Emergent Algebraic Gravity: Deriving the MOND and Vainshtein Scales from the Connes--Kreimer Factorization of Time Series

Recent empirical analyses of pulsar timing residuals have exhibited a systematic spatial decoherence, characterized by an anti-correlation between a multi-scale fluctuation exponent B and distance. While this behavior highlights a spatial variation in the temporal scaling of astrophysical signals, it has lacked a formal theoretical framework. In this work, we develop a mathematically consistent algebraic construction that formalizes this empirical behavior. We define a translation-invariant mapping from discrete time series to a dyadic Hopf algebra of rooted trees. Applying a canonical Rota-Baxter operator, we show that the multi-scale fluctuation amplitude corresponds to the regular component of a Connes-Kreimer Birkhoff factorization. Within this setting, the statistical exponent B naturally admits an interpretation as the anomalous dimension of a scale-dependent field governed by a combinatorial Renormalization Group (RG) flow. By embedding this RG flow into a holographic five-dimensional bulk, B can be interpreted as a conformal scalar field that modifies the effective macroscopic geometry. Under controlled assumptions (weak-field limit and one-loop truncation), the resulting dimensional transmutation yields a logarithmic contribution to the effective potential, producing asymptotically flat rotation curves v² (R) const at large radii. Imposing a thermodynamic UV-IR boundary condition at the horizon yields an effective acceleration profile exhibiting the characteristic deep-MOND scaling a₄ₗₓₑ₀ (R) M/R, and reproduces the associated Baryonic Tully-Fisher scaling relation without introducing galaxy-specific free parameters. The present framework does not claim to provide a complete alternative theory of gravity. Rather, it illustrates how algebraic renormalization structures arising from multi-scale astrophysical fluctuations may geometrize into an effective macroscopic description with phenomenological consequences for galactic dynamics.