Metric Tensor Folding and Discrete Structural Alignment: Bypassing GEMM via T112 Lattice Geometry and Inversion Transduction Grammars

Abstract:State-of-the-art natural language processing architectures remain constrained by continuous floating-point matrix multiplications (GEMM) and quadratic attention scaling, imposing severe memory bandwidth and thermal taxes on silicon substrates. We present a unified framework that replaces continuous matrix projections with deterministic discrete geometry and formal structural alignment. By mapping natural language sequences onto an order-112 triangular lattice (T112)—anchored by the topological checksum identity T73 + C3627 = 2701 + 3627 = 6328 and bounded by a Hausdorff F2 -> SO(3) free group embedding (cos θ = 1/3)—we establish a zero-FLOP dynamic routing substrate. Syntactic compositionality is preserved by integrating Inversion Transduction Grammars (ITG; Wu, 1997). Operating entirely inside CPU L2/L3 cache lines via O(1) pointer offsets, our framework achieves a pure graph routing throughput of 329.17M tok/s and an end-to-end decoding rate of 6,979 to 41,119 tok/s at 0 FLOPs dynamic GEMM tax. Verified Hardware Telemetry & Invariants:- Lattice Checksum: T_73 (2701) + C_3627 (3627) = 6328 Vertices- Hausdorff Invariant: cos(θ) = 1/3 (Free Group F2 -> SO(3) Embedding)- Pure Graph Routing Throughput: 329,169,359 tok/s (0.0498 ms/pass)- End-to-End Decoding Speed: 6,979 - 41,119 tok/s (0.0243 ms/token)- Dynamic GEMM Tax: 0 FLOPs (Pure Cache Pointer Offsets)- Packaged Artifacts: Contains pre-computed PyTorch LUT tensors (.pt) and JSON telemetry manifest.