Tradeoffs Between Rate and Conditional Content with Encoder-Observed Context

Version 1.7 (2026-09-23): the review manuscript as submitted to the IEEE Transactions on Information Theory, a snapshot of paper/tit-cr-context at tag tit-cr-context-1.7 of github.com/ahb-sjsu/geometric-observation. Single-column review format, 33 pages; figures set in Type 1 fonts through the pgf backend; the AI-assistance acknowledgment names the systems, sections and level of assistance. Title and mathematics unchanged from 1.5. Files: the manuscript PDF, the executed reproduction notebook (reproduce.ipynb), and a zip of the manuscript directory with its LaTeX source, figures, figure scripts, independent numerical and symbolic verification scripts, MATLAB checks, and archived earlier versions.

An encoder observes a jointly Gaussian pair (Y, V) and describes Y for a decoder that sees the description alone; a third party retains a noisy copy S = V + U of the context and never decodes. The paper evaluates Steinberg's common-reconstruction functional on this augmented source: it gives the minimal conditional content in closed form, the exact rate-content region with its two-noise-fraction Pareto frontier and a strict tradeoff exactly when the described variable and the context are partially correlated, non-determination by the reduced (Y, S) law, the vector-context frontier, the binary rate-content frontier through a tilt equation, and a direct Gaussian operational theorem.