Cosmology Cosmic Inflation From Jcost Cosmic Inflation Cert

The certification

Cosmic inflation is the leading cosmological idea that the early universe expanded exponentially for a brief instant, smoothing out irregularities. The standard picture names five canonical models: chaotic, natural, Starobinsky, Higgs inflation, and axion monodromy. Each proposes a different field or potential to drive the expansion.

The Recognition Science framework models a universe that keeps a discrete record of recognition events, where recognition is the act of matching a new observation to a stored pattern. Its cost function J(x) = (x + 1/x)/2 - 1 measures how expensive a recognition ratio is. The framework's machine-checked library of formal theorems defines a structure called CosmicInflationCert. This certificate bundles three facts: the count of the five canonical models, the condition that inflation ends when the cost J reaches zero at ratio 1, and a reference to a separate canonical certificate. The library proves the count is 5 and proves J(1) = 0, so the certificate is fully checked with no gaps.

The certificate's reheating condition is a statement about the cost function, not about physics. It says that when the inflaton ratio approaches 1, the cost approaches 0, which the framework identifies with inflation ending. This is a formal link between the framework's abstract cost and a cosmological phase transition. The framework does not claim that any of the five models is correct, nor that inflation happened. The certificate is a consistency check: within the framework, the five named models fit the same cost dynamics, and reheating lands at the zero-cost threshold.

What the certificate does not do is more important than what it does. It does not derive the expansion history, the amplitude of primordial fluctuations, or the tensor-to-scalar ratio. It does not select among the five models. It does not prove that the framework's cost function governs the real inflaton field. The certificate is a formal object, not an observational claim. A reader should take it as a demonstration that the framework can encode a standard cosmological classification, not as evidence that inflation occurred.

structure CosmicInflationCert where
  five_models : Fintype.card InflationModel = 5
  reheating_condition : J 1 = 0
  threshold : CanonicalCert

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

/-- Inflation ends when J-cost crosses the canonical threshold. -/
theorem inflation_ends_at_threshold : J 1 = 0 := J_one

What this page does not claim

The certificate does not prove that cosmic inflation occurred. The certificate does not select among the five inflation models. The certificate does not derive any observational cosmological parameter.

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/CosmicInflationFromJCost.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: