Anomaly-Induced Horizon Memory and Topological Signatures in Unitary Black Hole Evaporation within Algebraic Quantum Gravity

This manuscript presents a rigorous, non-perturbative, and background-independent resolution to the black hole information paradox and the AMPS firewall problem within the twin frameworks of Algebraic Quantum Gravity (AQG) and Non-Commutative Geometry (NCG). By abandoning continuous semiclassical coordinate manifolds at the Planck scale, the event horizon is formulated algebraically as a unique Type $\text{III}_1$ von Neumann factor over an indefinite Lorentzian Krein space spectral triple $(\mathcal{A}_{QG}, \mathcal{H}_{\text{Krein}}, \mathbb{D}_{\mathbb{A}})$. We introduce the Horizon Memory Principle, demonstrating that the global gravitational chiral anomaly—extracted via the Wodzicki residue—acts as an exact topological information conduit governed strictly by the Jaffe-Lesniewski-Osterwalder (JLO) entire cyclic cocycle.

By invoking the algebraic Longo-Doplicher split property to construct an intermediate Type $\text{I}_\infty$ factor, we bypass mathematically ill-defined local partial traces and prove the existence of an exact, phase-preserving quantum isometry ($V_{\mathcal{H}}$) that preserves the inner product of infalling states. A non-autonomous modular flow mapped to asymptotic observer time via a Connes Radon-Nikodym clock intertwiner formally derives a unitary Page curve, culminating in an inert, stable, zero-entropy topological ground-state remnant. Finally, the framework offers explicit, falsifiable astrophysical directives to break the quantum gravity empirical deadlock, predicting a $16\%$ parity-violating helicity asymmetry in terminal primordial black hole radiation and localized, phase-inverted gravitational-wave echo packets in binary mergers such as GW190521.