Attention at the Theoretical Minimum: A Mathematics of Arrays Framework for Memory-Optimal Transformer Kernels

The attention mechanism is the dominant computational bottleneck in modern transformer-based AI. Its standard implementation incurs quadratic memory traffic in the sequence length~n, and DRAM accesses cost 100--1000× more energy than arithmetic operations on contemporary hardware, so any analysis focused solely on FLOP counts fundamentally mischaracterises the bottleneck. We present a Mathematics of Arrays (MoA) reformulation of scaled dot-product attention and its numerically stable softmax, deriving a Denotational Normal Form (DNF) that eliminates all intermediate arrays -- including the implicit transposed-key buffer and every softmax temporary -- by algebraic construction rather than empirical tuning. The DNF achieves O(ndk + ndv) data movement versus O(n2 + ndk + ndv) for the standard implementation, where n is the sequence length, dk is the key dimensionality and dv the value dimensionality, and is verified numerically against PyTorch at full double-precision floating-point on concrete inputs. Unlike hardware-specific accelerators or empirical tiling schemes such as FlashAttention, MoA simultaneously provides array fusion, shape-transformation correctness, and predictive cost models from a single algebraic framework. Memory minimality is a theorem established before any code is written. A predictive performance model projects 2--100× speedup and 2--50× energy reduction, with the advantage widening at exascale. The derivation establishes a formally verified pipeline from Python specification through (ONF) Operational Normal Form, and dimension-lifted hardware mapping, providing performance-portable AI kernels of direct relevance to DARPA edge-deployment and DOE exascale priorities.