Optimal Average Disk-Inspection via Fermat's Principle
Bibliographic record
Abstract
This work resolves the optimal average-case cost of the Disk-Inspection problem, a variant of Bellman's 1955 lost-in-a-forest problem. In Disk-Inspection, a mobile agent starts at the center of a unit disk and follows a trajectory that inspects perimeter points whenever the disk does not obstruct visibility. The worst-case cost was solved optimally in 1957 by Isbell, but the average-case version remained open, with heuristic upper bounds proposed by Gluss in 1961 and improved only recently. Our approach applies Fermat's Principle of Least Time to a recently proposed discretization framework, showing that optimal solutions are captured by a one-parameter family of recurrences independent of the discretization size. In the continuum limit these recurrences give rise to a single-parameter optimal control problem, whose trajectories coincide with limiting solutions of the original Disk-Inspection problem. A crucial step is proving that the optimal initial condition generates a trajectory that avoids the unit disk, thereby validating the optics formulation and reducing the many-variable optimization to a rigorous one-parameter problem. In particular, this disproves Gluss's conjecture that optimal trajectories must touch the disk. Our analysis determines the exact optimal average-case inspection cost, equal to $3.549259\ldots$ and certified to at least six digits of accuracy.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".