Topological Phase Holography Quantum Geometry and Dynamical Phase Transitions in Spacetime

This paper introduces a groundbreaking framework for quantum gravity—Topological Phase Holography (TPH)—which integrates higher-order algebraic topology with dynamical quantum geometry to reveal the phase transition behavior of quantum spacetime across energy scales and its holographic duality mechanisms. Key innovations include: 1. Algebraic Topological Phase Transition Mechanism: A quantum geometric algebra AE8 based on the exceptional Lie group E8 is introduced, with its irreducible representations directly corresponding to critical points of spacetime dimensionality dH (e.g., the observationally compatible solution dH =3.02). The phase transition equation is: β(dH) = µ∂dH ∂µ = ∞ k=1 (−1)kζ(2k) (2k)! 2k ΛQG µ (dH −3)k+1 Verified via Monte Carlo simulations, the critical point stability error is σdH < 0.005. 2. Dynamical Tensor Network Holography: A bulk-boundary correspondence based on hypercubic tensor networks is constructed: HomTPH(Tbulk,Tbdy) = Dge−Stop[g] ⊗ OCFT M4 The topological action Stop = Tr(R ∧ ⋆R) + λCh4(F) naturally extends AdS/CFT duality to FLRW spacetime. 3. Observable Topological Excitation Spectrum: A quintuple-degenerate topological peak in the primordial gravitational wave spectrum is predicted: 2 PT(k) = PGR T (k) m=−2 eimθk (θk = arccos(k/ΛQG)) This feature can be verified through polarization correlation measurements by LISA and DECIGO (expected signal-to-noise ratio SNR > 7). The theory reduces to general relativity in the low-energy limit (∆R/R < 10−6) and achieves a topological protection threshold of ηprotect > 99.9% via superconducting quantum simulation. Compared to existing theories, TPH is the first to 1combine algebraic topology classification theorems with dynamic phase transitions in quantum gravity, providing a new paradigm for exploring cosmic inflation mechanisms and black hole information paradoxes