A new characterization of the analytic surfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> that satisfy the local Phragmén-Lindelöf condition
Bibliographic record
Abstract
We prove that an analytic surface <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> in a neighborhood of the origin in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> satisfies the local Phragmén-Lindelöf condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>PL</mml:mi> <mml:mi>loc</mml:mi> </mml:msub> </mml:math> at the origin if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> satisfies the following two conditions: (1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is nearly hyperbolic; (2) for each real simple curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the (algebraic) limit variety <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -dimensional analytic variety <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> to satisify <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>PL</mml:mi> <mml:mi>loc</mml:mi> </mml:msub> </mml:math> .
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".