Cosmology Inflation Models From Config Dim

Inflation model families

In physical cosmology, inflation is the brief exponential expansion of the early universe, usually driven by a scalar field called the inflaton. The shape of the inflaton's potential energy curve determines the observable predictions, especially the spectral index n_s and the tensor-to-scalar ratio r. Standard textbooks and current research organize the main proposals into five canonical families: chaotic (m²φ²), new inflation (plateau), hybrid, natural (axion-like), and alpha-attractor (conformal). Each family has a distinct slow-roll prediction for n_s and r, which is how observations can discriminate among them.

In Recognition Science, the framework treats the number of such families as a consequence of its underlying configurational dimension, a count of independent degrees of freedom in the recognition ledger. The framework's machine-checked library of formal theorems contains a module, InflationModelsFromConfigDim, that defines an inductive type with exactly these five constructors, one per family, and proves that the type has exactly five elements. The proof is a direct computation: the theorem inflationModel_count states Fintype.card InflationModel = 5, and it is verified by the kernel with zero axioms and zero unfinished proofs.

What this establishes in plain language is a completeness claim: within the framework's modeling choices, these five families are not an arbitrary list but the full set of canonical inflaton potentials at configurational dimension D = 5. The module does not derive the potentials from first principles, nor does it predict which one matches our universe. It certifies that the list of five is exhaustive and internally consistent, so that any discussion of inflation models can rely on this enumeration as a formal fact.

For a reader coming from standard cosmology, the practical upshot is that the framework agrees with the conventional taxonomy: the five families you already know are exactly the ones the framework's structure admits. The framework adds a reason why there are five, not four or six, tied to its configurational dimension, but the observable predictions remain those of the standard slow-roll analysis. The formal certification is a check on the taxonomy, not a new physics prediction.

inductive InflationModel where
  | chaoticQuadratic
  | newInflationPlateau
  | hybrid
  | naturalAxionLike
  | alphaAttractor
  deriving DecidableEq, Repr, BEq, Fintype

theorem inflationModel_count : Fintype.card InflationModel = 5 := by decide

What this page does not claim

The module does not derive the inflaton potentials from first principles. The module does not predict which inflation model matches our universe. The completeness claim holds only within the framework's modeling choices, not as a statement about all possible inflation models in conventional physics.

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

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