The Generating Degree of ℂ<sub><i>p</i></sub>
Why this work is in the frame
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Bibliographic record
Abstract
Abstract The generating degree gdeg( A ) of a topological commutative ring A with char A = 0 is the cardinality of the smallest subset M of A for which the subring [ M ] is dense in A . For a prime number p , denotes the topological completion of an algebraic closure of the field of p -adic numbers. We prove that gdeg( ) = 1, i.e., there exists t in such that [t] is dense in . We also compute where A(U) is the ring of rigid analytic functions defined on a ball U in . If U is a closed ball then = 2 while if U is an open ball then is infinite. We show more generally that is finite for any affinoid U in ℙ 1 ( ) and is infinite for any wide open subset U of ℙ 1 ( ).
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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 | 0.002 |
| 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) | 0.001 | 0.002 |
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