Explain the Lean structure `InflatonPotentialCert` in module `IndisputableMonolith.Cosmology.InflatonPotentialFromJCost`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) Plain-English meaning

InflatonPotentialCert is a Lean structure (record type) that bundles five proved properties of the inflaton potential V on the recognition manifold. In plain terms it asserts:

• V(0) = 0 (vacuum energy is zero),
• V is nonnegative for all physical displacements phi_inf > -1,
• V is strictly positive away from the vacuum,
• V obeys the same reciprocal symmetry as the underlying J-cost,
• V admits the exact quadratic-form expression phi_inf² / (2(1 + phi_inf)).

The companion definition inflatonPotentialCert supplies an explicit inhabitant of this structure by wiring in the supporting lemmas.

(2) Relevance to Recognition Science

Recognition Science derives the inflaton field as the deviation of the Z-coordinate from the consciousness-gap reference rung and sets its potential to the universal J-cost: V(phi_inf) := J(1 + phi_inf). The certificate therefore shows that every structural property previously proved for J-cost (non-negativity, reciprocal symmetry, quadratic small-field behaviour) transfers directly to the cosmological inflaton. This supplies the missing scalar-field foundation for the earlier gap-45 inflation predictions without introducing new parameters.

(3) How to read the formal statement

structure InflatonPotentialCert where
  vacuum          : V 0 = 0
  nonneg          : ∀ {phi_inf}, -1 < phi_inf → 0 ≤ V phi_inf
  pos_off_vacuum  : ∀ {phi_inf}, phi_inf ≠ 0 → -1 < phi_inf → 0 < V phi_inf
  reciprocal_symm : ∀ {phi_inf}, -1 < phi_inf → V phi_inf = V ((1 + phi_inf)⁻¹ - 1)
  quadratic_form  : ∀ {phi_inf}, -1 < phi_inf → V phi_inf = phi_inf ^ 2 / (2 * (1 + phi_inf))

Each field is a theorem (a Prop). The curly-brace syntax means the fields are implicit arguments when the structure is used; the ∀ quantifiers range over real displacements phi_inf subject to the stated inequalities. The structure is therefore a compact “certificate” that all five properties hold simultaneously.

(4) Visible dependencies and certificates

• V is defined as V phi_inf := Cost.Jcost (1 + phi_inf).
• The five fields are populated by the lemmas V_vacuum, V_nonneg, V_pos_off_vacuum, V_reciprocal_symm and V_eq_quadratic.
• All lemmas ultimately rest on the J-cost theorems imported from IndisputableMonolith.Cost (non-negativity, positivity off unity, reciprocal symmetry, and the squared-form identity).
• The module itself carries the status annotation “Lean status: 0 sorry, 0 axiom.”

(5) What the declaration does not prove

The certificate stops at the algebraic and positivity properties of V. It does not derive the slow-roll parameters ε_V and η_V, does not compute the number of e-folds N_e, does not obtain the spectral index n_s or tensor-to-scalar ratio r, and does not connect the potential to observational data. Those steps remain in the earlier (not supplied) module Cosmology/InflationFromRecognitionCurvature or in downstream analysis.