Slot swap `i` on the geometry (round six, `swE_geo`)

Proved

NLogN.KFX.swE_geo

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

complexity-theoryinteger-multiplicationkappa-improvement

Slot swap i on the geometry with expression parameters.

The following holds (Lean statement, in the scope of module Solutions.KFX1Sw):

theorem NLogN.KFX.swE_geo :
    ∃ T q₀ : ℕ, ∀ q, q.Prime → q₀ < q → ∃ K : ℝ, 0 ≤ K ∧
      ∀ (i : Fin 3) (g : Geo), g.Pos → ∀ u : ℕ, 1 ≤ u → g.Q = 2 ^ u →
      ∀ (I : List Sym) (t : IState HS) (zs : List ℤ) (R : List Item),
      KFL.Bits zs → t.G 0 = ints zs ++ R → zs.length = g.V →
      ∀ (eP eG eS eU : Ex HS) (Bb : ℕ), Bb ≤ zsz (g.V : ℤ) + 3 →
      eP.size + eG.size + eS.size + eU.size ≤ 50 →
      NoK eP → NoK eG → NoK eS → NoK eU →
      ESp eP t Bb (svP g i : ℤ) → ESp eG t Bb (svG g i : ℤ) → ESp eS t Bb (svS g i : ℤ) →
      ESp eU t Bb (u : ℤ) →
      ∃ c : ℕ, IRuns I l12prog (swE q eP eG eS eU) t
          (t.upd 0 (ints (rixT g (swT i) zs) ++ R)) c ∧
          (c : ℝ) ≤ K * ((g.V : ℝ) + 1) * (u : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 50)

Context. KFX §1: the Lemma 12 swap wrapper with expression parameters

swE q eP eG eS eU pushes the four Lemma 12 parameters as the values of arbitrary expressions (read from registers at run time), runs l12F, pops. swE_spec is KFL.swI_spec with the literals replaced by expressions (KFL.ESp facts), and swE_geo gives the three slot swaps rixT g (swT i) on the 8-axis geometry.

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

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFX
open IntMul NLogN.RS NLogN.KIL NLogN.KFD NLogN.KFE NLogN.KFB NLogN.KFL AModel AModelL12 NetProp7

Formal statement

theorem NLogN.KFX.swE_geo :
    ∃ T q₀ : ℕ, ∀ q, q.Prime → q₀ < q → ∃ K : ℝ, 0 ≤ K ∧
      ∀ (i : Fin 3) (g : Geo), g.Pos → ∀ u : ℕ, 1 ≤ u → g.Q = 2 ^ u →
      ∀ (I : List Sym) (t : IState HS) (zs : List ℤ) (R : List Item),
      KFL.Bits zs → t.G 0 = ints zs ++ R → zs.length = g.V →
      ∀ (eP eG eS eU : Ex HS) (Bb : ℕ), Bb ≤ zsz (g.V : ℤ) + 3 →
      eP.size + eG.size + eS.size + eU.size ≤ 50 →
      NoK eP → NoK eG → NoK eS → NoK eU →
      ESp eP t Bb (svP g i : ℤ) → ESp eG t Bb (svG g i : ℤ) → ESp eS t Bb (svS g i : ℤ) →
      ESp eU t Bb (u : ℤ) →
      ∃ c : ℕ, IRuns I l12prog (swE q eP eG eS eU) t
          (t.upd 0 (ints (rixT g (swT i) zs) ++ R)) c ∧
          (c : ℝ) ≤ K * ((g.V : ℝ) + 1) * (u : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 50) := 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.KFX.swE_geo`, Solutions/KFX1Sw.lean (lines 168–227) of the round-six development.

View graph