Random reals, the rainbow Ramsey theorem, and arithmetic conservation
Why this work is in the frame
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Bibliographic record
Abstract
Abstract We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2- RAN be the principle that for every real X there is a real R which is 2-random relative to X . In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA 0 and so RCA 0 + 2- RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA 0 for arithmetic sentences. Thus, from the Csima–Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2- RAN has non-trivial arithmetic consequences. In Section 4, we show that 2- RAN is conservative over RCA 0 + B Σ 2 for -sentences. Thus, the set of first-order consequences of 2- RAN is strictly stronger than P − + I Σ 1 and no stronger than P − + B Σ 2 .
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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 | 0.002 | 0.001 |
| 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 | 0.001 |
| 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