Nondegeneracy of half-harmonic maps from ℝ into 𝕊¹
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Bibliographic record
Abstract
We prove that the standard half-harmonic map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U colon double-struck upper R right-arrow double-struck upper S Superscript 1"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">U:\mathbb {R}\to \mathbb {S}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined by <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow StartBinomialOrMatrix StartFraction x squared minus 1 Over x squared plus 1 EndFraction Choose StartFraction minus 2 x Over x squared plus 1 EndFraction EndBinomialOrMatrix"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtable rowspacing="0.8em 0.4em" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mfrac> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} x\to \begin {pmatrix} \frac {x^2-1}{x^2+1} \\[4pt] \frac {-2x}{x^2+1} \end{pmatrix} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> is nondegenerate in the sense that all bounded solutions of the linearized half-harmonic map equation are linear combinations of three functions corresponding to rigid motions (dilation, translation, and rotation) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 it