On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a translation surface of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 g minus 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2g-2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conical points of angle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 pi"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi> π </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma prime"> <mml:semantics> <mml:msup> <mml:mi> γ </mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\gamma ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be two homologous saddle connections of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> joining two conical points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and bounding two surfaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">S^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript minus"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mo> − </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">S^-</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with boundaries <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper S Superscript plus Baseline equals gamma minus gamma prime"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi> γ </mml:mi> <mml:mo> − </mml:mo> <mml:msup> <mml:mi> γ </mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial S^+=\gamma -\gamma ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper S Superscript minus Baseline equals gamma prime minus gamma"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msup> <mml:mi>S</mml:mi> <mml:mo> − </mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> γ </mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo> − </mml:mo> <mml:mi> γ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial S^-=\gamma ’-\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Gluing the opposite sides of the boundary of each surface <inline-formula content-type="math/mathml"> <mml:math
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.013 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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