A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of<i>K</i>–algebras
Bibliographic record
Abstract
Abstract In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. LetAbe a finitely generated commutativeK–algebra over a field of characteristic 0, and letσbe aK–algebra automorphism ofA. Given idealsIandJofA, we show that the setSof integersmsuch that Jis a finite union of complete doubly infinite arithmetic progressions inm, up to the addition of a finite set. Alternatively, this result states that for an affine schemeXof finite type overK, an automorphismσ∊ 2 AutK(X), andYandZany two closed subschemes ofX, the set of integersmwith Yis as above. We present examples showing that this result may fail to hold if the affine schemeXis not of finite type, or ifXis of finite type but the fieldKhas positive characteristic.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".