New certified lower bounds for the moving sofa problem in a three-dimensional snaky corridor
Abstract
Consider the "snaky" corridor K4 ⊂ ℝ3 from Problem 6.64 of arXiv:2511.02864: a tube of unit-square cross-section whose two right-angle turns lie in perpendicular planes, with a straight middle section of clear length $4$. We exhibit explicit rigid motions of connected bodies through K4 and certify their volumes with the problem's own evaluation pipeline, unmodified, at the tight motion tolerances of its final scoring stage (translation $0.005$, rotation $0.05$ degrees). The certificates give V*(K4) ≥ 1.8355162 at grid fineness $0.002$ and V*(K4) ≥ 1.8558404 at grid fineness $0.001$, against the previously reported $1.81$; rerunning the pipeline's published best path under the identical configuration yields $1.8179428$, so every comparison is path against path under one fixed certifier. Three stages produce the record motion: a simple "staggered twist" backbone in which the whole 90-degree twist about the tube axis happens inside the second turn; evolutionary (CMA-ES) refinement, including a mixed-fidelity scheme that re-scores each generation's leading candidates on a grid twice as fine and lets those ranks steer recombination; and a gradient polish carried out directly at the finer carve resolution, guarded by periodic rebuilding of the frozen interpolation structure and by a penalty on control-pose spacing. These guards close three documented failure modes in which an optimizer inflates its own objective: frozen-structure inflation, spacing stretch, and exploitation of a coarse grid by optimizing and selecting on it. A staircase-based predictor with two grid bases, anchored by matching grid sensitivity, forecast the certificates to about $0.001$ before the runs. With the companion note's upper bound V*(K4) ≤ 3.4083, the problem now sits in the bracket 1.8558404 ≤ V*(K4) ≤ 3.4083. All paths, notebooks and certified results are public.
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BibTeX
@misc{tang2026certified,
title = {{New certified lower bounds for the moving sofa problem in a three-dimensional snaky corridor}},
author = {Yuzhe Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14051},
eprint = {2610.14051},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14051},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - New certified lower bounds for the moving sofa problem in a three-dimensional snaky corridor
AU - Yuzhe Tang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.14051
UR - https://arxiv.org/abs/2610.14051
AB - Consider the "snaky" corridor K_4 ⊂ ℝ^3 from Problem 6.64 of arXiv:2511.02864: a tube of unit-square cross-section whose two right-angle turns lie in perpendicular planes, with a straight middle section of clear length $4$. We exhibit explicit rigid motions of connected bodies through K_4 and certify their volumes with the problem's own evaluation pipeline, unmodified, at the tight motion tolerances of its final scoring stage (translation $0.005$, rotation $0.05$ degrees). The certificates give V^*(K_4) ≥ 1.8355162 at grid fineness $0.002$ and V^*(K_4) ≥ 1.8558404 at grid fineness $0.001$, against the previously reported $1.81$; rerunning the pipeline's published best path under the identical configuration yields $1.8179428$, so every comparison is path against path under one fixed certifier. Three stages produce the record motion: a simple "staggered twist" backbone in which the whole 90-degree twist about the tube axis happens inside the second turn; evolutionary (CMA-ES) refinement, including a mixed-fidelity scheme that re-scores each generation's leading candidates on a grid twice as fine and lets those ranks steer recombination; and a gradient polish carried out directly at the finer carve resolution, guarded by periodic rebuilding of the frozen interpolation structure and by a penalty on control-pose spacing. These guards close three documented failure modes in which an optimizer inflates its own objective: frozen-structure inflation, spacing stretch, and exploitation of a coarse grid by optimizing and selecting on it. A staircase-based predictor with two grid bases, anchored by matching grid sensitivity, forecast the certificates to about $0.001$ before the runs. With the companion note's upper bound V^*(K_4) ≤ 3.4083, the problem now sits in the bracket 1.8558404 ≤ V^*(K_4) ≤ 3.4083. All paths, notebooks and certified results are public.
ER -