On the Causal Incompleteness of Entropic Information: A Theory of Rooted Causal Blocks

Quantum information is usually characterized by a state available at a given time and by entropic or channel-theoretic quantities derived from it. We develop a broader operational framework in which entropic content is one coordinate of physical information rather than its exhaustive causal description. Relative to an accessibility restriction C_X, we distinguish entropic content S, generative capacity G, accessible causal efficacy, causal shielding, and the local, distributed, recoverable, and blocked records of a physical history. We introduce rooted causal blocks as history-labelled points in fibres of the restriction map over a local carrier state, and prove a Causal-Fibre Shielding and Recovery Theorem: equal local carrier states force the locally accessible historical record to vanish, whereas global distinction implies a positive record stored outside the carrier, quantitatively lower-bounded by binary distinguishability; any later common map that restores local distinction certifies a recoverable fraction of that record. Thus local identity of a carrier need not imply identity of its causal history. We further prove a Bekenstein causal non-exhaustiveness result: a bounded carrier with fixed local Hamiltonian and radius has fixed local entropy, energy expectation, and Bekenstein bound, yet may belong to distinct rooted blocks with different distributed records and causal futures. This does not violate the Bekenstein entropy bound; it rejects only its interpretation as a complete taxonomy of causal information. The proofs use established quantum-information tools. A separately labelled, spectrally constrained completion and an experimental falsification hierarchy are proposed as conjectural extensions, kept formally distinct from the proven results. Version 2 adds certification at a sufficient causal cut, a dynamical transition portal for the proposed completion with bounds from published atomic lifetime data and from solar energy loss, and a two-ensemble transfer test across a causal cut.