tensor_of_lines with the explicit witnesses tensCLM (kerA A), tensCLM (kerB B)

tensor_of_lines with the explicit witnesses tensCLM (kerA A), tensCLM (kerB B).

The following holds (Lean statement, in the scope of module Solutions.KFMR19):

theorem NLogN.KFM.tensor_of_lines_expl (s t : Fin d → ℕ) [∀ i, NeZero (s i)] [∀ i, NeZero (t i)] (c : ℂ)
    (A : (i : Fin d) → (ZMod (s i) → ℂ) →L[ℂ] (ZMod (t i) → ℂ))
    (B : (i : Fin d) → (ZMod (t i) → ℂ) →L[ℂ] (ZMod (s i) → ℂ))
    (hA : ∀ i, ‖A i‖ ≤ 1) (hB : ∀ i, ‖B i‖ ≤ 1)
    (h : ∀ i u, IntMul.OpenAI.RH.permS (s i) (t i) ((s i : ℂ)⁻¹ • ZMod.dft u) =
      c • B i (IntMul.OpenAI.RH.permT (s i) (t i) ((t i : ℂ)⁻¹ • ZMod.dft (A i u)))) :
    ‖tensCLM (kerA A)‖ ≤ 1 ∧ ‖tensCLM (kerB B)‖ ≤ 1 ∧
      ∀ u, Rperm s t (dftM s u) =
        c ^ d • tensCLM (kerB B) (Qperm s t (dftM t (tensCLM (kerA A) u)))

Context. The lemma belongs to the part Item 12 (math side): Lemma 19/20 with the base witnesses and a permutation-free B̃₀: The resampling identities of Resample19/ResampleTensor/ResampleFinal are existential in the maps A, B₀. The machine computes the base thread's fixed-point maps Sapprox and Bapprox. This file pins the witnesses to those maps.

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.