<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -sets and finite automata
Bibliographic record
Abstract
It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a k -automatic subset of the natural numbers to that of an F -automatic subset of a finitely generated abelian group Γ equipped with an endomorphism F . Applied to the Mordell–Lang context, where F is the Frobenius action on a commutative algebraic group G over a finite field, and Γ is a finitely generated F -invariant subgroup of G , it is shown that the “ F -subsets” of Γ introduced by the second author and Scanlon are F -automatic. It follows that when G is semiabelian and X ⊆ G is a closed subvariety then X ∩ Γ is F -automatic. Derksen’s notion of a k -normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F -subsets are F -normal. In particular, the X ∩ Γ appearing in the Mordell-Lang problem are F -normal. This generalises Derksen’s Skolem–Mahler–Lech theorem to the Mordell–Lang context.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.038 | 0.016 |
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".