Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures
Bibliographic record
Abstract
We develop a framework that systematically casts solvability and uniqueness conditions for linearized overdetermined boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the underlying geometric structure and provides tools to study the resulting cohomology explicitly. To achieve this generality, we develop the notion of an elliptic pre-complex, which generalizes the machineries of Hodge theory to sequences of Douglis--Nirenberg systems in the pseudodifferential calculus of boundary-value problems that interact through Green's formulae, the notion of overdetermined ellipticity, and a condition we call the order-reduction property, which relaxes the rigid requirement that the sequence form a cochain complex. This property typically arises from linearized symmetries and constraints, as we demonstrate through several geometric examples that have long resisted analysis, including exterior covariant derivatives, the Killing and Hessian equations, and the Riemann curvature equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".