On the solution space of the Golomb recursion
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Bibliographic record
Abstract
We explore the nature of the solution space for the Golomb nested recursion g(n)=g(n-g(n-1))+1. On this solution space, we define a natural equivalence relation and restrict our attention to non-equivalent solutions. We describe and prove an algorithm that determines whether a given set of initial conditions generates a solution. Up to equivalence, there is a unique solution whose forward differences are eventually either 0 or 1, namely, the Golomb sequence g0=1,2,2,3,3,3,4,4,4,4,⋯, generated by the initial condition g0(1)=1. This sequence is asymptotic to 2n; we conjecture that this is true of every solution. We further conjecture that each solution has what we call a generational structure that abstracts combinatorial properties of g0. It appears that for any given solution, its generations are composed of only a finite number of building blocks.
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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 | 0.000 |
| 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