An Application Note: Registrability-Based Structural Classification of the SPT2349-56 Proto-Cluster Event

SPT2349-56 (z ≈ 4.3) is a compact, high-redshift proto-cluster complex exhibiting extreme star formation rates, strong multi-wavelength emission, and elevated intracluster medium temperature inferred via the Sunyaev–Zeldovich effect. These signatures have prompted discussion regarding accelerated structure formation, modified feedback mechanisms, or non-standard cosmological amplification. In this application note, we apply a registrability-based structural classification framework to SPT2349-56 without modifying cosmological parameters or invoking anomalous physics. The framework enforces ordered evaluation of duration, chromatic behavior, multi-channel persistence, geometric admissibility, dynamical stability, recurrence, and identity preservation prior to interpretive inference. We demonstrate that SPT2349-56 is naturally classified as an identity-preserved, over-constrained structural basin (Cluster-Class IV analogue). The observed thermal excess and strong Sunyaev–Zeldovich signal are interpreted as downstream diagnostics of early structural compression and boundary-mediated routing rather than as evidence of modified gravitational dynamics. This analysis illustrates how registrability-first classification provides a conservative and reproducible diagnostic tool for extreme high-redshift systems, preventing escalation based solely on thermal amplitude or compactness while remaining fully compatible with ΛCDM cosmology.