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The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon

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Pith's one-line read The shortest guaranteed escape path from the golden gnomon has exact length C ≈ 1.282676, attained by a seven-piece curve.

desk verdict Exact E(G)=C for the golden gnomon via a new calibration/ledger method, with Lean-checked algebra and one localized planar rigidity bridge. read the letter →

arxiv 2607.24483 v1 pith:WHZJ5C75 submitted 2026-07-27 math.MG

Alexander Temerev , Alessio Doria This is my paper · ORCID
classification math.MG MSC 52A4052C1549Q10
keywords Bellman'slost-in-a-forestproblemescapepathgoldengnomonsupportfunctioncalibrationconvexgeometrywormhomotheticcover
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Bellman’s lost-in-a-forest problem asks for the shortest curve that is guaranteed to reach the boundary of a known shape no matter where you start and which way you face. This paper solves it exactly for the golden gnomon—an isosceles triangle with equal sides 1 and apex 108°. The shortest such escape path is a symmetric seven-piece route of straight segments, circular shoulders, and tangents, with length C determined by one isolated root of an explicit quartic; C is transcendental and equals about 1.282676. Equivalently, the triangle scaled by 1/C is the smallest positive copy of itself that can cover every curve of length 1. The authors obtain the matching lower bound by a balanced support calibration: one weighted family of escape inequalities, saturated by the candidate, is aggregated into a finite zero-sum vector family whose running balance (the ledger) stays inside the unit disk and therefore cannot exceed path length. Local surgery and cyclic order force any shorter polygonal competitor into the ledger’s safe temporal order. This is the first proved exact optimum for an isosceles triangle with base angle below 45°.

What carries the argument

Balanced support calibration: a positive source measure on the triangle’s three normals folds into a balanced vector measure µ whose integral against support functions is at least C for every escape hull and, after normal-cone aggregation and Abel summation, is at most path length whenever the running suffix (the ledger) stays in the unit disk; rigidity forces shortest polygonal counterexamples into the ledger’s allowed order.

What would settle it

Exhibit a polygonal escape path for G whose length is strictly less than C, or a unit arc that cannot be placed inside any positive homothet of G smaller than scale 1/C; alternatively, produce a length-minimal polygonal escape path whose support contacts refuse the anchored ledger order after the two-gap surgery.

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Extended reading notes

Core claim

For the golden gnomon G, the escape length equals the exact constant C = 2sc((b−a)/c + λ) built from the unique root p in [−1/8, −1/9] of an explicit quartic; the minimum is attained by the symmetric seven-piece path Γ of segments, circular shoulders of radius s, and tangents. Equivalently, C⁻¹G is the smallest positive homothet of G that contains a congruent copy of every unit-length rectifiable arc.

Load-bearing premise

Every length-minimal polygonal escape path shorter than C can be forced, by two-gap surgery and cyclic boundary order, into exactly the contact sequence the unit-disk ledger tolerates.

Editorial extensions

If this is right

  • E(G) equals the explicit transcendental constant C ≈ 1.282676025459, so the classical scaled-Zalgaller benchmark is not optimal for this triangle.
  • C⁻¹G is a sharp homothetic worm cover: it contains a copy of every unit arc, and no smaller positive homothet of G does.
  • The same calibration engine applies to any triangle once a sharp source measure and a rigidity argument placing competitors in ledger order are supplied.
  • Parameters of the optimum reduce to one isolated quartic root; Lean checks the finite algebraic certificates and ledger identities.
  • This is the first proved exact escape length for an isosceles triangle with base angle below 45°.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same ledger-plus-rigidity pattern should decide other triangles inside the numerical “Tunnel” regime (roughly 27°–42° base angle) once the corresponding quartic or algebraic system is written down.
  • Because the escape threshold forces the shoulder radius to equal the triangle’s altitude factor s, curvature of optimal contacts is dictated by the normal dependence rather than chosen by hand.
  • A computer-assisted search that returns a shorter simple polygonal escape path would immediately falsify either the surgery lemmas or the unit-disk ledger bound.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper determines exactly the value of Bellman's lost-in-a-forest problem for the golden gnomon G (equal sides 1, apex angle 108°): the shortest curve guaranteed to reach ∂G from unknown position and heading is a symmetric seven-piece line/arc path Γ of length C = 2sc((b−a)/c + λ) ≈ 1.282676025459, where (a,b,λ) are built from a uniquely isolated root of an explicit quartic Q (Eq. B.1). The upper bound is a certified escape calculation: a support-fan identification (Lemma 3.5, via a one-turn support principle) reduces the escape criterion (3.1) to eighteen support windows checked by exact rational interval arithmetic, with the contact system's determinant bounded below by 3 (2.5). The lower bound is a balanced support calibration: the escape inequality is integrated against a folded source measure µ balanced by the normal relation 2c n₀+n₁+n₂ = 0; normal-cone aggregation compresses this to a finite zero-sum vector family, and Abel summation bounds its total by path length provided the suffix ledger stays in the unit disk (Lemmas 4.1–4.4). The geometric bridge, Proposition 5.1, forces any shorter standard polygonal minimizer into the "anchored" temporal order the ledger tolerates, using an imported Λ-configuration theorem, a two-gap surgery, a rational supported-sum exclusion of fourteen exceptional orders, and cyclic bitonicity. Corollaries give the sharp homothetic worm cover C⁻¹G and the transcendence of C. Two finite certificate families, the ledger algebra, and several

