V_vacuum
plain-language theorem explainer
The theorem shows that the inflaton potential on the recognition manifold vanishes at the canonical reference point. Cosmologists deriving slow-roll parameters from the J-cost would cite this to anchor the vacuum energy. The proof is a term reduction that unfolds the potential definition, rewrites the argument via ring, and applies the J-cost unit lemma.
Claim. Let $V(φ) := J(1 + φ)$ where $J$ is the recognition cost function. Then $V(0) = 0$.
background
The module defines the inflaton field as the deviation φ_inf of the Z-coordinate from the consciousness-gap reference rung on the recognition manifold. The potential is given by V(φ_inf) := Jcost(1 + φ_inf), the same J-cost that appears throughout Recognition Science. The module doc states that this choice yields V(0) = 0, V'(0) = 0, and V''(0) = 1, grounding the earlier inflation observables N_e = 44 and n_s ≈ 0.9556 in an explicit scalar field.
proof idea
The proof is a term-mode reduction. It unfolds the definition of V, rewrites 1 + 0 to 1 by ring, and applies the lemma Jcost_unit0 which states Jcost(1) = 0.
why it matters
This supplies the vacuum component of inflatonPotentialCert, the structure that collects the potential identities used for slow-roll calculations. It realizes the zero of J at unity (T5 J-uniqueness) at the origin of the phi-ladder, closing the gap between the gap-45 inflation predictions and an explicit potential on the recognition manifold. The parent is inflatonPotentialCert.
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