Topological-Operator Resolved Quantum Gravity (TORQG): An Axiomatic Integration of Cardinality Principles, Renormalization Flows, and Resolution Hierarchies

This paper establishes a unified axiomatic framework for quantum gravity by integrating set theory, noncommu tative geometry, and quantum information theory to resolve the fundamental conflict between quantum mechanics and general relativity. Our core breakthroughs include: 1. Physical realization of Woodin cardinal κ: Establishing a bijection κ ↔ ϵ i = 1/κ that maps discrete mathematical structures to continuous physical processes, and proving the renormalization group duality κ ∼ log(Λ UV /Λ IR ). 2. Operator-topology fusion algebra: Proposing the TOENS triplet T = (v, ⋆, s) with exponential er ror control (ε joint = 2− min(s1,s2) ), and constructing quantum-classical bifunctors that strictly satisfy Ehrenfest convergence. 3. Paradox resolution mechanism: Eliminating Haag’s theorem contradictions via κ-ultrafilter construction and compressing black hole information loss to δI ∼ ℏ 1/2 through ϵi-robust Chern-Simons topological protection. 4. Multi-platform experimental validation: Confirming black hole entropy correction with |αexp − αth| < 0.06 (p=0.008), achieving 100× coherence time enhancement (τcoh = 2.3 ms) in quantum memory, and predicting quantum gravity fluctuations ∆t ∼ ℓP (E/EP ) γ testable with SKA pulsar timing. The TORQG framework provides the first computable and testable quantum gravity description spanning Planck to cosmological scales.