Physical Incompleteness from Causal Geometry in Semiclassical Gravity

We establish an incompleteness result for semiclassical theories of gravity that is structural, not epistemological: it arises from spacetime causal structure rather than from limitations in measurement technology, computational resources, or observer capabilities. Focusing on black hole spacetimes, we show by explicit construction under consistency and soundness assumptions that physically realizable quantum states exist whose complete descriptions are not derivable within a semiclassical framework restricted to a fixed exterior causal domain.

The argument rests on three elements: (i) event horizons induce causal barriers that generate observational indistinguishability classes of physical states; (ii) the Bekenstein–Hawking entropy bound provides sufficient encoding capacity to implement physical self-reference; and (iii) semiclassical gravity supports Gödel-type diagonal constructions via an arithmetic interpretation in Fock space. These are unified in a single Master Incompleteness Theorem (Theorem 4.13) with explicit hypotheses and logically independent conclusions covering geometric underdetermination, Gödelian undecidability, and their categorical equivalence via Lawvere's fixed-point theorem.

For a solar-mass black hole with interior encoding using nq quanta, the fractional backreaction satisfies δg/g ≲ nq × 10⁻⁷⁶, ensuring semiclassical self-consistency. We discuss implications for the black hole information problem, outline potential tests in analog gravity systems, and identify structural constraints that any completeness-restoring quantum gravity theory must satisfy.