History-Dependent Gravity LXIX: From Memory Reduction to Emergent Dynamics

Abstract

We establish the conceptual and mathematical foundation of the History-Dependent Gravity (HDG) program by formulating a rigorous reduction chain for causal history-dependent theories. Moving beyond model-specific constructions, we prove four general theorems: 1. **Reduction:** Any causal history functional admitting a finite-memory factorization reduces to a quotient state space via a physical state map.2. **Observables:** If the state map is complete, the physical observable algebra is isomorphic to the algebra of functions on the reduced state space.3. **Markovianization:** Dynamics compatible with the reduction induces a local-in-time Markov evolution family on the state space.4. **Variational Principle:** The reduced Markov dynamics unconditionally admits a first-order variational representation on the cotangent bundle, which yields a regular second-order action principle when the Helmholtz integrability conditions are satisfied. **Physical Realizations:*** **Few-Body Systems (Papers LXVII–LXVIII):** We apply the reduction chain to the rank-2 separable model, proving that the off-shell parameter $\lambda_{12}$ is a pure gauge redundancy. The physical interaction is entirely encoded in the invariant Schur complement $\lambda_{\mathrm{eff}}$. This explains the previously observed ultraviolet stability as a structural consequence of gauge quotienting, eliminating the need for an independent three-body counterterm.* **Gravitational Sector:** We propose a model-dependent gravitational realization where the physical state is represented by a scale-dependent effective action $\Gamma_k[g]$ with spectral memory $\rho_k(\mu) \geq 0$. The connection to the Functional Renormalization Group (FRG) is formulated as a specific realization of the general reduction principle. **Core Principle:** Memory defines the physical state; the state defines the reduced dynamics; the dynamics generates the variational action. The mathematical principle (Theorems 1–4) and the physical realization (FRG/Spectral HDG) are strictly logically separated, providing clear, independent falsifiability criteria for the HDG framework.