Spectral Analysis of the MIXMAX Random Number Generators
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Bibliographic record
Abstract
We study the lattice structure of random number generators of the MIXMAX family, a class of matrix linear congruential generators that produces a vector of random numbers at each step. The design of these generators was inspired by Kolmogorov K-systems over the unit torus in the real space, for which the transition function is measure preserving and produces a chaotic behavior. In actual implementations, however, the state space is a finite set of rational vectors, and the MIXMAX has a lattice structure just like linear congruential and multiple recursive generators. Its matrix entries were also selected in a special way to allow a fast implementation, and this has an impact on the lattice structure. We study this lattice structure for vectors of successive and nonsuccessive output values in various dimensions. We show in particular that for coordinates at specific lags not too far apart, in three dimensions, or if we construct points of k+2 or more successive values from the beginning of an output vector of size k, all the nonzero points lie in only two hyperplanes. This is reminiscent of the behavior of lagged-Fibonacci and add-with-carry/subtract-with-borrow generators. And even if we skip the output coordinates involved in this bad structure, other highly structured projections often remain, depending on the choice of parameters. We show that empirical statistical tests can easily detect this structure.
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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 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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