OCTOPUS: Optimized KV Cache for Transformers via Octahedral Parametrization Under optimal Squared error quantization

The key-value (KV) cache dominates memory bandwidth and footprint in long-context autoregressive inference. Recent rotation-preconditioned codecs (TurboQuant, PolarQuant) show that a structured random rotation followed by a per-coordinate scalar quantizer matched to an analytically tractable marginal is a near-optimal recipe for KV compression. OCTOPUS advances this paradigm through joint quantization of rotated coordinate triplets. Each triplet's direction is mapped to a square via an octahedral parameterization, and the two resulting coordinates and the triplet norm are Lloyd-Max quantized against implementation-matched marginals. Optimizing the per-triplet squared error gives a strictly non-uniform bit allocation depending only on the total dimensionality of the keys. We find the finite-dimensional quality optimum with sweeps to be constant on every real decoder we test. The codec is data-oblivious, online, and deterministic given a seed. Across text, video, and audio, OCTOPUS matches or beats every prior rotation codec at every reported bit width and metric, with a lead that grows as bits drop for extreme compression. Furthermore, a fused Triton implementation reconstructs keys on the fly without materializing the uncompressed key, so the codec adds no decode-time bandwidth or latency over the existing dequantization.