P-Cast Precision in FP8 Attention: Sink-Induced Collapse and the Optimality of S=2^8

arXiv:2606.06521v1 Announce Type: new Abstract: FP8 (E4M3) acceleration for attention computation offers significant throughput gains, but the 3-bit mantissa introduces precision challenges when the softmax probability matrix P is cast to FP8 before the P*V matrix multiplication. We analyze two implementation choices that affect output precision under the Attention Sink phenomenon: (1) the KV block iteration order, and (2) the static scaling factor applied to P before casting. We show that forward KV iteration causes "P-collapse" -- to leading order, a fraction Phi(Delta + delta_k - 6.93 - ln S) of non-sink P values underflow to zero, where the small shift delta_k ~ 1 (for k_sink = 4) is the expected within-sink-block score maximum -- and that reverse iteration removes it, with a zero-underflow guarantee when reverse is combined with S = 256. We further give a constructive characterization of S = 256 = 2^8 as the static scale that simultaneously satisfies (i) bit-exact IEEE 754 scaling, (ii) the lower envelope of a sawtooth function dp(S) over the E4M3 number line (dp = 2^-4, the minimum worst-case quantization step), and (iii) the maximum normal-range coverage among bit-exact (2^k) scales (a non-bit-exact scale such as 448 attains slightly higher coverage). Both optimizations are already deployed in FlashAttention-3/4 on engineering grounds; our contribution is a quantitative account of why these choices are good and a closed-form threshold Delta_c = 6.93 + ln S - delta_k for predicting kernel-level precision loss. Kernel-faithful experiments (Q, K, V in FP32 to isolate the P-cast effect) show 3-10x MSE improvement at moderate sink strengths, and paired tests confirm both fixes saturate to the same precision floor when combined.