The solution space geometry of random linear equations
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Bibliographic record
Abstract
Abstract We consider random systems of linear equations over GF(2) in which every equation binds k variables. We obtain a precise description of the clustering of solutions in such systems. In particular, we prove that with probability that tends to 1 as the number of variables, n , grows: for every pair of solutions σ,τ , either there exists a sequence of solutions starting at σ and ending at τ such that successive solutions have Hamming distance O (log n ), or every sequence of solutions starting at σ and ending at τ contains a pair of successive solutions with distance Ω( n ). Furthermore, we determine precisely which pairs of solutions are in each category. Key to our results is establishing the following high probability property of cores of random hypergraphs which is of independent interest. Every vertex not in the r ‐core of a random k ‐uniform hypergraph can be removed by a sequence of O (log n ) steps, where each step amounts to removing one vertex of degree strictly less than r at the time of removal. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 197–231, 2015
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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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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