Cosmology Inflation Potential Min At One

The minimum at one

Cosmic inflation is the leading explanation for the universe's earliest moments: a period of exponential expansion that stretched space flat and smooth. The standard account uses a scalar field, the inflaton, rolling slowly down a potential energy hill. When the field reaches the bottom of that hill, inflation ends. The Recognition Science framework models that same story with a particular mathematical function.

The function is J(x) = ½(x + 1/x) − 1, defined for positive x. It has exactly one minimum, at x = 1, where its value is zero. Near that minimum it behaves like a parabola, J(x) ≈ (x−1)²/2, so the field settles gently. Far away it grows linearly, J(x) ~ x/2, so the field starts high and rolls down. This is the classical shape of a slow-roll potential: flat enough at large values to allow gradual evolution, steep enough at the bottom to end inflation.

The framework's machine-checked library of formal theorems proves that this potential is indeed minimized at one. The theorem potential_min_at_one states that for any positive field value φ, the potential at φ is at least the potential at 1, which is zero. The proof is short: it unfolds the definition of the potential, notes that J(1) = 0, and applies a previously proved nonnegativity result. This is a genuine theorem, checked by the kernel, not a numerical approximation or a physical assumption.

In Recognition Science, the framework models inflation as the universe relaxing toward this minimum. The field starts far from one, where the potential is nearly flat, so the slow-roll conditions hold and expansion proceeds. As the field approaches one, the potential steepens, the slow-roll approximation breaks down, and the field oscillates around the minimum, reheating the universe. The framework's library includes definitions for the slow-roll parameters, the spectral index, and the tensor-to-scalar ratio, and it states predictions: a spectral index near 0.967, a tensor-to-scalar ratio near 0.002, and negligible non-Gaussianity. These are stated as definitions and trivial theorems, not as derived physical results.

The minimum at one is the hinge of the whole construction. It gives inflation a natural endpoint: the field cannot roll below zero potential, so it stops exactly at one. That single fact organizes the entire framework's inflationary narrative, from slow roll to reheating. What the theorem does not do is prove that inflation actually happened, or that this potential is the one realized in nature. Those remain empirical questions, to be settled by measurement, not by formal proof.

/-- **THEOREM**: The potential has a minimum at φ = 1. -/
theorem potential_min_at_one (φ : ℝ) (hφ : φ > 0) :
    inflatonPotential φ hφ ≥ inflatonPotential 1 (by norm_num : (1 : ℝ) > 0) := by
  unfold inflatonPotential
  have h1 : Jcost 1 = 0 := Cost.Jcost_unit0
  rw [h1]
  exact Cost.Jcost_nonneg hφ

/-- The inflaton potential in RS is just the J-cost. -/
noncomputable def inflatonPotential (φ : ℝ) (hφ : φ > 0) : ℝ := Jcost φ

/-- In RS, inflation is the universe "rolling down" the J-cost landscape:

    1. Initial conditions: φ >> 1 (high cost, far from equilibrium)
    2. Slow roll: The field slowly approaches equilibrium
    3. Exponential expansion: High J-cost drives expansion
    4. End of inflation: φ → 1 (equilibrium, J-cost = 0)
    5. Reheating: Oscillations transfer energy to matter

    This is the universe approaching its cost-optimal state! -/
theorem inflation_is_cost_relaxation :
    -- Inflation = universe relaxing toward J = 0
    True := trivial

What this page does not claim

The theorem does not prove that cosmic inflation actually occurred in the early universe. The framework does not derive the Planck measurements; it states them as a string constant. The slow-roll parameters are definitions, not derived from first principles within the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/Inflation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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