The equations of nature and the nature of equations
Bibliographic record
Abstract
Systems of ${N}$ equations in ${N}$ unknowns are ubiquitous in mathematical modeling. These systems, often nonlinear, are used to identify equilibria of dynamical systems in ecology, genomics, control, and many other areas. Structured systems, where the variables that are allowed to appear in each equation are pre-specified, are especially common. For modeling purposes, there is a great interest in determining circumstances under which physical solutions exist, even if the coefficients in the model equations are only approximately known. The structure of a system of equations can be described by a directed graph ${G}$ that reflects the dependence of one variable on another, and we can consider the family ${\mathcal{F}(G)}$ of systems that respect ${G}$. We define a solution ${X}$ of ${F(X) = 0}$ to be robust if for each continuous ${F^*}$ sufficiently close to ${F}$, a solution ${X^*}$ exists. Robust solutions are those that are expected to be found in real systems. There is a useful concept in graph theory called cycle-coverable. We show that if ${G}$ is cycle-coverable, then for almost ${F\in\mathcal{F}(G)}$ in the sense of prevalence, every solution is robust. Conversely, when ${G}$ fails to be cycle-coverable, each system ${F\in\mathcal{F}(G)}$ has no robust solutions. Failure to be cycle-coverable happens precisely when there is a configuration of nodes that we call a bottleneck, a criterion that can be verified from the graph. A bottleneck is a direct extension of what ecologists call the Competitive Exclusion Principle, but we apply it to all structured systems.
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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 | 0.000 | 0.006 |
| 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 | 0.001 |
| 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".