MIsCORE–RIUD: Paper III Partially Shared Dynamics in Recurrent Gravitational-Wave Undiscovered Paths

**MIsCORE–RIUD: Paper III** tests whether the connected multiscale paths established in the preceding RIUD analyses can be described by recurrent family-level dynamics and, more strongly, whether those dynamics can be compressed into a single compact governing law. The analysis uses a frozen 13-event path panel with **13,000 control-native realizations** and **1,000 valid population draws**. Unlike a coordinate-only null, each control is allowed to construct its own feature map, search freely for a path, form a trajectory, and receive a frozen family classification before the population statistic is evaluated. This preserves the higher-order search freedom relevant to the Paper-III question. Three prelocked family-level hypotheses survive the canonical selection-aware null: **H1 multistep raw/M5, p = 0.005994; H2 oscillatory raw/M5, p = 0.002997; and H3 quasi-stationary raw/M3, p = 0.026973**. A separately regenerated internal computational run, using newly generated controls and a distinct randomization namespace, returns corrected p-values of **0.009990, 0.002997, and 0.045954**, respectively. The canonical and regenerated nulls remain separate and are not pooled. The stronger compression attempt fails. A development corridor containing **58 candidate laws**, reduced to **7 consensus candidates** and then to one selected relation, does not survive the required selection-aware test. The compact-law audit gives **search-aware p = 0.089910** and **fixed-law p = 0.677323**. The corresponding universal-law claim is therefore rejected rather than retuned. Paper III consequently preserves a deliberately asymmetric result: **recurrent partially shared path-generating dynamics survive, while one compact universal trajectory law does not**. The failed compression becomes the reason to move to a higher-order population representation rather than to continue fitting the same family-level model. Related integrated volumehttps://zenodo.org/records/22254049 Standalone Papers MIsCORE–RIUD: Paper IDefining and Falsifying Recurrent Merger-Local Morphology in Gravitational-Wave Post-Model Data* https://zenodo.org/records/22270204 MIsCORE–RIUD: Paper II Connected Multiscale Paths in Gravitational-Wave Post-Model Data* https://zenodo.org/records/22278310