The chunk-store cost is within the per-axis budget (round six, `chunkStore_cost_ev`)

Proved

J6Seg.PreT.chunkStore_cost_ev

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

complexity-theoryinteger-multiplicationkappa-improvement

The chunk-store cost is within the per-axis budget (Cp = 1).

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

theorem J6Seg.PreT.chunkStore_cost_ev (P : Prm) (hP : P.Ok) (aA aC mA mC : ℕ) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A M : ℕ,
      A ≤ aA * J6Seg.Q6 (pP6 P n) (dP6 P n) (src k) + aC →
      M ≤ mA * J6Seg.Q6 (pP6 P n) (dP6 P n) (src k) + mC → ∀ A' : ℕ,
      (cCS2 (J6Seg.Q6 (pP6 P n) (dP6 P n) (src k)) (R6 (pP6 P n)) A (lamP (dP6 P n)) M
        (tF6 P n k / lamP (dP6 P n)) : ℝ) ≤ cU6 P n A' / (D + 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_R6_Defs68

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PreT
open IntMul NLogN IntMul.Kappa.R6 J6Seg J6Seg.Side

Formal statement

theorem J6Seg.PreT.chunkStore_cost_ev (P : Prm) (hP : P.Ok) (aA aC mA mC : ℕ) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A M : ℕ,
      A ≤ aA * J6Seg.Q6 (pP6 P n) (dP6 P n) (src k) + aC →
      M ≤ mA * J6Seg.Q6 (pP6 P n) (dP6 P n) (src k) + mC → ∀ A' : ℕ,
      (cCS2 (J6Seg.Q6 (pP6 P n) (dP6 P n) (src k)) (R6 (pP6 P n)) A (lamP (dP6 P n)) M
        (tF6 P n k / lamP (dP6 P 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.PreT.chunkStore_cost_ev`, Solutions/J6SegILPre28.lean (lines 96–198) of the round-six development.

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