The Cold Component Keeps Its Collapse Mass: A Derived Dark-Sector Rule for a Lambda-Tied MOND Law, Tested with Weak-Lensing Collapse Masses

The programme ties the MOND acceleration scale to the vacuum, a0 = kappa c sqrt(G rho_Lambda), with kappa = 1/2 fitted, not derived. Its best current version adds a cold fluid with the cosmic amount Omega_c h^2 = 0.12. Inside bound systems the fluid supplies the law's phantom density, and its cosmic share is written against the baryons a system holds today. That bookkeeping leaves baryon-depleted systems short: twenty X-ray groups by 1.4x at R500 and 1.9x inside R2500, tracking their lost baryons (Spearman rho = -0.96), and seven X-ray-bright ellipticals by a factor 1.9. The rule. The fluid is pressureless and conserved, so feedback cannot remove it: a bound system keeps the cold mass it collapsed with. The fluid first builds the law's phantom, and any leftover stays as collapse debris with the collapse profile. No constant is added, but the collapse mass must be measured. Here it comes from weak lensing (Mandelbaum et al. 2016), which separates passive from star-forming central galaxies. The tests. With a colour-blind stellar-to-halo relation the rule breaks the most massive spirals (up to +0.30 dex in rotation speed). With the measured colour split, every star-forming spiral is left on the law. Sixteen passive early types with extended HI discs follow the law (+0.026 +- 0.085 dex) and receive no debris. The X-ray ellipticals half-close (+0.125 dex, 1.0 sigma). The globular-cluster dispersions of 19 SLUGGS early types, which the law leaves +0.080 dex short (3.3 sigma), are fitted at +0.007 +- 0.017 dex (0.4 sigma). Costs and limits. Added to the programme's 48-gate harness, the rule moves SPARC by 0.0009 dex and the cold budget by at most 0.007 of Omega. One massive S0 in SPARC disagrees by +0.14 dex. LambdaCDM on the same measured halos sits within about 2 sigma, so these data do not favour the framework over LambdaCDM. Every quoted number is re-derived from committed outputs by PAPER36_audit.py (48 of 48). AI-assisted research programme; not peer reviewed; nothing is claimed closed.