Formation Versus Merger Writing in a Hypothetical Cosmological Memory Field: Parametric and Simulation-Measured Halo Histories

Paper I showed that a hypothetical two-field scalar model (B3) can hold history-dependent,localized states in dark-matter halos. It also found a limit: halo formation meets the samewriting threshold as later mergers, so a formation overshoot comparable to merger excursionswrites most halos early and removes most of the merger-history signal. Here we ask whetherrealistic halo histories impose that limit. A generator of core-density histories, calibrated onpublished accretion, concentration and merger relations, shows that in a fixed physical coreregion smooth growth never writes and formation writes only through an overshoot. In aparametric study, the fraction of band halos whose final state depends on merger history, I,is set by two quantities: the overshoot amplitude, and how assembled the core is at the time.For formation overshoots at least as large as merger overshoots, I is 0.03–0.08 if the overshootcomes at 90% core assembly and 0.34–0.39 if it comes at the Ludlow formation epoch. Writingis quasi-static to good accuracy (26 of 27 direct integrations agree with the threshold rule).Erasure can, however, leave a long-lived trapped excitation, a residual dynamical memorythat occasionally rewrites a state. We then measured both quantities in 72 IllustrisTNGdark-matter-only halos, with definitions, thresholds and analysis code pre-registered andhashed before the 32 evaluation halos were measured. The pre-registered label is late overshoot(median peak-to-final core-mass ratio 1.09, amplitude ratio 1.01). Propagated through thefrozen model, however, the measured transients give I = 0.21 (0.15–0.31), excluding theI ≲0.1 that the label was expected to imply; robustness tests keep I between 0.17 and 0.24.Applied directly to the measured histories, the rule (confirmed by direct integration in 10 of10 selected trajectories) writes a similar fraction of halos, but at different times. Formationwindows overlap major-merger windows in 70% of TNG halos against 37% in the generator.I is therefore conditional on an independence assumption the measured histories do notsatisfy. We thus show how cosmological assembly histories could write persistent states ina hypothetical scalar field, and that realistic formation–merger correlations substantiallychange how that history dependence must be interpreted. The field is hypothetical; all resultsare model-dependent predictions. Reproducibility archive included: generator, parametric study, direct-integration and IllustrisTNG measurement code, pre-registration and amendments, event tables, analysis and robustness reports, and SHA-256 freeze records. B3 is a hypothetical field; all results are model-dependent predictions, not evidence that such a field exists.