The freezing threshold for k-colourings of a random graph
Why this work is in the frame
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Bibliographic record
Abstract
We rigorously determine the exact freezing threshold, rkf, for k-colourings of a random graph. We prove that for random graphs with density above rkf, almost every colouring is such that a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below rkf, then almost every colouring has at most o(n) frozen variables. This confirms hypotheses made using the non-rigorous cavity method. It has been hypothesized that the freezing threshold is the cause of the "algorithmic barrier", the long observed phenomenon that when the edge-density of a random graph exceeds hf k ln k(1+ok(1)), no algorithms are known to find k-colourings, despite the fact that this density is only half the k-colourability threshold.
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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 | 0.001 | 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