`ε` can be taken rational (round six, `rat_eps`)

Proved

IntMul.Kappa.R6.rat_eps

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

complexity-theoryinteger-multiplicationkappa-improvement

ε can be taken rational: lowering ε to a nearby rational keeps P.Ok and κ < gap (all constraints are monotone or open in ε ≥ 1/2).

The following holds (Lean statement, in the scope of module Solutions.R6MFinal):

theorem IntMul.Kappa.R6.rat_eps (P : Prm) (hP : P.Ok) (κ : ℝ) (hκ : κ < P.gap) :
    ∃ q : ℚ, (({ P with ε := (q : ℝ) } : Prm)).Ok ∧ κ < ({ P with ε := (q : ℝ) } : Prm).gap

Context. Round six: the top-level assembly kappa_final6

The frozen KFM.kappa_final chain at the round-six parameters, with every stack piece not yet proved taken as one of the fixed interfaces of R6Ifaces.lean:

SetupNeed6 (§9.2 setup, sources), CrtNeed6 (Φ with the E′ CRT reversal), FineNeed6 (fine-bit exposure), LineNeed6 (the line maps), P14Need6 (the uniform Prop 14, P14UH), P18IL6 (Prop 18 from P14UH).
kappa_final6_at: for one parameter record with rational ε.
rat_eps: ε can be lowered to a rational one keeping P.Ok and κ < P.gap.
kappa_final6: the target.

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
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open Finset IntMul NLogN NLogN.KIL NLogN.KFB NLogN.KFT NLogN.KFU NLogN.RS IntMul.OpenAI.Glue IntMul.OpenAI.RCost NLogN.KFM

Formal statement

theorem IntMul.Kappa.R6.rat_eps (P : Prm) (hP : P.Ok) (κ : ℝ) (hκ : κ < P.gap) :
    ∃ q : ℚ, (({ P with ε := (q : ℝ) } : Prm)).Ok ∧ κ < ({ P with ε := (q : ℝ) } : Prm).gap := 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 `IntMul.Kappa.R6.rat_eps`, Solutions/R6MFinal.lean (lines 112–163) of the round-six development.

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