Bibliographic record
Abstract
We model each 4\(\mathrm{\times}\)4 magic square by encoding its 16 integers as magnitudes of repelling positive point charges on a fixed 2D lattice, evolved under Coulomb forces with linear damping and a harmonic pinning to anchor sites. We simulate all 880 magic squares and compare them with equally sized ensembles of random permutations of \(\{1,{\dots},16\}\). Three readouts differentiate the ensembles. (i) Final positions: magic cases form sixteen tight, index-specific clusters on an annulus, whereas random cases show broader arcs and central accumulation. (ii) Displacement–correlation structure: across cases, many pair-of-pairs of inter-index displacements in magic squares are near-linearly dependent; the random ensemble exhibits only moderate relationships, with |r| and R\(^2\) distributions shifted to weaker correlation. (iii) Center potential: the Coulomb-type potential at the geometric center collapses to a single value at the anchors for all magic squares and remains narrowly distributed after dynamics, while random squares remain broad. Sensitivity analysis reveals a broad damping–stiffness region with high convergence, indicating that the results are robust to parameter choice.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".