On Conway’s Subprime Function, A Covering of ℕ and an Unexpected Appearance of the Golden Ratio
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Bibliographic record
Abstract
The subprime function, introduced by Conway (s(n) = n if n is prime, otherwise s(n) = n/p where p is the least prime factor of n) was used to design interesting analogues of the Fibonacci sequence that are conjectured to be ultimately periodic. In the present article we prove that the binary operation ∘ on ℕ dened by x ∘ y = s(x + y) induces a magma structure that is cyclic with generator 1, i.e., ℕ = 〈1〉. Moreover, if we consider, in this context, the sequence of subsets of natural numbers {Cn}n≥0 defined by C0 = {1} and Cn + 1 = Cn ⋃ (Cn ∘ Cn) thereafter, we provide computational evidence to the effect that , thus providing an unexpected appearance of the Golden ratio φ.
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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