History-Dependent Scalar-Field Memory in Cosmological Environments: Writing, Retention, Localization, and Observability

We ask whether two cosmological regions with identical present-day environments can carry different, spatially localized, observable scalar-field states solely because of their histories. Using numerical simulations with reported equations, boundary conditions and convergence tests, we follow a progression of models. A free, minimally coupled linear field on a ΛCDM background, with non-negative mass and smooth sources, left no appreciable localized record in any tested configuration: its history leaves in outgoing radiation, and a localization–excitation bound forbids smooth histories from exciting localized modes when m²(x) ≥ 0. Linear fields with trapping potentials can retain history: near-critical, locally tachyonic wells hold it in stable bound modes, but only under fine tuning. A nonlinear φ⁶ field supports bistable, localized memory at matched final environment; its static fold, R_fold = 10.4583 ± 1.3×10⁻⁴ at m = 0.10, is found by a direct zero-eigenvalue solve. In an action-derived two-field model with mediator back-reaction (B3), the bistable band widens by about 70% and a held-out synthetic-population history signal rises 2.6× over the model without back-reaction, within an admissible region set by vacuum localization. Writing is a threshold crossing whose dynamic offset is consistent with τ⁻¹ scaling through 3000 wall times. A growth-integral criterion calibrated at 30–300 wall times reproduces this scaling but, in a prospective test, underestimates writing at longer durations and for non-Gaussian excursions; all observed errors made writing easier than predicted. Extrapolated to astrophysical durations, writing is expected to become quasi-static. Halo formation then meets the same threshold as later mergers: if collapse overshoot is comparable to merger excursions, about 60% of otherwise quiet halos are written at formation and the history signal falls about fourfold. The field’s ability to remember is established numerically within the model. Whether the Universe supplies histories that make that memory observationally informative is unresolved. Contents: the manuscript (PDF) and a reproducibility archive (ZIP) with 39 Python scripts, their logs, data tables, 13 figures, METHODS.md (equations, initial and boundary conditions, convergence, results), MANIFEST.md (SHA-256 hashes and run order) and REFERENCES.md (reference verification). Environment: Python 3.13.18, NumPy 2.5.3, SciPy 1.18.1, Matplotlib 3.11.2.