The filtering host over an oracle of time n^2.9955609 runs in time O(n^2.99556095)

Proved

Light.Filter2.filterTime_bound

by xuanji · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

apsp-2995561apsp-exponentfine-grained-complexityword-ram

Let $T_o$ be such that for all real $c,\kappa$ , eventually for every $U\le cn^{\kappa}$ and every $p\le$ PmaxOf n, $T_o(n,U,s(n),p)\le n^{2.9955609}$ , where $s(n)=$ Light.Filter2.sOf n. Then for all real $c,\kappa$ there is $K$ such that eventually, for every $U\le cn^{\kappa}$ , $\mathrm{filterTime}(T_o,n,U)\le K\,n^{2.99556095}$ .

filterTime is the time of the filtering host: parameters, prime sieve, marks, roundsOf n rounds of rOf n steps with one oracle call per pool prime, and the final answer.

Preamble

/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/
-- Generated by skeleton subtraction from the ThreeSumApsp library at commit 3b45b374640a.

import Definitions.Def_APSP2995561_Defs2

set_option autoImplicit false
set_option relaxedAutoImplicit false

open Filter ThreeSumApsp
open Light
open Light.Filter2
open ThreeSumApsp.Spec Light.Sec3
set_option maxHeartbeats 2000000

Formal statement

theorem Light.Filter2.filterTime_bound {To : List ℕ → ℕ}
    (hT : ∀ c κ : ℝ, ∀ᶠ n : ℕ in atTop, ∀ U : ℕ, (U : ℝ) ≤ c * (n : ℝ) ^ κ →
      ∀ p : ℕ, p ≤ Light.Filter2.PmaxOf n → (To [n, U, Light.Filter2.sOf n, p] : ℝ) ≤ (n : ℝ) ^ (2.9955609 : ℝ))
    (c κ : ℝ) : ∃ K : ℝ, ∀ᶠ n : ℕ in atTop, ∀ U : ℕ, (U : ℝ) ≤ c * (n : ℝ) ^ κ →
      (Light.Filter2.filterTime To n U : ℝ) ≤ K * (n : ℝ) ^ (2.99556095 : ℝ) := by sorry

Source

Developed for the prove2.me APSP exponent campaign (https://prove2.me/campaigns/apsp-exponent) on top of the Lean formalization anthropics/formal-math, folder 3sum-apsp, commit e1a4e65 (https://github.com/anthropics/formal-math/tree/e1a4e6508154ea59f030480661590a9fe3018011/3sum-apsp; Lean v4.33.1, Mathlib v4.33.1). The statement comes from the file 3sum-apsp/ThreeSumApsp/Filter2/Final.lean of the campaign's modified copy of that library, which is not published; the new definitions it uses are in the platform definition bundles APSP2995561_Defs1 and APSP2995561_Defs2 (and APSP2996001_Defs1-2, APSP299791_Defs1-5 of the earlier entries). It follows the technical note "Deterministic APSP via thin products" (revised 7 October 2026) refining J. Alman and V. Vassilevska Williams, "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs", arXiv:2610.06783v1 (https://arxiv.org/abs/2610.06783v1), Lemma 3.1 and Section 3.

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