Void-Amplified Dimensional Flow (VADF): Curvature-Driven Anomalous Dimension, RG-Improved Propagator and Emergent Fractional Kinetics(v7.1)

**Void-Amplified Dimensional Flow (VADF) v7. 1** Scalar-Selective Extension with GW Safety We present a significantly updated version of the VADF framework. The mathematical core remains clean and rigorous: a curvature-dependent anomalous dimension η (R) arises naturally from the functional renormalization group (FRG) in the Einstein–Hilbert truncation with Litim regulator. This produces a well-defined RG-improved variable-order fractional kinetic operator \ (K ₄₅₅ (x) M^ (x) (-g) ^s (x) \) with \ (s (x) = 1 - (x) /2\). All formal properties — causal light-cone support of the retarded Green function, positivity of the Källén–Lehmann spectral density, adiabatic criterion, pseudodifferential construction, and commutator estimates — are verified and hold for η ≲ 0. 1. Extensive tests on real observational data revealed that the original global implementation (where the fractional operator acts on both scalar and tensor perturbations) is strongly disfavoured. It predicts frequency-dependent dispersion and time delays of gravitational waves of order \ (t (D/c) (₁/₂) \). Direct comparison with GW170817 and 15-year NANOGrav pulsar-timing data excludes the global version by 14–17 orders of magnitude (\ (10^-17\) ). A **scalar-selective realisation** successfully resolves this tension: the modification is applied exclusively to scalar density perturbations through an effective \ ( (k, a) \) in the Poisson equation, while tensor modes remain strictly General Relativity. This eliminates all GW/PTA constraints while preserving the ability to suppress structure growth on non-linear scales. Numerical tests (post-processing + preliminary CLASS patches) demonstrate that suitable choices of parameters (\ (0. 02\) –\ (0. 15\), \ (k ₏₈ₕ₎ₓ 0. 09\) –\ (0. 18\, h\, Mpc^-1\) ) bring \ (S₈\) into excellent agreement with weak-lensing measurements (DES Y3 / KiDS) with BAO shifts \ (2\%\). The scalar-selective version is now ready for full self-consistent implementation in Boltzmann codes and complete likelihood analyses. **Key advances in v7. 1**- Clean separation of scalar and tensor sectors (tensor modes untouched) - Explicit demonstration of GW/PTA safety via direct comparison with real data- Working scalar-only prototype inside CLASS- Clear, realistic roadmap for Planck + BAO + RSD + WL fits The mathematics of VADF is solid, elegant, and of independent interest as a concrete example of RG-improved fractional kinetics in curved spacetime. The cosmological application requires further development (primarily a production-grade CLASS implementation and full MCMC analysis), but the scalar-selective direction opens a promising path toward resolving the \ (S₈\) tension while remaining fully compatible with all existing data. This deposit contains the updated preprint, all analysis scripts, CLASS patches, parameter scans, reproducibility materials, and detailed comparison with GW170817 and NANOGrav data. **Keywords**: modified gravity, renormalization group, fractional operators, \ (S₈\) tension, scalar-selective modification, gravitational waves