Xf_le (KFM 4c: the closed per-round bound (MACHINE-PLAN item 4, §5 (C)))

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

theorem NLogN.KFM.Xf_le (r Z v w : ℕ) (hv : v ≤ 2 * w + 1) (hvZ : v + 1 < 2 ^ Z)
    (hr : 2 * (2 * Z + 4) ^ 3 + 2 ≤ r) : Xf r (PsiR r Z) v ≤ 6 * r * (w + 1)

Context. The lemma belongs to the part KFM 4c: the closed per-round bound (MACHINE-PLAN item 4, §5 (C)): The cost of one Proposition 14 round in the merged table (prop14Ph_kTable) is kC w + cWrapPh …, with the row constant F1 = 2((k+1)m(10^5 + k a + 2 k b CR) D^{τ50} + 1) + 10^6(71k+2), k = ⌊log_m D⌋, m = 100^3, CR = Kc(K+2)+1, and the width w' = w + Ew(D) inside Xf r Ψ.

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.