Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702
Definition, evaluation, and provenance
The fixed-kernel problem and its specified inputs › Definition, evaluation, and provenance · lines 289–353
Three different qualifications are used below:
- Specified but unevaluated.
A finite, unambiguous formula is available, but its component expansion or its contraction in the characteristic calculation has not been performed.
- Fixed input not instantiated here.
An array or distribution is part of the mathematical data of the paper. Its particular physical realization is not independently derived here.
- Inherited interpretation.
A predecessor identifies a displayed expression with a branch of the physical graph. That interpretation is not inferred merely because two manuscripts use similar notation.
A reproducible symbolic specification suffices to define coefficients. An explicit component list is not inherently required. Conversely, a component expansion does not replace a missing boundary-value prescription.
| Object | Mathematical status | Treatment here |
|---|---|---|
| Curvature and cubic density | Finite specification | Reproduced in Appendix app:source, without changing the unintegrated representative. |
| Weighted Euler source | Finite specification | Derived by coefficient extraction and converted explicitly to the propagated component convention. |
| Specified jet coefficients | Determined by (eq:jet-Hessian); their complete graph contraction is unevaluated. | |
| Explicit linear maps | Their matrices and their component contractions are fixed below. | |
| Fixed constant bilinear map | Retained as an input; no physical component realization is independently derived. | |
| Crossed array | Fixed distributional input | Its grouped tensor-product form is retained; physical component boundary values are not evaluated. |
| Complete characteristic coefficient | Open calculation | Not supplied by the source rule or the scalar model. |
The source and the covariance package are separate inputs. Fixing the curvature convention does not instantiate the crossed covariances.