Bibliographic record
Abstract
We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> without boundary, <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta Subscript g Baseline u Subscript i Baseline equals upper H Subscript i Baseline left-parenthesis u 1 comma midline-horizontal-ellipsis comma u Subscript m Baseline right-parenthesis on double-struck upper M comma"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mtext> </mml:mtext> <mml:mtext>on</mml:mtext> <mml:mtext> </mml:mtext> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} -\Delta _g u_i = H_i(u_1,\cdots ,u_m) \ \ \text {on} \ \ \mathbb {M}, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript g"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Delta _g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stands for the Laplace-Beltrami operator, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript i Baseline colon double-struck upper M right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">u_i:\mathbb {M}\to \mathbb R</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 H Subscript i Baseline element-of upper C Superscript 1 Baseline left-parenthesis double-struck upper R Superscript m Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_i\in C^1(\mathbb R^m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i less-than-or-equal-to m"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>i</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1\le i\le m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This system is called symmetric if the matrix of partial derivatives of all components of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H left-parenthesis u right-parenthesis equals left-parenthesis partial-differential Subscript j Baseline upper H Subscript i Baseline left-parenthesis u right-parenthesis right-parenthesis Subscript i comma j equals 1 Superscript m"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi mathvariant="normal">
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.
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| 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 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".