A polynomial per-axis budget (round six, `budget_ev`)

Proved

J6Seg.PJ.budget_ev

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

complexity-theoryinteger-multiplicationkappa-improvement

A polynomial per-axis budget.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJC12):

theorem J6Seg.PJ.budget_ev (P : Prm) (hP : P.Ok) (K e : ℕ) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A' : ℕ,
      ((K * bP6 n ^ e * tF6 P n k ^ e * pP6 P n ^ e : ℕ) : ℝ) ≤ cU6 P n A' / (D + 1)

Context. B5 set-up, part 12: a polynomial per-axis budget

budget_ev K e: eventually K · b^e · t^e · p^e ≤ cU6 P n A' / (D + 1) on every axis (t^{e+1} ≤ 2^e T once d ≥ e + 1, and (K b^{e+1} p^e)^{e+1} 2^{e²} ≤ T).

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 Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_SApprox
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
import Definitions.Def_IntMul_Kappa_R6_Defs14

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul NLogN IntMul.Kappa.R6

Formal statement

theorem J6Seg.PJ.budget_ev (P : Prm) (hP : P.Ok) (K e : ℕ) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A' : ℕ,
      ((K * bP6 n ^ e * tF6 P n k ^ e * pP6 P n ^ e : ℕ) : ℝ) ≤ cU6 P n A' / (D + 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 `J6Seg.PJ.budget_ev`, Solutions/J6SegILPreJC12.lean (lines 23–128) of the round-six development.

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