Universality without Totality: Diagonal Constraints on Information, Coherence, and Holography

This work develops a unified logical and physical framework for understanding the black-hole information problem as a limitation of decoding, rather than a problem of information loss. We argue that in sufficiently expressive and self-referential quantum systems, information about the system can exist and be preserved in correlations without admitting a single, globally coherent, and uniformly decodable internal representation.

The analysis proceeds in three complementary layers.

First, we isolate a semantic no-go principle based on diagonal arguments (Cantor, Gödel, Turing, Lawvere): once a system can internally represent and act on descriptions of itself, no total internal decoder can exist that decides all semantic predicates about that system. This constraint applies to information about the system (truths, predicates, observables), not to the existence or conservation of correlations.

Second, we translate this logic into categorical holography, modeling bulk–boundary relations as encoding functors. We show that while holography can be universal in a precise abductive sense (robust, representation-independent cores), it cannot be total. Modern holographic mechanisms—quantum error-correcting code subspaces and entanglement wedge reconstruction—already realize this “universality without totality” by making reconstruction relative to regions, sectors, and states.

Third, we interpret the island / quantum extremal surface prescription as a physical regularization of non-totality. Island formation makes the reconstructible bulk region a state- and regime-dependent output of a variational principle, resolving Page-curve and monogamy tensions without invoking information loss. Black holes emerge as maximally self-referential systems, where horizons enforce observer separation, Hawking radiation acts as an internal encoding, and decoding is itself a physical process within the same system—rendering total internal decoding impossible in principle.

Finally, we connect these semantic constraints to a physical mechanism, Higher Categorical Coherence Breakdown (HCCB). In this framework, global linear/unitary descriptions fail to glue coherently across contexts due to higher-order coherence obstructions, while local consistency is preserved. The operational manifestation is sectorization, history dependence, and completely positive (CP) effective dynamics on accessible algebras, rather than a single global decoder.

The paper concludes that the black-hole information paradox is best understood as a totality paradox, not a paradox of information destruction. The same structural limitation appears beyond gravity—in algebraic quantum field theory (Type-III algebras), gauge theories with edge modes, quantum measurement with nested observers, and open quantum systems—indicating a universal constraint on information in self-referential quantum theories.

Key message:
Black holes do not destroy information; they expose a universal limit of information itself—namely, that in sufficiently self-referential quantum systems, information about the system can exist and be preserved without ever being totally decodable.