Event Horizon Geometry Beyond Metrics: Rigidity, Dual Scaling, and the Firewall Paradox

The black hole information paradox, sharpened by the AMPS firewall argument, highlights a tension between the smoothness of the event horizon implied by general relativity and the unitary evolution required by quantum mechanics. In this work, we address this tension by introducing a rigorous framework in which the horizon is defined as a bi-structured space (𝑋,𝐷,Γ), where intrinsic geometric structure and dynamical propagation are treated as independent primitives. We replace the standard random-matrix ansatz for black hole microstates with a deterministic matrix word algebra subject to unitary constraints. Using inverse distance rigidity results—specifically the Inverse Falconer Theorem over rings—we show that the requirement of a stable, low-entropy vacuum enforces arithmetic rigidity in the microstate distance spectrum. This rigidity confines the microstate degrees of freedom to a low-rank active simple block, producing an algebraic trap that restricts admissible propagation Γwhile preserving the underlying metric geometry. We derive a holographic emergence mechanism associated with this constrained propagation, leading to a dual-geometry scaling law at the event horizon. At short times 𝑡≪𝑡×, the horizon exhibits a reduced spectral dimension 𝑘, corresponding to firewall-like behavior in dynamical probes. At long times 𝑡≫𝑡×, diffusion relaxes to the smooth intrinsic shell dimension 𝑛. Identifying the crossover scale 𝑡×with the scrambling time, we show that the firewall arises as a spectral effect rather than a metric singularity, reconciling horizon smoothness with structured quantum microstates.