JaDE: Grounding Discrete Logic in a Topology-Invariant Continuous Substrate

AbstractA primary mathematical constraint in scaling multimodal architectures for real-time physical simulationis the difficulty autoregressive models face when evaluating continuous dynamics, often bounded by thesequential computational overhead of discrete token prediction. The JaDE (Joint Asynchronous DifferentialEngine) framework addresses this limitation by analytically decoupling abstract semantic reasoning fromcontinuous physical simulation. Deployed on centralized compute clusters, JaDE utilizes a dual-hemispherearchitecture: an autoregressive Transformer manages discrete logic and topological hypothesis generation,while an Infinite-Dimensional Equilibrium (IDE) manifold resolves arbitrary-dimensional physics via atopology-invariant, 1-dimensional computational substrate.By introducing Block-Sparse Modality-Routed Cross-Attention and a Perceiver IO Latent Bottleneck, JaDEpartitions the continuous domain into orthogonal physical subspaces, rendering the global cross-modalityJacobian strictly block-diagonal and analytically tractable. Supported by a formal implementation of hardware-level constraints—including Gather-Scatter Active Set Compaction, Ghost Cell Halo Padding, Strang-Marchuk Symmetrized ADI Splitting with Matrix-Free Krylov fallbacks, Deterministic FP32 AndersonAcceleration, and Amortized Spectral Normalization—the engine explicitly bounds floating-point truncationdrift, global DRAM bandwidth saturation, and Streaming Multiprocessor (SM) warp divergence. Directed byActive Free-Energy optimization to mitigate irreducible entropy states, JaDE provides a continuous generativeengine capable of simulating infinite-depth, ∞-dimensional physical environments with a strictly boundedO(1) memory footprint, establishing a computational paradigm invariant to the structural complexity of themodeled domain.This manuscript establishes the formal mathematical boundaries, algorithmic routing protocols, and theoreticalhardware constraints of the JaDE architecture. The realization of this framework—specifically the nativeC++ memory allocators, asynchronous CUDA execution graphs, and the associated LLVM MLIR dialectsrequired to lower these topology-invariant operations to available compute equipment—is currently in activedevelopment, with empirical validation and cluster-scale benchmarks deferred to a subsequent publication.[Update V2: Expanded with rigorous mathematical boundaries, hardware-level compiler constraints, and theoretical roofline analysis.]