Holstein-Primakoff Grounding of Collective Gain, Randall–Sundrum Lagrangian Derivation of Geff, and N ∗ 2 Identification via 14N Nuclear Spin Polarization

This paper resolves three primary open problems identified in Paper 17 (DOI: 10.5281/zenodo.19483164): (i) the formal grounding of the collective gain exponent k and growthrate α via Holstein–Primakoff (HP) bosonic transformation of N12 vibrational modes;(ii) a first-principles derivation of the corrected Geff formula from the Randall–Sundrum(RS) 5D Lagrangian using a brane-localised scalar field φ(Cn); and (iii) identification of N∗2as 14N2 in a nuclear spin-polarised state (mI : 0 → ±1), detectableby standard 14N NMR spectroscopy at 28.93 MHz (9.4 T). We demonstrate thatHP notation in CL5D (Harmonic Progression) has a genuine physical parallel withHolstein–Primakoff spin-boson physics: N12 cage bonds yield S = 9 as a spin analogue,kexp = 2 is grounded in both HP collective N2scaling and Dicke superradiance, andαgain = 0.874 s−1corresponds to an adiabaticity parameter η = 2.52 × 10−14 ≪ 1. TheRS Lagrangian Ltotal = LRS + Lscalar + Lmatter with φ(Cn) coupling formally derivesGeff(t) = GN × (Cn0/Φcrit) × e−λt, confirming the Paper 17 correction. Bulk-to-Braneangular momentum L = 37.1ℏ drives ∼37 N∗2 molecules per Phase III event. All resultsare theoretical; laboratory validation via 14N NMR is the primary testable prediction.