Monge’s transport problem on a Riemannian manifold
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Bibliographic record
Abstract
Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript plus Baseline not-equals mu Superscript minus"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> μ </mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo> ≠ </mml:mo> <mml:msup> <mml:mi> μ </mml:mi> <mml:mo> − </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu ^+ \ne \mu ^-</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , find the measure-preserving map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s colon upper M long right-arrow upper M"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false"> ⟶ </mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s:M \longrightarrow M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> — and assuming absolute continuity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript plus"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> — an optimal map will be shown to exist. Aspects of its uniqueness are also established.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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 it