Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau--De Gennes Study
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Bibliographic record
Abstract
We prove the existence of a well order reconstruction solution (WORS)-type Landau--de Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parameters---the outer square edge length $\lambda$, and the aspect ratio $\rho$, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and all temperatures, and is globally stable for either $\lambda$ small enough or for $\rho$ sufficiently close to unity. We study three different types of Landau--de Gennes critical points in this setting: critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom, and critical points with five degrees of freedom. We identify the competitors for the WORS in the two- and three-dimensional settings. In the three-dimensional setting, we numerically find up to $28$ critical points for moderately large values of $\rho$, of which diagonal solutions are global energy minimizers when they exist. We find two nonenergy minimizing critical points with five degrees of freedom for very small values of $\rho$, with an escaped profile around the isotropic square inclusion.
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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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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