Existing attention accelerators often trade exact softmax semantics, depend on fused Tensor Core kernels, or incur sequential depth that limits FP32 throughput on long sequences. We present \textbf{ELSA}, an algorithmic reformulation of online softmax attention that (i)~preserves exact softmax semantics in real arithmetic with a \emph{provable}
O(ulogn) FP32 relative error bound; (ii)~casts the online softmax update as a prefix scan over an associative monoid
(m,S,W), yielding
O(n) extra memory and
O(logn) parallel depth; and (iii)~is Tensor-Core independent, implemented in Triton and CUDA C++, and deployable as a \emph{drop-in replacement} requiring no retraining or weight modification. Unlike FlashAttention-2/3, which rely on HMMA/GMMA Tensor Core instructions and provide no compatible FP32 path, ELSA operates identically on A100s and resource-constrained edge devices such as Jetson TX2 -- making it the only hardware-agnostic exact-attention kernel that reduces parallel depth to
O(logn) at full precision. On A100 FP32 benchmarks (1K--16K tokens), ELSA delivers
1.3--
3.5× speedup over memory-efficient SDPA and
1.97--
2.27× on BERT; on Jetson TX2, ELSA achieves
1.5--
1.6× over Math (64--900 tokens), with
17.8--
20.2% throughput gains under LLaMA-13B offloading at
≥32K. In FP16, ELSA approaches hardware-fused baselines at long sequences while retaining full FP32 capability, offering a unified kernel for high-precision inference across platforms. Our code and implementation are available at https://github.com/ming053l/ELSA.