Maximal monotone operators are selfdual vector fields and vice-versa
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Bibliographic record
Abstract
If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $X\times X^*$, then the vector field $x\to \bar\partial L(x):=\{p\in X^*; (p,x)\in \partial L(x,p)\}$ is maximal monotone. Conversely, any maximal monotone operator $T$ on $X$ is derived from such a potential on phase space, that is there exists a selfdual Lagrangian $L$ on $X\times X^*$ (i.e, $L^*(p, x) =L(x, p)$) such that $T=\bar\partial L$. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form $T=\partial ϕ$ for some convex lower semi-continuous function on $X$. This representation allows for the application of the selfdual variational theory --recently developed by the author-- to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form $Λx\in Tx$ for a given map $Λ: D(Λ)\subset X\to X^*$, can now be obtained by minimizing functionals of the form $I(x)=L(x,Λx)-< x, Λx>$.
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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 | 0.000 | 0.000 |
| 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 | 0.001 | 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 it