Round-six lemma `big_ev`

Proved

J6Seg.PreT.big_ev

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPre28):

theorem J6Seg.PreT.big_ev (K x : ℕ) : ∃ B₀ : ℕ, ∀ b, B₀ ≤ b →
    2 * ((K * b ^ 5 * (6 * b ^ (1 + x)) ^ 9) * (K * b ^ 5 * (6 * b ^ (1 + x)) ^ 9)) * b ^ (1 + x) ≤
      2 ^ (b + 1)

Context. B-line IL, pre-tables (B5′), part 28: the chunk store is within the per-axis budget

chunkStore_cost_ev: for widths A ≤ aA·Q + aC, M ≤ mA·Q + mC (Q = Q6 p d s), eventually cCS2 Q R A λ M nc ≤ cU6 P n A' / (D + 1) on every axis (Cp = 1, any A').

The ℕ core: cCS2 ≤ K b⁴ t p¹⁰ (cCS2_le, λ ≤ 4b², nc λ ≤ t), d ≤ b, and K b⁵ p⁹ t ≤ T because t² ≤ 2T (ℓ ≤ (L+1)/2 for d ≥ 2) and 2 (K b⁵ p⁹)² ≤ T (T b' ≥ 4n > 2^(b+1)).

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

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul NLogN IntMul.Kappa.R6

Formal statement

theorem J6Seg.PreT.big_ev (K x : ℕ) : ∃ B₀ : ℕ, ∀ b, B₀ ≤ b →
    2 * ((K * b ^ 5 * (6 * b ^ (1 + x)) ^ 9) * (K * b ^ 5 * (6 * b ^ (1 + x)) ^ 9)) * b ^ (1 + x) ≤
      2 ^ (b + 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.PreT.big_ev`, Solutions/J6SegILPre28.lean (lines 78–94) of the round-six development.

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