The Supply Edge: A Mass-Conserving Rule Without Tuned Radii Under a Vacuum-Tied MOND Law Keeps Particle Clustering Within 10%, but Not the Gravitating Matter Spectrum at 512^3 (Working Note)

Working note, version 2.2 (2026-10-10). AI-assisted research programme; not peer reviewed. Version 2.2 corrects the growth claim of versions 2.0 and 2.1: their numbers were measured on the particle field; on the gravitating field the rule fails at 512^3. Correction (2026-10-10). Every growth number in v2.0/v2.1 was measured on the particle density. The rule moves cold fluid within each catchment and that mass gravitates, so the matter power spectrum (what weak lensing, cosmic shear and sigma_8 respond to) is that of the particles plus the settled source. Re-measured on that density from the same saved z = 0 states (lane CFG555): all twelve 256^3 runs pass (max|P-1| = 0.063-0.082, sigma_8 ratio 1.014-1.019), but all four 512^3 runs fail the 10% spectrum cut: canonical 1.0405 / 0.163 (the headline row), alternative 1.0459 / 0.193, a0 tracking dark energy 1.0429 / 0.167, and the second 512^3 realisation 1.0333 / 0.184. The mass-conservation control (no compensation) is 1.1730 / 0.505 on this density. The comparison rows (edge set by hand, edge from census retention) and the +15.3% / +19.3% / +17.4% 'law everywhere' numbers are also particle-field numbers and were not re-measured. We study the law a0 = kappa c sqrt(G rho_Lambda) (kappa = 1/2, fitted) with the kernel nu_mono (equal to nu(y) = 1/(1 - exp(-sqrt y)) for y <~ 2.5, with a monotonic tail above; it differs only in the near-Newtonian regime, by <= 2.4% at y = 10-30) 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 particle 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 in the particle (baryon-tracing) power spectrum is 2.7% at 256^3 and 3.3% at 512^3, with sigma_8 within 0.6% of a control run without the law (cold fluid evolved as ordinary cold matter, i.e. LCDM-equivalent growth): 15 of 15 runs pass on the particle density. In the gravitating matter density the excess is 6.3-8.2% at 256^3 but 16-19% at 512^3 (sigma_8 +3-5%), above the 10% limit, in every 512^3 run; 12 of 15 pass. Structure growth as seen by lensing is therefore not fixed by this rule, and the gap grows with resolution. The rule adds no tuned radius or width; its framework inputs are kappa and the cosmic baryon fraction, and it inherits standard ingredients (LCDM initial spectrum, spherical-collapse turnaround density, halo finder, pass thresholds). The cold fluid is handled as bookkeeping, not as simulated particles; its settling is imposed, not derived from an action; 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 (44/44). AI assistance: Anthropic's Claude and DeepSeek.