The Supply Edge: A Mass-Conserving Rule Without Tuned Radii Fixes Structure Growth Under a Vacuum-Tied MOND Law (Working Note)

Working note, version 2.0 (2026-10-08). AI-assisted research programme; not peer reviewed. Version 1.0 (2026-10-07) stated the supply-edge hypothesis; this version adds the simulations that test it. We study the law a0 = kappa c sqrt(G rho_Lambda) (kappa = 1/2, fitted) with the kernel nu(y) = 1/(1 - exp(-sqrt y)) and a cold fluid of fixed cosmic amount. In particle-mesh simulations, letting the law act wherever matter has collapsed makes small-scale structure too clumpy: the excess in the matter power spectrum at k <= 1 h/Mpc is +15.3% / +19.3% on the two footings at 256^3 and +17.4% at 512^3, against an allowed 10%. We test one rule with no tuned radius. The law's extra (phantom) mass is the cold fluid itself, so a halo can build the phantom only until its own cold fluid is used up (for an undepleted halo at r_edge = r_M / ln[1/(1 - f_b)] = 5.85 r_M), and the cold fluid used is drawn from the same halo's turnaround sphere, so mass is conserved halo by halo. With this rule the excess is 2.7% at 256^3 and 3.3% at 512^3, with sigma_8 within 0.6% of the Newtonian control: 15 of 15 runs pass, across three random realisations, both footings, a0 tracking dark energy, and variations of the two settings inherited from earlier work. Turning the mass conservation off restores the excess (15.4%). Earlier rules that set the edge by hand, or from measured baryon censuses, sit at the 10% limit at 512^3. The rule's only inputs are kappa and the cosmic baryon fraction. The cold fluid is handled as bookkeeping, not as simulated particles; its settling is imposed, not derived from an action; one realisation was run at 512^3; and kappa, the cold fluid's amount and rho_Lambda remain undetermined. No dark-matter particle is identified. Every number is checked against committed outputs by PAPER45_audit.py (19/19). AI assistance: Anthropic's Claude and DeepSeek.