Dark Matter as Temporal Nonlocality: Variational Derivation of the Memory Kernel (Paper XV in the History-Dependent Gravity Series)

This work constitutes Paper XV in the History-Dependent Gravity (HDG) series and represents a structural closure of the HDG framework. While Papers I-XIV established the phenomenological and dynamical consequences of temporal nonlocality — including relaxation dynamics, gravitational wave signatures, vacuum selection, and kernel phenomenology — the present work provides the first derivation of the memory kernel directly from a variational principle. We construct a causal, nonlocal modification of gravity in which dark matter phenomenology emerges from a retarded memory kernel encoded in the gravitational action S = (16πG) ⁻¹ ∫ d⁴x √ (-g) R + α R F (□⁻¹) R + Sₘ. The theory introduces an infrared modification of the gravitational propagator regulated by the cosmological horizon scale, characterized by a scale- and time-dependent enhancement factor μ (k, t) = 1 + (H₀/H (t) ) · kₛat/√ (k² + kₛat²), with kₛat = αH₀/c. The key result is that the temporal memory kernel K (t-t') ~ 1/ (t-t') · exp (- (t-t') /τ) emerges necessarily from action-level consistency, subject to three constraints: causality (retarded Green's function), dimensional consistency (scale-free in the IR limit), and absence of ghosts (no new poles in the propagator). This elevates HDG from an effective framework to a structurally grounded theory with a complete logical chain: Action → Kernel → Dynamics → Observables. The framework naturally reproduces: • Flat galactic rotation curves via a logarithmic correction to the Newtonian potential: Φₑff (r) ~ -GM/r - √ (GMa₀) ln (r/rₛ) • The baryonic Tully-Fisher relation v⁴ = GMa₀ with acceleration scale a₀ ~ cH₀ ≈ 1. 2 × 10⁻¹⁰ m/s²• Consistency with early-universe cosmology: the modification is dynamically suppressed at high redshift (H ≫ H₀), preserving CMB acoustic peaks and BAO• Gravitational wave constraints: cGW = c in the high-frequency limit• Lensing consistency: Φ = Ψ ensures equality of dynamical and lensing masses Rotation curve fits to M33 (Triangulum Galaxy) demonstrate quantitative agreement with observed asymptotic velocities, while revealing mild tension (~25%) with the canonical acceleration scale, suggesting possible scale-dependence of the coupling α or systematic uncertainties in baryonic mass estimates. This work establishes that dark matter phenomenology can emerge from the nonlocal, causal structure of spacetime itself, without introducing new particle degrees of freedom. The memory kernel is simultaneously a mathematical necessity and a falsifiable prediction of the theory.