The algorithmic hardness threshold for continuous random energy models
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Bibliographic record
Abstract
We prove an algorithmic hardness result for finding low-energy states in the so-called continuous random energy model (CREM) , introduced by Bovier and Kurkova in 2004 as an extension of Derrida’s generalized random energy model . The CREM is a model of a randomenergy landscape (X_v)_{v \in \{0,1\}^N} on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of p -spin models including the Sherrington–Kirkpatrick model. The CREM is parameterized by an increasing function A \colon [0,1]\to[0,1] , which encodes the correlations between states. We exhibit an algorithmic hardness threshold x_* , which is explicit in terms of A . More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any \varepsilon > 0 a linear-time algorithm which finds states v \in \{0,1\}^N with X_v \ge (x_*-\varepsilon) N . Second, we show that the value x_* is essentially best-possible: for any \varepsilon > 0 , any algorithm which finds states v with X_v \ge (x_*+\varepsilon)N requires exponentially many queries in expectation and with high probability. We further discuss what insights this study yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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