Explicit shift vectors and certified loneliness gaps for the Shifted Lonely Runner Conjecture

Abstract

For integer speeds v=(v₁,…,vₙ) and shifts s=(s₁,…,sₙ), the loneliness gap γ(v;s)=sup_(t ∈ ℝ)minᵢdist(sᵢ+vᵢt) measures how far from every integer the whole pack of runners can simultaneously be pushed; the Shifted Lonely Runner Conjecture asserts γ(v;s) ≥ 1/(n+1) whenever the |vᵢ| are distinct and nonzero. Blanco, Criado and Santos recently refuted it for every n with 5 ≤ n ≤ 17, but their paper prints an attaining shift vector for only three instances, and their companion code stops at n=13. We give explicit rational shift vectors s^((n)) ∈ [0,1)ⁿ for every n with 7 ≤ n ≤ 17, together with the exact value of γ((1,2,…,n);s^((n))) and an exact rational maximising time. At n=7 and n=8 the vectors reach the published optimal values 25/214 and 1/10; for 9 ≤ n ≤ 14 they certify γᵐⁱⁿ(1,…,n) ≤ 1/(n+2); and for n=15,16,17 — the range where no exact value of γᵐⁱⁿ is published — they certify γᵐⁱⁿ(1,…,n) ≤ 2/(2n+3), that is, 2/33, 2/35 and 2/37. Every gap is proved on both sides by a purely integral certificate: a finite interval cover for the upper bound and one rational time for the lower bound. We also reproduce, as upper bounds with explicit witnesses, the published values of γᵐⁱⁿ(1,…,n) for 5 ≤ n ≤ 13, and give the corresponding maximising times, which neither source prints. Every one of these certificates has been checked by the Lean 4 kernel, together with the soundness of the certificate scheme itself and the controls that make it falsifiable.

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