
      ((s.upd 0 (ints (gflat (List.zipWith (gmulT Δ) zs ts)) ++ R0)).upd pW RW)
      (zs.length * (4 * fmB (2 * w)) + 10 ^ 6 * (zs.length + 1) * (w + Δ + 1) +
        2 * cCnt (2 * zs.length))

Context. The lemma belongs to the part KFM U1: uniform counted repetition and uniform pointwise passes: UNIFORM.md: the count of a repetition is run-time data. rptX X A C D b is KFT's rpt with the literal count replaced by a copy (dup) of the top of stack X.

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.