Lemma 21 (36) with kernel a = chirp and endpoint phases ā

Lemma 21 (36) with kernel a = chirp and endpoint phases ā.

The following holds (Lean statement, in the scope of module Solutions.KFMErr3):

theorem NLogN.KFM.chirp_exact (s t : Fin d → ℕ) [∀ i, NeZero (t i)] (ht : ∀ i, Even (t i))
    (x : ((i : Fin d) → ZMod (t i)) → ℂ) :
    (fun j => (starRingEnd ℂ) (chirp s t j)) *
      (fun j => (∏ i, (t i : ℂ))⁻¹ * cconv t (chirp s t)
        ((fun j => (starRingEnd ℂ) (chirp s t j)) * x) j) = Qperm s t (dftM t x)

Context. The lemma belongs to the part KFM 14: the exact side of the error chain: The exact maps of §9.4–9.7 as LinC / BilC instances, and the identities chain_close needs.

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_Defs20.