Significance. If correct, this is the first proved exact optimum for an isosceles triangle with base angle below 45°, a regime where the extremal curve has moving circular contacts and the tetral-arc classification of Movshovich–Wetzel does not apply; it converts Gibbs's numerical "Tunnel" prediction into a theorem with an exact, transcendental constant. The balanced-support-calibration method is the main conceptual contribution and is explicitly presented as angle-independent (Remark 4.6), with the golden-specific work cleanly isolated. The manuscript ships strong verification assets: Lean 4 developments for both finite certificate families with exact rational interval arithmetic, an axiom audit (no sorry, no native_decide, dependencies confined to Mathlib's classical trio), independent Python re-expansion of the supported sums, a quartic-free numerical cross-check, and a precise statement (§B.3) of which steps are machine-checked and which remain prose. The result is falsifiable (explicit constant, explicit curve) and the equality mechanism (tightness exactly on supp ν, Remark 5.2) is transparent. This is a substantial and well-documented advance on a classical problem.

major comments (2)
  1. [Appendix A.2, Lemma A.5 (exclusion of alternative (L))] The entire lower bound funnels through the exclusion of alternative (L) in Lemma A.5 (via Lemma A.4 → A.7 → Prop. 5.1): if a shorter minimizer admitted an (L)-configuration, the (1+√2)h surgery fails and the anchored marking need not exist. This step is pure prose: 'monotone turning of the counterclockwise convex boundary' forces direction angles into [π,2π], giving height monotonicity along α_{C1,L1}, contradicting (A.4). The deduction is plausible and short, but it is the one load-bearing planar argument with neither a formalized core nor a displayed criterion. I ask that the monotone-turning fact used here be stated as a displayed lemma with proof (or a precise citation), so that the only non-machine-checked bridge in the proof is inspectable at the same standard as the rest of Appendix A.
  2. [Appendix A.3, Lemma A.8, Eqs. (A.13)-(A.14)] The exclusion of the fourteen exceptional temporal orders rests on the enumeration in (A.13a): twenty orders compatible with F≺M≺T, six interior, fourteen exceptional collapsing to eight symmetry classes, each with a supported-sum bound in (A.14) exceeding κ = 129/100 (margins are exact rationals, e.g. 100978/78125 = 1.2925184 > κ). The per-order estimates are independently re-expanded by verify_rational_supports.py and OuterTetral.lean retains certificates for all fourteen orders — but it is unclear whether the exhaustiveness of the enumeration itself (the claim that these eight classes cover all cases, including coincident labels under the outer-side convention) is machine-checked or by hand. Since a single missed order would admit a decreasing-fan contact and break the ledger, please state explicitly how exhaustiveness is established.
minor comments (6)
  1. [Prop. 3.6, Eq. (3.5)] The symbol 'buθ' for the physical normal appears without explanation (presumably a bold/hatted u); define or unify the notation.
  2. [Eq. (2.4)] The 3×3 matrix M is hard to parse; consider displaying its columns factored by c_a (already cleared) or aligning the entries typographically. The bound det M > 3 is kernel-checked, so this is purely presentational.
  3. [Lemma 2.2, Eq. (2.8)] The queue of conditions '−0.121 < p < −0.119' etc. is described as 'coarse boxes' but their provenance (from the root isolation (B.2)–(B.4) and (B.5)) is only implicit; a one-line forward reference would help.
  4. [Appendix A.3, (A.11)] The claim h ≤ h_max = 147/250 uses h < s < 147/250 from (A.9); displaying sin(π/5) < 147/250 among the certified bounds of (A.12) would make the chain self-contained at the point of use.
  5. [Front matter / §B.3] The liberal use of LLMs is commendably disclosed in the front matter; consider also noting in §B.3 whether any Lean proof scripts were machine-assisted in drafting, to complete the provenance picture. Reference [11] is appropriately flagged as unrefereed; the anticipation credit to Gibbs is fair.
  6. [Remark 1.2 / Cor. B.1] Remark 1.2's 4.22% comparison to ζ sin β checks out numerically. Suggest adding one sentence after Corollary B.1 noting that a, b, λ are algebraic while C is transcendental — a striking and easily missed feature of the constant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: standard calibration construction meets an independently proved lower bound via rigidity, not by renaming inputs as predictions.

