Bi-Structured Horizon Geometry and Spectral Dimension Running at Black Hole Horizons

The black hole information paradox arises in part from an implicit identification: the metric structure governing spacetime geometry and the propagation structure governing quantum dynamics are assumed to coincide. We argue that relaxing this identification resolves the tension between the Equivalence Principle and unitarity without modifying either. Treating the event horizon as a bi-structured space carrying an intrinsic metric geometry and an independent propagation graph, we show that the spectral dimension of the horizon is not a fixed invariant but a scale-dependent observable that runs from a reduced value at short times—identified with the scrambling regime—to the classical spacetime dimension at late times. This running is consistent with dimensional reduction phenomena established in causal dynamical triangulations, asymptotically safe gravity, and Hořava–Lifshitz gravity. In this framework, the AMPS firewall corresponds to the short-time spectral regime: a scale-dependent phase associated with constrained propagation, not a curvature singularity of the metric geometry. A freely falling observer probing the horizon at macroscopic timescales traverses the smooth metric structure and perceives no drama. Unitarity is preserved because the constrained propagation is generated by a Hermitian Laplacian. We present a lattice simulation demonstrating the spectral crossover explicitly, confirming the coexistence of a low-dimensional spectral trap and a smooth macroscopic geometry within a single object. Implications for gravitational wave echoes, holographic complexity, and Swampland-type consistency criteria are discussed.