The orbit intersection problem for linear spaces and semiabelian varieties
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Bibliographic record
Abstract
Let f_1 and f_2 be affine maps of the N-th dimensional affine space over the complex numbers, i.e., f_i(x):=A_i x + y_i (where each A_i is an N-by-N matrix and y_i is a given vector), and let x_1 and x_2 be vectors such that x_i is not preperiodic under the action of f_i for i=1,2. If none of the eigenvalues of the matrices A_i is a root of unity, then we prove that the set of pairs (n_1,n_2) of non-negative integers such that f_1^{n_1}(x_1)=f_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1, m_2, \ell_1, \ell_2 are given non-negative integers, and k is varying among all non-negative integers. Using this result, we prove that for any two self-maps Φ_i(x) := Φ_{i,0}(x)+y_i on a semiabelian variety X defined over the complex numbers (where Φ_{i,0} is an endomorphism of X and y_i is a given point of X), if none of the eigenvalues of the induced linear action DΦ_{i,0} on the tangent space at the identity 0 of X is a root of unity (for i=1,2), then for any two non-preperiodic points x_1,x_2, the set of pairs (n_1,n_2) of non-negative integers such that Φ_1^{n_1}(x_1) = Φ_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1,m_2,\ell_1,\ell_2 are given non-negative integers, and k is varying among all non-negative integers. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the p-adic exponential map for semiabelian varieties.
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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 | 0.005 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| 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