The Recovery Problem: Objectivity-Graded Reconstruction of Physical Structure by an Embedded Observer

Papers 1–18 typed the world-side structures of a consistency-constrained information history and, in the fourth triptych, causation, lawhood, and explanation. This paper inverts the program: given only the finite, local, record-mediated access available to an embedded observer, how much of that structure can be reconstructed, under what non-uniqueness, and at what cost? We separate two type-distinct senses of recovery — carrier/information recovery and structural reconstruction — and pose structural reconstruction as an inverse problem governed by a protocol-level forward map (a transcript distribution over whole experiments), a representation-invariant fit rule, and a declared recovery specification. The central negative principle, the No-Global-Decoder Constraint, adds one term to the series' “no global X” sequence: no physically privileged subsystem possesses complete-history access together with a uniquely authoritative decoding map. The central positive result is an elementary but explicit Protocol-Kernel Recovery-Limit Theorem; its value is not mathematical difficulty but the placement of causation, lawhood, explanation, objectivity, and horizon recovery inside one inverse-problem formalism. We distinguish three notions of uniqueness (global, observation-relative, finite-record); show that protocol refinement is monotone while finite-sample compatibility is not; define target-relative correctness through an external adequacy relation to the actual structure; profile recovered structure along the representational, evidential, and corrective objectivity axes of Paper 15; and exhibit the framework on two worked models, a linear-Gaussian structural-causal triple separated in stages by interventions and a conformal-versus-metric spacetime example. We emphasize that, with finite or noisy records, the output of reconstruction is in general not a single equivalence class but a specification-relative compatible set or indistinguishability region, and that no reconstruction certifies its own completeness, uniqueness, or correctness.