Born Attention is Linear

Softmax self-attention costs O(T^2) in the sequence length T because the exponential kernel has no finite-dimensional feature map, so linear-time variants approximate it. We observe that Born-rule attention, whose weights are quantum-measurement probabilities a_kj ∝ | |^2 over complex queries and keys, is an exact finite kernel with the rank-one density-matrix feature map ρ_x = x x†, since | |^2 = tr(ρ_Qk ρ_Kj). As the complex degree-two case of polynomial-kernel linear attention it admits an exact O(T d^2) causal recurrence and constant-state decoding (d^2 (d_v + 1) reals per head, with d_v the real value width, independent of T), and it keeps rotary positions exactly. For GPT-2's head geometry (64 real channels paired into d = 32 complex) the feature dimension is d^2 = 1024. Measured on an A100, compiled constant-state decoding runs from 6.4 MB per layer at 0.12 ms per token at any length, against a 3.2 GB FlashAttention-2 cache at 2.26 ms per token at 1M tokens; unfused prefill overtakes FlashAttention-2 between 64k and 128k tokens (11x at 1M), and a fused form (Triton feature lift plus a chunkwise linear-attention kernel) between 16k and 32k (4x at 128k), reaching training parity at 64k. The linear and quadratic forms are two implementations of one operator; whether that operator matches softmax is a separate modeling question, answered here only at small scale: the p = 1 operator bolted into a real-valued GPT trails softmax by 0.12 nats at context 1k and 0.77 at 16k (attention dilution; a decay kept exactly in the d^2 state restores the 0.12 gap at 16k, and p = 2 sharpening with decay recovers 74% of the 1k gap), whereas inside a fully complex residual stream it matches softmax at 1k and keeps a quarter of the real-stream gap at 4k; gated linear attention beats both there. At 128k context on PG-19 the variable that matters is forgetting: plain Born dilutes, Born with a fixed decay carried in its state is 0.32 nats ahead of softmax and within 0.04 of gated linear attention, which stays 1.8x faster. Streamed with their constant state over the PG-19 test partition in 5M-byte windows, two 4M-complex-parameter byte models of the source framework score 1.66 (linear Born with decay) and 1.70 (gated) bits per byte, where a 350M-parameter hierarchical Mamba–Transformer stack built for that window reports 2.45. Preprint, October 2026. Also submitted to arXiv (identifier to be added on announcement). Code: the CNet framework, https://github.com/crasmarum/CNet .