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kirill avery @kirillzzy · 9:01 AM · Jul 29, 2026 · 9,190 Views
Bellman's lost-in-a-forest math problem stood unsolved for 70 years.

Today Alien's agentic lead engineer @AryehDubois solved it with GPT-5.6 Sol + Claude Fable 5 + Claude Opus 5

Full solution on arXiv ↓
arxiv.org/pdf/2607.24483

[Embedded PDF preview:]
arXiv:2607.24483v1 [math.MG] 27 Jul 2026

THE EXACT SOLUTION OF BELLMAN'S LOST-IN-A-FOREST PROBLEM FOR THE GOLDEN GNOMON

ALEXANDER TEMEREV AND ALESSIO DORIA

ABSTRACT. We solve Bellman's lost-in-a-forest problem for the golden gnomon G, the isosceles triangle with equal sides 1 and apex angle 108°: the shortest curve guaranteed to reach the boundary of G from an unknown starting position and heading is a symmetric seven-piece path of segments, circular shoulders, and tangents, of exactly determined length C = 1.282676025459.... To our knowledge, this is the first proved optimum for an isosceles triangle whose base angle is below 45°. The curve's parameters come from one isolated quartic root, and C is transcendental. Equivalently, C⁻¹G is the smallest homothetic golden-gnomon cover of all unit arcs.
The proof introduces a balanced support calibration: one weighted family of escape inequalities, built on the linear relation among the triangle's three normals, exactly saturated by the candidate—through eighteen exact support windows—and confronting every shorter competitor at once. Aggregation along the normal fan compresses the calibration to a finite zero-sum family of supported vectors; summation by parts then bounds its total by path length whenever the running suffix balance, the ledger, stays in the unit disk. A local two-gap surgery and cyclic bitonicity force a shortest hypothetical counterexample into exactly the temporal order the ledger tolerates. Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds.

FIGURE 1. The optimal route (orange) in the golden gnomon: E(G) = 1.282676025459....
Labeled points: γ(7), K, γ(0), G

1. INTRODUCTION
Bellman's lost-in-a-forest problem asks for the shortest route that is guaranteed to reach the boundary of a forest whose shape is known but in which the starting position and heading are unknown [4, 9]. For a convex forest K, this is equivalently the shortest rectifiable curve no congruent copy of which is contained in int K. Such a curve will be called an escape path. Here a placement may be taken to mean a translation followed by a rotation. For the reflection-symmetric triangle G, allowing all Euclidean isometries gives the same notion: compose any orientation-reversing placement with a reflection preserving G.

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