chirpArr_near (KFM 9d: the `d`-axis chirp array — outer products of the axis tables)
ProvedNLogN.KFM.chirpArr_nearThe following holds (Lean statement, in the scope of module Solutions.KFMTabQ):
theorem NLogN.KFM.chirpArr_near (Δ T : ℕ) : ∀ (d : ℕ) (s t : Fin d → ℕ), (∀ i, s i ≤ t i) →
(∀ i, 0 < t i) → (∀ i, t i ≤ T) → d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ ≤ 1 →
∀ x ∈ chirpArr Δ d s t, NearU Δ (d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ) x
Context. The lemma belongs to the part KFM 9d: the d-axis chirp array — outer products of the axis tables.
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 (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine -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.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_K_Defs22 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 set_option linter.unusedSimpArgs false set_option linter.unusedVariables false set_option linter.unusedSectionVars false set_option linter.unnecessarySeqFocus false set_option maxRecDepth 20000 open NLogN open NLogN.KFM open IntMul KFC KFB KFT KIL
theorem NLogN.KFM.chirpArr_near (Δ T : ℕ) : ∀ (d : ℕ) (s t : Fin d → ℕ), (∀ i, s i ≤ t i) →
(∀ i, 0 < t i) → (∀ i, t i ≤ T) → d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ ≤ 1 →
∀ x ∈ chirpArr Δ d s t, NearU Δ (d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ) x := by sorry