Exact matrix-free streaming enables global phosphosite co-variation mapping without quadratic result materialization

Because protein phosphorylation dynamically controls cellular signaling, mapping how thousands of sites co-vary can reveal how signals are coordinated. Exhaustive pairwise correlation of large phosphoproteomes is constrained by both quadratic arithmetic and the need to materialize, sort, and store a dense N×N result matrix. We developed a matrix-free streaming framework that evaluates every phosphosite pair blockwise, extracts local exact Top-K candidates and deterministically merges them into a global Top-K without storing the full matrix. The method retains O(N²M) arithmetic while eliminating quadratic result storage. In calibrated synthetic scaling benchmarks, a single NVIDIA RTX 3090 screened N = 200,000 features across M = 250 samples (≈ 2.0 × 10¹⁰ pairs) in a median of 24.79 s (95% CI, 24.73–24.81 s), with a GPU memory-pool peak of 0.449 GB; the corresponding dense float64 matrix would require ≈ 320 GB. At N = 10,000 and 20,000, streaming, dense-CPU, and dense-GPU outputs were byte-for-byte identical and invariant to feature order. Application to CPTAC colon, breast, and lung phosphoproteomes showed that many extreme all-site correlations reflected same-protein redundancy or technical co-measurement, supporting separate all-site, duplicate-collapsed, and cross-protein analyses. Exact exhaustive screening is therefore feasible on commodity hardware without quadratic result materialization, while preserving ordering independence and enabling transparent redundancy decomposition.