alpha_attractor
plain-language theorem explainer
The α-attractor parameter is fixed to φ² in Recognition Science inflation models. Cosmologists studying slow-roll dynamics with self-similar potentials cite this to eliminate the free α parameter in favor of the golden-ratio curvature scale. The definition follows at once from the quadratic character of the J-cost near x=1 together with the identity φ²=φ+1.
Claim. The α-attractor parameter is defined by $α=φ²$, where $φ$ is the golden-ratio fixed point satisfying $φ²=φ+1$ and arising from the self-similarity condition on the cost functional $J$.
background
The Gravity.Inflation module formalizes RS inflationary predictions from the Universe-Origin Paper. It imports phi from Constants and builds observables on the J-cost functional, whose quadratic character near unity supplies the inflaton curvature scale. The module states that α=φ² follows from cost-functional self-similarity, that the spectral index is n_s≈1-2/N, and that the tensor-to-scalar ratio takes the RS-specific form r≈12φ²/N².
proof idea
One-line definition that directly assigns alpha_attractor to phi raised to the power two, using the upstream definition of phi in Constants.
why it matters
This definition supplies the fixed α value required by every downstream inflation certificate and spectral formula in the module. It is invoked by InflationCert to enforce alpha_derived=phi+1, by tensor_to_scalar to obtain r≈12φ²/N², and by alpha_from_curvature in JCostInflaton to equate the curvature scale with the golden-ratio identity. It realizes the T6 self-similar fixed point inside the inflationary sector and removes the free α parameter from the α-attractor model while remaining consistent with the eight-tick octave and D=3.
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