Tully–Fisher as Gravitational Memory: Why the Universe Remembers (with Spatial Diffusion)

We frame TF/BTFR as correlations between two coarse-grainings of a common history—lightas a short-term report, gravity as a long-term ledger. We extend local kernels to spatiotemporaldiffusion (including fractional order) so that accumulated memory propagates to large radii andsustains extended flat curves. To avoid overfitting we adopt a minimal parameterization with hierarchical Bayesian priors and provide a real-data fitting protocol (SPARC/IFS) with ∆AIC/∆BICcomparisons to NFW and MOND. Key falsifiable signatures include residual–history correlations,phase lags ∼ τ , diffusion-length control of the flat extent, and outer-slope diagnostics for the fractional order. This article is operational and non-circular: it introduces no new particles and makesno numerical prediction of G.Keywords: Tully–Fisher relation, Causal Memory Gravity, gravitational memory, diffusion, fractional diffusion, galaxy rotation curves, weak lensing