A unique shared slowest repair with two faster insertions forces strictness

Proved

LonelyRunner.SharedSlowest.strict_of_shared_slowest

by Whunt003 · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

formalizationlonely-runnerlrc-shared-repair-20261007number-theory

Write $d(x)=|x-\operatorname{round}(x)|$ for distance to the nearest integer. For finite sets $R,W\subseteq\mathbb N$ , let $S(n,R,W)=\{v\in\mathbb N:0<v<n,\ v\notin R\}\cup W$ . A strict time means a real $t$ such that $d(vt)>1/n$ for every $v\in S(n,R,W)$ . Let $n,r,s,m,k,q,u\in\mathbb N$ . Assume $r\ge7$ , $n\le2r$ , $r<s<n$ , $r\in R$ , $m\ge2$ , $k>0$ , $mr=ks$ , $mr<q<u$ , $r\nmid q$ , and $r\nmid u$ . Then $S(n,R,\{mr,q,u\})$ has a strict time.

Preamble

import Mathlib
import Mathlib.Data.ZMod.Units
import Mathlib.GroupTheory.Index
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.RingTheory.Int.Basic
import Definitions.Def_LRC_SharedRepair_Defs

Formal statement

open LonelyRunner
open LonelyRunner.SharedSlowest
open CollectiveUnits CollectiveSecondFastest CollectiveFibreNormalForm

/-- **Shared slowest repair theorem.** No upper bound on either faster speed
or the multiplier is imposed. The conclusion is a real strict lonely time
for every retained baseline speed and all three insertions. -/
theorem LonelyRunner.SharedSlowest.strict_of_shared_slowest {n r s m k q u : ℕ} {R : Finset ℕ}
    (hr : 7 ≤ r) (hlarge : n ≤ 2*r) (hrs : r < s) (hsn : s < n)
    (hrR : r ∈ R) (hm : 2 ≤ m) (hk : 0 < k) (hshared : m*r=k*s)
    (hpq : m*r < q) (hqu : q < u) (hnq : ¬r ∣ q) (hnu : ¬r ∣ u) :
    SeparatedMulti.HasStrictTime n R {m*r,q,u} := by sorry

Source

Wade Hunter, lonely-runner research repository, https://github.com/huntrontrakkr/lonely-runner; local source snapshot 9b5f51aa235e321a3ec794c9bf60358933a8e6dc, lean/LonelyRunner/SharedSlowest.lean, lines 70–107, declaration LonelyRunner.SharedSlowest.strict_of_shared_slowest. AI-assisted formalization. The snapshot identifier records the submitted local source; it is not a claim that this commit is publicly hosted.

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