Composition in a Bounded-Observer Projection Model: Serial Loss, Joint Observability, and Recursive Identifiability

This paper studies what is preserved when finite-dimensional linear observation maps are composed in series, combined laterally, and embedded in a recursively declared observer-locale hierarchy. The mathematically secure core is linear-algebraic: composition cannot restore distinctions already erased by a map; joint observation is represented by a product map whose kernel is the intersection of the component kernels; and an invariant is computable by an aggregate exactly when it factors through that aggregate observation map. The paper distinguishes these statements from stronger interpretations. Principal-angle separation is a geometric complementarity measure but is not, by itself, Shannon-information gain. A communication channel is a map from an observer's internal representation into a message space, not a new source subspace. The recursive epistemic boundary is conditional on level-wise channel-sufficiency assumptions. Low-dimensional examples at dimensions 1, 2, 3, 4, and 8 are separated into elementary geometric facts and conjectural capability interpretations; in particular, dimension 4 is not treated as a dimension-only Born-rule threshold, and dimension 8 supplies room for orthogonal subspaces rather than proving observer existence. The result is a compositional calculus for declared projection models together with an explicit research program for determining which biological, cognitive, or physical systems are adequately represented by that calculus.