potential_positive
plain-language theorem explainer
The J-cost serving as inflaton potential is strictly positive away from its minimum at field value one. Researchers deriving inflation from Recognition Science would cite this positivity to validate the slow-roll regime. The proof is a one-line reduction that unfolds the potential definition and applies the established J-cost positivity lemma.
Claim. Let $V(phi)$ denote the inflaton potential for $phi > 0$. Then $V(phi) > 0$ whenever $phi neq 1$, where $V$ coincides with the J-cost function.
background
The module derives cosmic inflation from Recognition Science's J-cost slow roll. The core mechanism identifies the inflaton with the J-cost field, which has a minimum at unity and grows linearly far away, enabling a nearly constant potential for exponential expansion. The inflatonPotential is defined directly as the J-cost of the field value. An upstream result states: J(x) > 0 for x ≠ 1 and x > 0. This positivity is key to ensuring the potential supports inflationary dynamics without sign changes.
proof idea
This is a one-line wrapper that unfolds the inflaton potential definition to the J-cost and then applies the lemma establishing J-cost positivity for positive arguments not equal to one.
why it matters
This result secures the basic positivity property required for the J-cost to act as a viable inflaton potential in the COS-001 framework. It underpins the slow-roll parameters and e-folding calculations in the module, consistent with the J-uniqueness and self-similar fixed point in the forcing chain. The module doc positions this as part of solving horizon, flatness, and monopole problems via slow roll.
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