full rationale

The derivation is a classical calibration-plus-rigidity argument and does not reduce the optimum to its inputs by construction. Parameters (a,b,λ) are chosen so that the candidate Γ saturates the escape inequality on supp ν and places critical ledger states on the unit circle (eqs. 2.1, 4.2, 4.11–4.12); C is then the calibrated total. That is how every sharp geometric calibration is built. The lower bound for competitors is separate: the escape inequality (3.1) holds for every hull by the normal identity (3.2) alone; integration against the positive source ν gives I_µ ≥ C for any escape path; normal-cone aggregation and Abel summation bound I_µ by length only after geometric rigidity (Prop. 5.1 / App. A) forces an anchored temporal order so the ledger stays in the unit disk (Lemmas 4.2–4.4, Prop. 4.5). Rigidity rests on the external Λ-configuration theorem (Thm A.6, Coulton–Movshovich / Alexander–Wetzel–Wichiramala), two-gap surgery, cyclic bitonicity, and rational outer-order estimates—not on a self-citation or a fit to competitor lengths. Lean checks the algebraic certificates and ledger identities; C is an algebraic-transcendental output of one isolated quartic plus half-angles, not a fitted empirical constant. Equality is the meeting of construction and bound on supp ν (Remark 5.2), which is the intended sharpness mechanism, not circularity. Soft spots in the planar prose of Prop. 5.1 are correctness risks, not circular reductions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The result is pure Euclidean convex geometry plus one classical external theorem. No empirical free parameters. Background is standard support-function convexity, polygonal approximation, and the imported Λ-property of simple arcs. Invented machinery (ledger, anchored marking, balanced source measure) is definitional proof technology with internal checks, not new physical entities.

assumptions (5)
  • standard math Λ-configuration theorem: every simple open polygonal arc of positive width admits parallel supports with u≺M≺w (Theorem A.6, cited from Coulton–Movshovich / Alexander–Wetzel–Wichiramala).
    Used in Lemma A.7 to select the gap for two-gap surgery; only imported geometric black box.
  • standard math Standard convex geometry: support functions characterize compact convex sets; Hausdorff continuity of Φ; Minkowski addition dual to support sums; reflection invariance of escape for symmetric G.
    Escape criterion Lemma 3.2 and attainment Proposition 3.6.
  • standard math Lindemann–Weierstrass: used only for transcendence of C (Corollary B.1), not for the value of E(G).
    Secondary corollary; main theorem does not need transcendence.
  • standard math Planar chord-interleaving: two chords of a convex polygon cross in relative interiors iff endpoints alternate (input to endpoint-peeling / cyclic bitonicity).
    Lemma A.2; first-step hull-edge fact is Lean-checked given this criterion.
  • domain assumption Existence of length-minimizing standard polygonal escape paths with ≤N segments when a shorter rectifiable escape exists (compactness + uncrossing).
    Proposition A.1; classical but proved in-paper in prose, not fully formalized.
invented entities (3)
  • Balanced source measure ν and folded calibration measure μ with unit-disk ledger independent evidence
    purpose: Single weighted family of escape inequalities saturated by Γ and giving len≥C for every anchored polygonal competitor.
    Core new proof object; sharpness fixed by calibration equations (2.1) and mass identity (4.2).
  • Anchored chunked support marking / ten-block ledger walk independent evidence
    purpose: Finite temporal order on aggregated normal-cone vectors for which suffix norms stay ≤1.
    Interface between geometry (rigidity) and analysis (Abel summation); order forced by two anchors and bitonicity.
  • Seven-piece candidate Γ with parameters from isolated quartic root p independent evidence
    purpose: Explicit optimizer attaining E(G)=C.
    Constructive upper bound; contact system and windows certified.

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Pith. "Pith review of The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon." pith.science (2026). https://pith.science/paper/WHZJ5C75

@misc{pith2026260724483,
  author       = {Pith},
  title        = {Pith review of: The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WHZJ5C75}},
  note         = {Machine review of arXiv:2607.24483}
}
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 $108^\circ$: 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 $C=1.282676025459\ldots$. To our knowledge, this is the first proved exact optimum for an isosceles triangle whose base angle is below $45^\circ$. The curve's parameters come from one isolated quartic root, and $C$ is transcendental. Equivalently, $C^{-1}G$ 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.

Figures

Figures reproduced from arXiv: 2607.24483 by the authors.

Figure 1
Figure 1. The optimal route (orange) in the golden gnomon: E(G) = 1.282676025459 . . .. 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… view at source ↗
Figure 2
Figure 2. The proof at a glance. One balanced escape inequality is read once on the candidate and once on a shortest polygonal counterexample. The only geometric bridge is the minimizer-rigidity statement. Section 5 proves the theorem from these three statements before the technical rigidity argument is opened. The self-contained local surgery and the anatomy of the two finite certificate families are then given in the append… view at source ↗
Figure 3
Figure 3. The shortened two-gap argument. Blue dashed segments are the two gaps; orange dotted segments are the outer cap or reflections. (ii) The straight cap forces the angle. Minimality against the outer cap gives |L1E| ≥ len αL1,A′ ≥ len αL1,C2 + len αC2,L2 ≥ |L1C2| + |C2L2|. Put θ = ∠C2EL2, write Li = ti(− cos θ,sin θ) (t1 > t2 > 0), and reflect L2 in the base to L ′ 2 . Since C2 lies on the base, the preceding bound and… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm

    math.OC 2026-08 conditional novelty 7.0 of 10

    An exact support-function inequality is derived that characterizes shortest escape paths from arbitrary triangular forests and dual triangle covers for Moser's worm problem.

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