G_second_deriv_at_zero
plain-language theorem explainer
The second derivative of the inflaton potential G at the vacuum equals one, confirming exact calibration to the constant A3. Cosmologists deriving alpha-attractor models from Recognition Science first principles would cite this result when linking J-cost to slow-roll parameters. The proof is a direct one-line application of the standard hyperbolic cosine identity at zero.
Claim. For the inflaton potential defined by $G(t) = 1 - 1$ where $G(t) = J(e^t)$ and $J(x) = (x + x^{-1})/2 - 1$, the second derivative satisfies $G''(0) = 1$.
background
The module treats J-cost as the inflaton potential. In log coordinates $t = ln(x)$, the potential becomes $G(t) = cosh(t) - 1$, which is exactly the J-cost evaluated on positive ratios. This identification rests on the Recognition Composition Law and the definition of cost for recognition events, with the identity event placed at the minimum $x=1$ where cost vanishes. The module records the vacuum conditions $G(0)=0$, $G'(0)=0$, and $G''(0)=1$, the last supplying the curvature scale A3 for slow-roll analysis.
proof idea
The proof is a one-line wrapper that applies the Mathlib lemma Real.cosh_zero. Since the second derivative of $G(t) = cosh(t) - 1$ is precisely $cosh(t)$, the identity $cosh(0) = 1$ directly yields the required curvature at the vacuum.
why it matters
This calibration anchors the inflaton sector to the J-uniqueness and phi fixed-point steps of the forcing chain, enabling the alpha-attractor identification alpha = phi^2 stated in the module doc-comment. It supplies the curvature input for sibling derivations of the spectral index and slow-roll parameters that recover standard inflationary predictions from Recognition Science constants. The result closes the link from the multiplicative recognizer cost to observable cosmology without additional hypotheses.
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