Cosmology Inflation Models From Config Dim Inflation Models Cert

The certified five

Cosmic inflation is the leading theory for the universe's first explosive growth, and cosmologists study it through a small set of standard model families. The Recognition Science framework's machine-checked library of formal theorems contains a declaration called InflationModelsCert. It establishes a simple counting fact: there are exactly five canonical inflaton-potential families in the framework's classification. The five are chaotic inflation with a quadratic potential, new inflation with a plateau, hybrid inflation, natural inflation with an axion-like potential, and alpha-attractor inflation with a conformal potential.

The certificate is a formal object, a machine-checked proof that the inductive type representing these models has exactly five members. The framework derives this count from a deeper principle: the configuration dimension, written configDim, equals 5 for this classification. The declaration does not invent new physics. It takes the five families that cosmologists already use and certifies, in the framework's formal language, that the classification is complete and non-redundant under its definitions.

Each of the five families makes a distinct prediction for two observable quantities: the spectral index n_s, which measures how density fluctuations vary with scale, and the tensor-to-scalar ratio r, which measures the strength of gravitational waves from inflation relative to density perturbations. These predictions are what let astronomers discriminate among the models using cosmic microwave background data. The certificate itself, however, only counts the families; it does not say which one matches the data.

In Recognition Science, this declaration is a small but load-bearing piece of a larger structure. The framework models the universe as a discrete record of recognition events, and from that starting point it derives constants and physical laws. The inflation certificate connects that abstract machinery to observational cosmology by showing that the framework's internal classification aligns with the standard model families physicists already work with. The alignment is a statement about the framework's definitions, not a new prediction about the sky.

The certificate's formal status is clean: it contains zero unproved assumptions and zero axioms beyond the standard logical foundations. That means a reader can trust the counting result as a matter of logic, independent of any physical speculation. What the certificate does not do is choose among the five models, predict which one the data favors, or claim that the framework's configDim principle is the reason the universe inflated. It is a counting certificate, nothing more and nothing less.

structure InflationModelsCert where
  five_models : Fintype.card InflationModel = 5

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

def inflationModelsCert : InflationModelsCert where
  five_models := inflationModel_count

What this page does not claim

The certificate does not claim that any one of the five inflation models matches observational data. The certificate does not claim that configDim = 5 is derived from the framework's foundational recognition principles. The certificate does not claim that the five-family classification is complete in the sense of covering 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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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