A new unconditional lower bound for shoreline search

Abstract

A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.

Disclosure

“the analytic compactness and integration arguments remain the paper proof rather than a claim of complete formalization. Disclosure of AI assistance Large language models were used in the preparation of this work: OpenAI GPT 5.6 Sol and Anthropic Claude Fable 5 assisted with numerical experiments, exploration of proof strategies, and editing of the text. All definitions, statements, and proofs were checked by the author, who takes full responsibility for the content. A Proof of Lemm”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-22T17:01:50.719755Z

Structural counts