BP+OSD Threshold of the 4D Cubic Toric Family Across Distance Classes d ∈ {4, 8, 12, 16}: d=4 Sub-Family Expansion and 100k-Trial Refinement of [[1152,6,12]] and [[1536,6,16]]

Extension of the 4D cubic toric BP+OSD threshold study (Zenodo 20304690) to six lattices spanning four distance classes: - d=4 sub-family at 10,000 trials/rate: [[96,6,4]] (2,2,2,2), [[192,6,4]] (2,2,2,4), [[384,6,4]] (2,2,4,4) - d=8: [[768,6,8]] (2,4,4,4) at 10,000 trials/rate - d=12: [[1152,6,12]] (3,4,4,4) at 100,000 trials/rate - d=16: [[1536,6,16]] (4,4,4,4) at 100,000 trials/rate All under BP+OSD-CS-7 (ldpc 2.3.9, min-sum 0.625, 50 iterations, parallel decoding). i.i.d. X-error noise model, Z-syndrome decoding, 9 noise rates p ∈ {0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08}. Three main findings: 1. Distance rule confirmed across the family. The minimum-pair-product rule d = min_{i