Lemma 13 discharges the hypotheses of KFN.prop14_il_complex_closed (m = 100^3): the row hypothesis with Ψ…

Lemma 13 discharges the hypotheses of KFN.prop14_il_complex_closed (m = 100^3): the row hypothesis with Ψ = PsiR r Z, CR = Kc (K+2) + 1, CR ≥ 0, Ψ monotone, Ψ ≥ Z+2.

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

theorem NLogN.KFR.closedHyps_of_lem13 {iprog : Fin 1 → IStmt HS 1} (H : Lem13ILP iprog) :
    ∃ Kc : ℝ, 0 < Kc ∧ ∀ (Isym : List Sym) {K r Z : ℕ}, 6 ≤ K → 1 ≤ r →
      80 * Z * (Z + 2) ≤ 2 ^ K →
      RowHyp Isym iprog K (100 ^ 3) r Z (PsiR r Z) (Kc * (K + 2) + 1) τ50 ∧
        0 ≤ Kc * (K + 2) + 1 ∧ Monotone (PsiR r Z) ∧ ∀ v, Z + 2 ≤ PsiR r Z v

Context. The lemma belongs to the part KFR: the bridge Lem13ILP iprog → RowHyp … τ50.

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.