Nested recursions with ceiling function solutions
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Bibliographic record
Abstract
Consider a nested, non-homogeneous recursion defined by with c initial conditions , where the parameters are integers satisfying , and . We develop an algorithm to answer the following question: for an arbitrary rational number , is there any set of values for and such that the ceiling function is the unique solution generated by with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to . The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if is the solution generated by any recursion of the form above, then r = 1. We also use our ceiling function methodology to derive the first known connection between the recursion and a natural generalization of Conway's recursion.
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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 | 0.000 | 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 | 0.000 | 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