Symbolic group varieties and dual surjunctivity
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Bibliographic record
Abstract
Let G be a group. Let X be an algebraic group over an algebraically closed field K . Denote by A=X(K) the set of rational points of X . We study algebraic group cellular automata \tau \colon A^G \to A^G whose local defining map is induced by a homomorphism of algebraic groups X^M \to X , where M is a finite memory. When G is sofic and K is uncountable, we show that if \tau is post-surjective, then it is weakly pre-injective. Our result extends the dual version of Gottschalk's conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When G is amenable, we prove that if \tau is surjective, then it is weakly pre-injective, and conversely, if \tau is pre-injective, then it is surjective. Hence, we obtain a complete answer to a question of Gromov on the Garden of Eden theorem in the case of algebraic group cellular automata.
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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 | 0.001 |
| 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