Spiral Renormalization Group Critical Exponents in Quadratic Gravity: A Derivational and Cosmological Analysis

We develop a quadratic gravity framework containing curvature-squared invariants � and � and demonstrate that coupled renormalization group (RG) beta functions naturally admit complex critical exponents. Linearizing the RG flow near a non-Gaussian fixed point, we derive the algebraic conditions under which spiral trajectories emerge in coupling space. We construct a minimal two-coupling RG system whose stability matrix produces complex eigenvalues constrained by cosmic microwave background (CMB) observations of the scalar spectral index and its running. The real part of the eigenvalue is fixed by the observed tilt, while the imaginary component controls log-periodic corrections to the primordial power spectrum. We derive the inflationary scalar and tensor spectra, compute spectral running, estimate non-Gaussianity, analyze reheating stability, and identify observational parameter bounds. The framework provides a phenomenological bridge between renormalization group flow geometry in quadratic gravity and inflationary cosmology while remaining consistent with current data.