On Monotonic Functionals Over Partially-Ordered Path Spaces
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Bibliographic record
Abstract
Abstract We study non-decreasing (non-increasing) monotonic functionals over unions of Skorokhod spaces of càdlàg paths which we show to have an equivalence to non-negative (non-positive) path-dependent spatial Dupire derivatives. These functionals provide an upper-bound for their Lie-bracket of non-commutative spatial and temporal Dupire operators. We provide a stochastic functional generalisation for the Lebesgue integral of derivatives of non-decreasing (non-increasing) functions. We also present a functional generalisation of Markov’s inequality. One can further associate monotonic functionals of order-preserving random paths to their stochastic differential equations. We encapsulate what we call buffered monotonic functionals on paths that never draw closer than a minimum distance over their lifetime. As an application, we generate path-dependent stochastic triangles that randomly change their location, shape and area, while embedding a minimum structure that ensures convexity of the geometry at every point in time—a construct for modelling temporal population cluster dynamics with memory.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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