Quadratic equations in hyperbolic groups are NP-complete
Bibliographic record
Abstract
We prove that in a torsion-free hyperbolic group Γ \Gamma , the length of the value of each variable in a minimal solution of a quadratic equation Q = 1 Q=1 is bounded by N | Q | 3 N|Q|^3 for an orientable equation, and by N | Q | 4 N|Q|^{4} for a non-orientable equation, where | Q | |Q| is the length of the equation and the constant N N can be computed. We show that the problem, whether a quadratic equation in Γ \Gamma has a solution, is in NP, and that there is a PSpace algorithm for solving arbitrary equations in Γ \Gamma . If additionally Γ \Gamma is non-cyclic, then this problem (of deciding existence of a solution) is NP-complete. We also give a slightly larger bound for minimal solutions of quadratic equations in a toral relatively hyperbolic group.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.024 | 0.004 |
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".