Sharp rounding of a near-unit value (round six, `Qp_unit`)

Proved

NLogN.KFZ.Qp_unit

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Sharp rounding of a near-unit value.

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

theorem NLogN.KFZ.Qp_unit (p : ℕ) {v X : ℂ} (hX : ‖X‖ ≤ 1) (he : ‖v - X‖ ≤ (2 : ℝ)⁻¹ ^ (2 * p + 2)) :
    ‖Butterfly.Qp p v - X‖ ≤ 2 * hp p ∧ ‖Butterfly.Qp p v‖ ≤ 1

Context. KFZ Phase: the rounded phase tables satisfy Phases

The tables are computed at precision Δ = p + g and rounded to precision p (KFH.gtr g). Phases asks for |θ̃ − θ| ≤ 2·2^{−p} and |θ̃| ≤ 1. The second needs more than the generic gtr_err: a grid point of 2^{−p}ℤ[i] of modulus above 1 has modulus at least √(1 + 4^{−p}), so the table error before rounding must be below about 4^{−p}.

Qp_unit: ‖v − X‖ ≤ 2^{−2p−2}, ‖X‖ ≤ 1 ⟹ ‖Q_p v − X‖ ≤ 2hp and ‖Q_p v‖ ≤ 1.
phases_QA: the array form.
norm_chirp_le, norm_twX, zeta_pow: the exact phases.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFZ
open IntMul NLogN.KIL NLogN.KFB NLogN.KFM Complex

Formal statement

theorem NLogN.KFZ.Qp_unit (p : ℕ) {v X : ℂ} (hX : ‖X‖ ≤ 1) (he : ‖v - X‖ ≤ (2 : ℝ)⁻¹ ^ (2 * p + 2)) :
    ‖Butterfly.Qp p v - X‖ ≤ 2 * hp p ∧ ‖Butterfly.Qp p v‖ ≤ 1 := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `NLogN.KFZ.Qp_unit`, Solutions/KFZPhase.lean (lines 38–98) of the round-six development.

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