Boundary Metric from Projection-Induced Finite Rank

This work analyzes how a notion of metric necessarily arises when observables are obtained through a finite-rank, non-isometric projection. Using the truncated singular value decomposition of the projection operator, it shows that boundary distinguishability is governed by a unique operational metric implemented by a finite-rank kernel and its associated rational projection filter. The framework is pre-geometric and pre-dynamical: the metric does not describe spatial distance, but the limits of resolvable structure imposed by projection. Finite rank selects a bounded set of logarithmic scale modes, defines a finite entropic window, and fixes a unique projection constant organizing boundary observables by scale. Revised version.This manuscript is made publicly available on Zenodo as a preliminary, pre-review work. The present version clarifies the scope and interpretation of the analysis. No technical results, definitions, or conclusions have been modified. The manuscript has not undergone peer review and may be subject to further revision.