Bibliographic record
Abstract
Let M be a hyperbolic 3-manifold such that ∂M has at least two boundary tori ∂0M and ∂1M. Suppose that M contains an essential orientable surface P of genus g with one outer boundary component ∂0P, lying in ∂0M and having slope λ in ∂0M, and p inner boundary components ∂iP, i = 1, � � � , p, each having slope α in ∂1M. Let β be a slope in ∂1M and suppose that M(β) is toroidal. Let ˆ T be a minimal essential torus in M(β), which means that ˆ T is pierced a minimal number of times by the core of the β-Dehn filling, among all essential tori in M(β). Let T = ˆ T ∩ M and denote by t the number of components of ∂T. In this paper we prove: (i) If t ≥ 3, then �(α, β) ≤ 6 + 10g − 5 p , (ii) If t = 2, then �(α, β) ≤ 13 + 24g − 12 p , (iii) If t = 1, then �(α, β) ≤ 1. Let M be a compact, connected, orientable and irreducible 3-manifold with a toral boundary component T. The unoriented isotopy class of a non-trivial simple closed curve on T is called its slope, and if α and β are slopes on T then �(α, β) will denote their minimal geometric intersection number. Let γ be a slope on T, and let M(γ) be the 3-manifold obtained by γ-Dehn filling. Thus M(γ) = M ∪ J, where J is a solid torus, glued to M along T in such a way that γ bounds a disk in J. A surface in 3-manifold is called essential if it is properly embedded and either (i) incompressible, not parallel to a subsurface of the boundary of the 3-manifold, and not a 2-sphere, or (ii) a 2-sphere that does not bound a 3-ball. A 3-manifold is said to be toroidal if it contains an essential torus. Recall that Thurston has shown that if M is hyperbolic then M(γ) is hyperbolic for all but finitely many slopes γ. Furthermore if M(γ) is not hyperbolic, then it is either reducible, toroidal or a Seifert fiber space. Let M be a hyperbolic 3-manifold such that ∂M has at least two boundary tori ∂0M and ∂1M. Suppose that M contains an essential orientable surface P of genus g with one outer boundary component ∂0P, lying in ∂0M and having slope λ in ∂0M, and p inner boundary components ∂iP, i = 1, � � � , p, each having slope α in ∂1M. We assume that the genus g of P is minimal (over all essential
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How this classification was reachedexpand
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 teacher head, 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".