Emergent Spatial Metric from Topological Adjacency in a Non-Spatial Information Substrate: Tensor Network Formulation of the Topological Rendering Model and Resolution-Distance Duality

The tension between quantum nonlocality and the local geometric structure of general relativity remains an open foundational challenge in quantum gravity. Dominant emergent spacetime frameworks, including ER=EPR correspondence, AdS/CFT duality and standard MERA tensor networks, universally assume primitive geometric manifolds or fixed lattice dimensions at the fundamental substrate level. Such preset background geometry creates persistent logical barriers to fully resolving the EPR paradox and observer-induced wavefunction collapse.

This work constructs the Topological Rendering Model (TRM), a graph-tensor hybrid information-theoretic framework where macroscopic 3D spatial metric and geometric distance act as coarse-grained emergent observables projected from a fully geometry-free discrete Fundamental Information Substrate (FIS). We reformulate FIS within generalized MERA tensor network formalism, mapping native topological adjacency between qubit tensors to entanglement coupling and generating macroscopic spacetime via hierarchical coarse-graining flow. Using graph min-cut theorems, we derive a discrete analog of the Ryu–Takayanagi holographic entropy law, proving area-law scaling in the continuum macroscopic limit without primitive bulk geometry.

We formalize the original Resolution-Distance Duality (RDD) principle via a perturbed low-dimensional SYK-inspired quantum toy model, performing tractable numerical simulation to quantify exponential informational resolution decay and distance-dependent decoherence timescales. We ground the coarse-graining “resolution optimization” mechanism rigorously within quantum Darwinism and the Bekenstein holographic information bound, removing informal computational analogies and establishing consistent physical ontology for RDD scaling behavior.

Within this tensor network formulation, quantum entanglement corresponds to direct topological tensor adjacency inside FIS. Bell nonlocality merely originates from mismatched intrinsic graph distance and emergent macroscopic geometric separation, requiring no superluminal signaling and fully preserving relativistic causality. The RDD duality supplies a unified information-theoretic origin for environment-independent decoherence, projective wavefunction collapse, and the quantum-to-classical transition, yielding three distinct falsifiable observational predictions for cosmological and quantum experiments. This framework eliminates primitive geometric structure at the fundamental physical layer and offers a self-consistent conceptual framework and novel information-theoretic perspective toward reconciling quantum information dynamics with emergent gravitational geometry.