Self-avoiding walks in a slab: rigorous results
Why this work is in the frame
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Bibliographic record
Abstract
A polymer in the confined spaces between colloid particles loses entropy and exerts a repulsive entropic force on the confining particles. This situation can be modelled by a self-avoiding walk confined in a slab between two parallel planes in the lattice. In this paper, we prove the existence of a limiting free energy for the general case that the walk is interacting with the parallel bounding planes. We also prove that the limiting free energy is strictly increasing with the distance between the bounding planes in some regions of the phase diagram. These results demonstrate the presence of a non-zero repulsive entropic force in the model. Finally, we also examine the relation between the limiting free energy in this model and the limiting free energy in a model of walks adsorbing onto a single plane. We prove that these limiting free energies are equal in some regions of the phase diagram in the limit that the width of the slab between the parallel bounding planes is taken to infinity.
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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 | 0.001 | 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 |
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