Radial Halo Structure Suppresses History Dependence in a Hypothetical Cosmological Memory Field

Static hysteresis in a field model does not imply a population of systems that preserves its assembly history. We test this distinction for B3, a hypothetical two-field scalar model that Papers I and II showed can hold bistable, history-dependent states in dark-matter halos, by replacing their uniform-core convention with the radial halo density profile. All quantities are simulated states of the hypothetical field, not astronomical observables. In a pre-registered radial recalibration, only a narrow halo-mass range can hold memory at a single transition density. 15.8% of synthetic halos are bistable: 36.9% at 10^12–10^13 solar masses, 9.0% at 10^13–10^14 and none below. When generator-based assembly histories are propagated through radial thresholds, formation writes most bistable halos. The final state differs between the full history and its no-merger counterfactual for only 1.7% of all halos as pre-registered (95% interval 1.1–2.4%). The pre-registered direct validation of the threshold rule failed (10 of 12). We trace the failure to a fault in the fold-threshold computation and repair it post hoc. The repair changes the final state of 3 of 1500 halos, gives 1.5% (0.9–2.2%; 9.7% of bistable halos), and in a post-hoc reassessment brings the rule into agreement with all 12 integrations. A separate pre-registered test validates the rule’s erasure in 6 of 6 selected prospective trajectories at tenfold time compression. This is not a population-wide or real-duration validation, and its convergence check failed in 1 of 8 classifications. Bistability survives several analytic density profiles, but its prevalence (0–13.7%), the mass range and the identity of the bistable halos all change with the profile. In particle-measured IllustrisTNG profiles, 4 of 32 holdout halos (4 of the 24 resolved) are nominally bistable and none passes the pre-registered robustness criterion. A robustly bistable population is therefore not established at the tested resolution: the deciding structure lies within 2–3 field wall widths of the centre, at the resolution limit. An energy identity gives a geometry-independent necessary condition: no static written state of the reduced model exists where the density stays below 0.385 times the transition density everywhere. In the calibrated model, static memory capacity exists in selected halo environments, but only a small fraction of the modelled population retains a final state informative about merger history. The archive contains all pre-registrations, deviation records, code, per-halo outputs, independent audits and SHA-256 freeze records. The 72 IllustrisTNG particle cutouts are not included; their checksums and download instructions are in the README. B3 is hypothetical.