Quantum Memory Matrix Framework Applied to Cosmological Structure Formation and Dark Matter Phenomenology

We show that the Quantum Memory Matrix (QMM)—a discretized, Planck‑scale register that stores the von‑Neumann entropy deposited by microscopic interactions—naturally reproduces the gravitational phenomena attributed to cold dark matter. Coarse‑graining the lattice of quantum imprints endows the macroscopic action with a single, dimensionless information–geometry coupling \texorpdfstring{$\lambda$}{lambda} multiplying the canonical gradient term \texorpdfstring{$(\partial S)^2$}{(∂S)\textsuperscript{2}}, where \texorpdfstring{$S(x)$}{S(x)} is the continuum entropy field. The induced, conserved, and ghost‑free stress–energy tensor \texorpdfstring{$T_{\mu\nu}^{\text{(QMM)}}=\lambda\bigl[(\nabla_\mu S)(\nabla_\nu S)-\tfrac12 g_{\mu\nu}(\nabla S)^2+g_{\mu\nu}\Box S-\nabla_\mu\nabla_\nu S\bigr]$}{T\_munu = ...} behaves as pressure‑less dust whenever \texorpdfstring{$\dot S^{\,2}\ll H|\dot S|$}{Ṡ² ≪ H|Ṡ|}. Implemented in a modified \textsc{CLASS} Boltzmann solver, the QMM component fits \emph{Planck} 2018 and BAO data for \texorpdfstring{$0.5\lesssim\lambda\lesssim2$}{0.5 ≲ λ ≲ 2} and keeps the linear‑growth factor within \texorpdfstring{$0.2\%$}{0.2\%} of \texorpdfstring{$\Lambda$CDM}{LambdaCDM} across all \emph{Euclid} scales. Pilot \texorpdfstring{$N$‑body}{N-body} simulations that evolve the entropy field on a staggered grid reproduce the halo mass function down to dwarf‑galaxy masses while alleviating the sub‑halo and cusp–core tensions. Using a holographically regulated imprinting prescription, we show that entropy deposited across causal surfaces accumulates sufficient gravitational mass to match observed dark matter halos without invoking new particles. Because the imprints are non‑baryonic and collision‑less, QMM lensing maps match those of particle dark matter in merging clusters yet predict percent‑level, scale‑dependent shifts in the convergence power spectrum—observable with the \emph{Roman} High Latitude Survey. An information‑geometric sector with no new particles thus offers a falsifiable alternative to particle dark matter, with clear signatures for forthcoming surveys.