Finding the Conjectured Sequence of Largest Small n-Polygons by Numerical Optimization
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
LSP(n), the largest small polygon with n vertices, is a polygon with a unit diameter that has a maximal of area A(n). It is known that for all odd values n≥3, LSP(n) is a regular n-polygon; however, this statement is not valid even for values of n. Finding the polygon LSP(n) and A(n) for even values n≥6 has been a long-standing challenge. In this work, we developed high-precision numerical solution estimates of A(n) for even values n≥4, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists in the efficient numerical solution of the model-class considered. This is followed by results for an illustrative sequence of even values of n, up to n≤1000. Most of the earlier research addressed special cases up to n≤20, while others obtained numerical optimization results for a range of values from 6≤n≤100. The results obtained were used to provide regression model-based estimates of the optimal area sequence {A(n)}, for even values n of interest, thereby essentially solving the LSP model-class numerically, with demonstrably high precision.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it