§9.5.1 with explicit witnesses: the source transform through the chirp identity

§9.5.1 with explicit witnesses: the source transform through the chirp identity.

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

theorem NLogN.KFM.source_transform_base {p α : ℕ} {s t : Fin d → ℕ} [∀ i, NeZero (s i)]
    [∀ i, NeZero (t i)] (M : AxisMaps (d := d) p α s t) (ht : ∀ i, Even (t i)) (u) :
    Rperm s t (dftM s u) = (2 : ℂ) ^ (2 * d * α ^ 2) • tensCLM (kerB M.B) (fun j =>
      (∏ i, (t i : ℂ))⁻¹ * ((starRingEnd ℂ) (IntMul.OpenAI.RC.chirp s t j) *
        ∑ k, IntMul.OpenAI.RC.chirp s t (j - k) *
          ((starRingEnd ℂ) (IntMul.OpenAI.RC.chirp s t k) * tensCLM (kerA M.A) u k)))

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.