The last-bit step on the geometry (round six, `xorU_geo`)

Proved

NLogN.KFX.xorU_geo

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

complexity-theoryinteger-multiplicationkappa-improvement

The last-bit step on the geometry, register-driven (j = ρ + (f-1)K, E = 2^{fK-j-1}).

The following holds (Lean statement, in the scope of module Solutions.KFX2Rot):

theorem NLogN.KFX.xorU_geo {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {K ρ f : ℕ} (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) {Bb : ℕ} (hB2 : 2 ≤ Bb) (zV : zsz (g.V : ℤ) ≤ Bb) (w : ℕ)
    (t : IState HS) (xs : List ℤ) (R : List Item) (hlen : xs.length = g.V) (hbw : Bnd w xs)
    (h0 : t.G 0 = ints xs ++ R) {eP eB ed eQ ej : Ex HS}
    (hsz : ((xorPl eP eB ed eQ ej).map fun p => p.2.size + 1).sum ≤ 100)
    (n1 : NoLow eP) (n2 : NoLow eB) (n3 : NoLow ed) (n4 : NoLow eQ) (n5 : NoLow ej)
    (E1 : ESp eP t Bb ((g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1) : ℕ) : ℤ))
    (E2 : ESp eB t Bb ((2 ^ (ρ + (f - 1) * K) * g.S1 : ℕ) : ℤ))
    (E3 : ESp ed t Bb ((g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1) : ℕ) : ℤ))
    (E4 : ESp eQ t Bb ((g.Q : ℕ) : ℤ)) (E5 : ESp ej t Bb ((2 : ℤ) ^ (ρ + (f - 1) * K))) :
    IRuns I iprog (xorU eP eB ed eQ ej) t
      (t.upd 0 (ints (rixT g (XLc K ρ f) xs) ++ R))
      (cXorU (g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1)) (2 ^ (ρ + (f - 1) * K) * g.S1)
        w Bb)

Context. KFX §2: register-driven table rotations (the lines (22) and the last bit)

pushSt_G_mem, pushOK_of: pushed parameter lists built from KFL.ESp facts.
tabRot_runs: a generic withPushI pl (fcode U ; rotF) where U pushes the rotation table.
tabGenU sgn b: KFL.tabGen with the literals P, Q, d0, d1, 2^ρ, f-1, 2^K read from the registers 1, 2, 3, 5, 6, 7, 8 (tabGenU_sub: the literal instance is RS.sub).
rotU_geo: one line rixT g (rot2 Q (amt K ρ f l)) with parameters from expressions.
xorGenU, xorU_geo: the last-bit step rixT g (XLc K ρ f).

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 NLogN.KFW AModel Butterfly

Formal statement

theorem NLogN.KFX.xorU_geo {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {K ρ f : ℕ} (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) {Bb : ℕ} (hB2 : 2 ≤ Bb) (zV : zsz (g.V : ℤ) ≤ Bb) (w : ℕ)
    (t : IState HS) (xs : List ℤ) (R : List Item) (hlen : xs.length = g.V) (hbw : Bnd w xs)
    (h0 : t.G 0 = ints xs ++ R) {eP eB ed eQ ej : Ex HS}
    (hsz : ((xorPl eP eB ed eQ ej).map fun p => p.2.size + 1).sum ≤ 100)
    (n1 : NoLow eP) (n2 : NoLow eB) (n3 : NoLow ed) (n4 : NoLow eQ) (n5 : NoLow ej)
    (E1 : ESp eP t Bb ((g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1) : ℕ) : ℤ))
    (E2 : ESp eB t Bb ((2 ^ (ρ + (f - 1) * K) * g.S1 : ℕ) : ℤ))
    (E3 : ESp ed t Bb ((g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1) : ℕ) : ℤ))
    (E4 : ESp eQ t Bb ((g.Q : ℕ) : ℤ)) (E5 : ESp ej t Bb ((2 : ℤ) ^ (ρ + (f - 1) * K))) :
    IRuns I iprog (xorU eP eB ed eQ ej) t
      (t.upd 0 (ints (rixT g (XLc K ρ f) xs) ++ R))
      (cXorU (g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1)) (2 ^ (ρ + (f - 1) * K) * g.S1)
        w Bb) := 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.xorU_geo`, Solutions/KFX2Rot.lean (lines 458–501) of the round-six development.

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