Exact recovery: the real part of the computed final array rounds to the integer coefficients
ProvedNLogN.KFM.chain_roundExact recovery: the real part of the computed final array rounds to the integer coefficients.
The following holds (Lean statement, in the scope of module Solutions.KFMErr2):
theorem NLogN.KFM.chain_round {κ : Type*} {X : κ → ℂ} {c : κ → ℤ}
(h : ∀ j, ‖X j - (c j : ℂ)‖ < 1 / 2) (j : κ) : round (X j).re = c j
Context. The lemma belongs to the part KFM 14: the whole error chain and exact recovery (§9.7), abstract form.
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_Defs20.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs20
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
open NLogN
open NLogN.KFM
open Complex
variable {ι κ : Type*} [Fintype ι] [Fintype κ]
theorem NLogN.KFM.chain_round {κ : Type*} {X : κ → ℂ} {c : κ → ℤ}
(h : ∀ j, ‖X j - (c j : ℂ)‖ < 1 / 2) (j : κ) : round (X j).re = c j := by sorry