A Homeostatic State-Space HRSM Test of Cosmic-Web Memory in SDSS and TNG300

Large-scale structure is usually treated as the passive outcome of gravitational instability, with galaxies responding to environment but without an explicit state-space test for historical dependence. This paper develops a cosmic-web HRSM state-space framework in which web cells are described by reserve, recoverability, stability, and memory coordinates, together with smoothed overdensity and T-web morphology. An SDSS $\hat{\phi}$ webcube first shows that the observed $H_{\rm obs}$--$M_{\rm obs}$ coupling is environment dependent and strongly modulated by stellar mass: raw correlations remain positive across sheets, filaments, and nodes, but mass-residualised partial correlations are small or negative in most well-populated environments. This observational result motivates a temporal test using TNG300-1. Native TNG300 group catalogues are streamed into $64^3$ periodic webcubes, smoothed at $R=8\,{\rm cMpc}/h$, and classified by T-web morphology. A four-snapshot bridge over persistent voxel IDs shows that a memory-augmented predictor modestly outperforms a Markovian baseline, reducing aggregate RMSE from $0.4018$ to $0.3946$, while $72.4\%$ of frontier cells move closer to the fitted HRSM band at the next step. A seven-snapshot sampled bridge confirms the same direction. A full seven-snapshot streaming evaluation over $1{,}048{,}576$ transition rows then compares Markov, one-lag, and two-lag memory models, with aggregate RMSE decreasing from $0.3555$ to $0.3493$ and $0.3481$, respectively. The lag-depth result persists after variance scaling, ablation tests localise the dominant historical contribution to $H_{\rm web}$ and $R_{\rm web}$ rather than to density or web morphology alone, and the result remains stable when the density field is rebuilt with stellar-mass weights instead of unit weights. Object-block bootstrap uncertainty propagation keeps the lag-two improvement separated from zero, and SubLink main-progenitor branches show the same Markov, one-lag, two-lag ordering along sampled galaxy lineages. A $3\times4$ robustness grid over $R=5,8,10\,{\rm cMpc}/h$ and $\lambda_{\rm th}=0.0,0.1,0.2,0.3$ favours the memory model in all twelve configurations. The lag-depth ordering also survives changes in the stellar-mass filter and replacement of the T-web labels by a density-Hessian web classifier. The result does not prove a literal cosmic homeostatic potential. It supports a narrower claim: under this coarse-grained HRSM representation, TNG300 web-cell evolution carries weak but reproducible lagged predictability consistent with short-depth non-Markovian memory, together with return-to-band behaviour that remains above matched-null expectation.