`EpsOK` for row exponents (round six, `epsOK_tabM`)

Proved

IntMul.Kappa.R6.IL3H.epsOK_tabM

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

complexity-theoryinteger-multiplicationkappa-improvement

EpsOK for row exponents on Fin m × Fin f (the frozen epsOK_tab, any m).

The following holds (Lean statement, in the scope of module Solutions.R6IL3Steps):

theorem IntMul.Kappa.R6.IL3H.epsOK_tabM {m f : ℕ} (hm : 0 < m) {σ : Fin HS → Option ℤ} {s : IState HS}
    (hs : Agree σ s) {eR KR NR fR : Fin HS} {e K : ℕ} (hR : AffReg σ eR KR NR e K (m * f))
    (hf : σ fR = some (f : ℤ)) (hfS : fR ≠ KFW.kS)
    (hlt : eR.val < 56 ∧ KR.val < 56 ∧ NR.val < 56 ∧ fR.val < 56)
    (hK : 0 < K) (hf0 : 0 < f) (hN : K * (m * f - 1) ≤ e) (a : Fin m → ℕ) (t Bε : ℕ)
    (hB : zsz (((e + t + K + m * f + f : ℕ)) : ℤ) ≤ Bε) (h3 : zsz (3 : ℤ) ≤ Bε) :
    EpsOK s (εTab eR KR NR fR (List.ofFn a)) t
      (aEof (fun p : Fin m × Fin f => e - K * (p.2.val + f * p.1.val)) (fun p => a p.1)) Bε

Context. Round six, item language (I3): the local steps of a node

A node on E bits (F = E / m) acts on W + T role streams of nN (B/W) t S₀ records, bits Fin m × Fin F at the positions IL3F.posR m K e F, strides stH. Parametric copies of the frozen KFWStep1/2/3/5 step lemmas (P8Lift.Wc → W + T, mC → m, mC^k → F):

stepH_gate (atoms), stepH_perm (role permutation), stepH_phase (with epsOK_tabM, the m-generic epsOK_tab), stepH_fixG (a ±1 role diagonal as phase steps), stepH_row (from RowUCG, a row interface with a general coefficient aK K).

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_MachineRef
import Definitions.Def_IntMul_KappaFramework

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly

Formal statement

theorem IntMul.Kappa.R6.IL3H.epsOK_tabM {m f : ℕ} (hm : 0 < m) {σ : Fin HS → Option ℤ} {s : IState HS}
    (hs : Agree σ s) {eR KR NR fR : Fin HS} {e K : ℕ} (hR : AffReg σ eR KR NR e K (m * f))
    (hf : σ fR = some (f : ℤ)) (hfS : fR ≠ KFW.kS)
    (hlt : eR.val < 56 ∧ KR.val < 56 ∧ NR.val < 56 ∧ fR.val < 56)
    (hK : 0 < K) (hf0 : 0 < f) (hN : K * (m * f - 1) ≤ e) (a : Fin m → ℕ) (t Bε : ℕ)
    (hB : zsz (((e + t + K + m * f + f : ℕ)) : ℤ) ≤ Bε) (h3 : zsz (3 : ℤ) ≤ Bε) :
    EpsOK s (εTab eR KR NR fR (List.ofFn a)) t
      (aEof (fun p : Fin m × Fin f => e - K * (p.2.val + f * p.1.val)) (fun p => a p.1)) Bε := 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.IL3H.epsOK_tabM`, Solutions/R6IL3Steps.lean (lines 29–77) of the round-six development.

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