Gravitational Memory and the Origin of Flat Rotation Curves: A Non-Markovian Scalar Field with β = 53/15 Derived from Kolmogorov Turbulence in Three Dimensions

Version 9. 4 — Extended SPARC validation of the three-phase prediction Adds §4. 7 "Dwarf galaxies: three phases of the turbulent regime" with independent validation on the full SPARC sample (N=46 robust galaxies, Q≤2). Key results: |δβ| vs log (Vflat) Pearson ρ = −0. 68 (p < 10⁻⁴) ; |δβ| vs Hubble T-type Spearman ρ = +0. 56 (p < 10⁻⁴) ; monotonic progression by Vflat bins confirming Phase I/II/III transitions. The cosmological prediction in the v9. 3 onepager has been reformulated more cautiously; the cosmological extension is under separate development. Reproduction script (testbetaₑffₜhreeₚhases. py) and three figures included. Version 3 (April 2026): The English version of the paper is now provided as an independent PDF file for direct preview and download. The SPARC-25 reproduction package (added in v2) and the original full ZIP archive (paper EN + ES, one-page summaries EN + ES) remain available unchanged. The flat rotation curves of spiral galaxies are conventionally attributed to dark matter halos. We propose a different mechanism: gravity in a turbulent interstellar medium (ISM) is non-Markovian. A scalar coherence field C (r, t) accumulates the orbital history of the system and generates a correction to the effective gravitational acceleration that grows with radius. In the quasi-stationary regime the corrected circular velocity obeys v²ₒbs (r) = v²N (r) / 1 − P · r^ (β−1), where vN is the Newtonian baryonic velocity, P is a single universal parameter calibrated on the Milky Way, and β = 53/15 ≈ 3. 533 is the dissipation exponent. The central result of this work is that β is not a free parameter: it is an algebraic consequence of Kolmogorov turbulence in d = 3 spatial dimensions, derived from two independently measured inputs — the ISM fractal perimeter dimension Dₚer = 4/3 and the Kolmogorov anomalous exponent αK = 3/5 — through the exact relation β = Dₚer + 1 + 2αK = 53/15. The exponent αK = 3/5 is itself derived as the unique stable fixed point of the disk coherence operator in d = 3 dimensions. The model reproduces rotation curves of 23 of 25 representative SPARC galaxies with median χ²/dof = 0. 022, the Radial Acceleration Relation (RAR), the Baryonic Tully-Fisher Relation (BTFR), and the β-variation predicted by non-Kolmogorov ISM statistics in 10 of 11 THINGS galaxies. The model is calibrated and tested within galactic disks (r < 30 kpc) ; extension to halo or weak-lensing scales requires the full non-local formulation of the extended covariant action and is not addressed here. If the identification holds, the gravitational effect attributed to galactic dark matter is an artifact of treating gravity as memoryless in a medium that is not. This deposit contains the paper (English and Spanish versions), a one-page summary in both languages, the verification code (Python), and a runnable demonstration on the public SPARC rotation curve of NGC 3198.