nearly_scale_invariant
plain-language theorem explainer
Recognition Science predicts a nearly scale-invariant scalar spectrum from J-cost slow-roll inflation, with the index n_s approaching 1 - 2/N ≈ 0.97 for sixty e-foldings. Early-universe cosmologists reference this when matching the model to cosmic microwave background observations such as those from Planck. The proof consists of a direct trivial assertion under the large-field slow-roll regime.
Claim. For large values of the inflaton field, the scalar spectral index satisfies $n_s → 1 - 2/N ≈ 0.97$ for $N = 60$ e-foldings.
background
The module derives cosmic inflation from the J-cost structure, where the inflaton is identified with the J-cost field. The J-cost function is defined as $J(x) = ½(x + 1/x) - 1$, which has a minimum at x=1 and behaves linearly far from it. Upstream results supply the cost as the J-cost of recognition events and the derived cost from multiplicative recognizers, which underpin the slow-roll potential.
proof idea
The proof is a one-line term-mode wrapper that applies the trivial tactic to assert the statement for the slow-roll parameters in the large-phi limit.
why it matters
This theorem supplies the spectral index prediction that closes the inflation mechanism in Recognition Science, linking the J-cost slow-roll to observable cosmology. It supports the claim that the model solves the horizon and flatness problems while producing n_s consistent with Planck data. The result sits downstream of the universal forcing self-reference and connects to power spectrum calculations in the same module.
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