InflatonPotentialCert
InflatonPotentialCert is a record bundling five algebraic properties of the potential V(φ) = J(1 + φ) on the recognition manifold. Cosmologists extending the gap-45 inflation analysis would cite it to supply an explicit scalar field whose vacuum and quadratic behavior reproduce the required slow-roll attractor. The declaration itself is a structure that simply assembles the already-verified lemmas V_vacuum, V_nonneg, V_pos_off_vacuum, V_reciprocal_symm and V_eq_quadratic.
claimA certificate that the potential defined by $V(φ) = J(1 + φ)$ satisfies $V(0) = 0$, $V(φ) ≥ 0$ whenever $φ > -1$, $V(φ) > 0$ whenever $φ ≠ 0$ and $φ > -1$, the reciprocity $V(φ) = V((1 + φ)^{-1} - 1)$ for all $φ > -1$, and the closed form $V(φ) = φ² / (2(1 + φ))$ for all $φ > -1$.
background
The module treats the inflaton field φ_inf as the dimensionless deviation of the Z-coordinate from the consciousness-gap reference rung. Its potential is obtained directly from the recognition cost: $V(φ_inf) := J(1 + φ_inf)$, where J obeys the Recognition Composition Law. This supplies the missing scalar-field description for the earlier derivation of $N_e = 44$, $n_s ≈ 0.9556$ and $r ≈ 0.017$ from the gap-45 identity.
proof idea
The declaration is a structure whose five fields are populated by the sibling lemmas V_vacuum, V_nonneg, V_pos_off_vacuum, V_reciprocal_symm and V_eq_quadratic. Each lemma is an independent algebraic verification that the corresponding property holds for the concrete function Cost.Jcost(1 + φ).
why it matters in Recognition Science
The certificate closes the gap between the abstract gap-45 structural identity and an explicit inflaton field on the recognition manifold. It feeds the constructor inflatonPotentialCert and thereby grounds the slow-roll analysis whose Taylor coefficients at φ = 0 reproduce the universal attractor. The construction rests on J-uniqueness (T5) and the eight-tick octave (T7) that fix the functional form of J; it leaves open the explicit computation of higher-order slow-roll parameters from the next terms in the J-expansion.
scope and limits
- Does not compute the slow-roll parameters ε_V or η_V.
- Does not derive numerical values for the spectral index or tensor-to-scalar ratio.
- Does not address reheating or the termination of inflation.
- Does not prove that this potential is the unique recognition-derived scalar field.
formal statement (Lean)
100structure InflatonPotentialCert where
101 vacuum : V 0 = 0
102 nonneg : ∀ {phi_inf : ℝ}, -1 < phi_inf → 0 ≤ V phi_inf
103 pos_off_vacuum :
104 ∀ {phi_inf : ℝ}, phi_inf ≠ 0 → -1 < phi_inf → 0 < V phi_inf
105 reciprocal_symm :
106 ∀ {phi_inf : ℝ}, -1 < phi_inf →
107 V phi_inf = V ((1 + phi_inf)⁻¹ - 1)
108 quadratic_form :
109 ∀ {phi_inf : ℝ}, -1 < phi_inf →
110 V phi_inf = phi_inf ^ 2 / (2 * (1 + phi_inf))
111
112/-- Inflaton-potential certificate. -/