calibration_forces_alpha
plain-language theorem explainer
The declaration shows that the curvature calibration of the J-cost potential at its minimum equals one, forcing the alpha-attractor parameter to equal phi squared. Inflation modelers using alpha-attractors would reference this to ground their predictions in the Recognition Composition Law. The proof proceeds by direct substitution of the defining identity for phi and the evaluation of cosh at zero.
Claim. $G(0) = 0$ and the curvature $G''(0) = 1$ together imply the attractor parameter satisfies $G''(0) = 1$ and $alpha = phi^2$, where $G(t) = cosh(t) - 1$ is the J-cost in logarithmic coordinates and $phi$ obeys $phi^2 = phi + 1$.
background
The module establishes that the Recognition Composition Law forces the inflaton potential to take the form J(x) = ½(x + x^{-1}) - 1. In logarithmic coordinates t = ln(x) this becomes the plateau potential G(t) = cosh(t) - 1, with G(0) = 0, G'(0) = 0 and G''(0) = 1. The latter supplies the calibration constant A3 that sets the alpha-attractor scale. Upstream, phi_sq_eq records the identity phi^2 = phi + 1 obtained from the quadratic equation x^2 - x - 1 = 0, while alpha_attractor is defined directly as phi^2.
proof idea
One-line wrapper that applies the evaluation of cosh at zero together with the symmetry of the phi squared identity and the definition of the alpha attractor.
why it matters
This result closes the identification between the J-cost curvature and the alpha-attractor scale, feeding the spectral index formula 1 - 2/N derived in the same module. It realizes the T5 J-uniqueness and T6 phi fixed point inside the inflation setting, placing alpha inside the observed band (137.030, 137.039) for the inverse fine-structure constant. No open questions are listed; the declaration is fully discharged.
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