Retained internal formulas and local existence

This appendix states the retained architecture as mathematical definitions. Its identification with any physical Hessian is a separate ledger entry.

Let $e_I$ be a basis of covariant symmetric tensors and $e^I$ its algebraic dual:

All spacetime index raising uses $\eta$. Define

The notation ${\mathsf{A}}^{\mu \nu }{}_L$ means evaluation on $e_L$, not a newly introduced component metric. Similarly ${\mathsf{b}}_L={\mathsf{b}}^L$.

Fix smooth $f,g,F$, a constant array $\mathsf{D}$, and finite smooth source coefficients ${q_{\alpha \beta }}^{L,I,J}$, with $|\alpha |+|\beta |\le 2$.