Cosmology Cosmic Inflation From Jcost Inflation Model

The Inflation Model Ledger

Cosmic inflation is the theory that the universe underwent a period of exponentially rapid expansion in its first fraction of a second. The five canonical models of inflation are chaotic inflation, natural inflation, Starobinsky inflation, Higgs inflation, and axion monodromy. Each model proposes a different physical mechanism to drive this expansion, but they all share a common mathematical structure when viewed through the lens of Recognition Science.

In Recognition Science, the framework models physical processes using a ledger, a discrete record of recognition events, where each event carries a forced cost. The cost function J(x) = (x + 1/x)/2 - 1 measures the deviation from a state of unity. The framework's library, a machine-checked collection of formal statements, proves that the number of these canonical inflation models is exactly five. This is not a physical measurement but a formal statement about a defined list: the statement inflationModelCount establishes that the list contains five entries and no more.

The framework also connects the end of inflation to this cost function. Its library proves that when the cost J equals zero, the argument equals one, written as J(1) = 0. The framework models this as the reheating condition: when the inflaton field reaches unity, the cost drops to zero and inflation ends. This is a formal threshold derived from the cost function's definition, not a physical derivation of how reheating occurs in detail.

What the declaration does not claim is more modest than it might appear. It does not prove that these five models are the only possible models of inflation in physics, nor does it derive the physical mechanism of reheating. The five-model count is a definitional choice about which models to include in the list, not a statement about the universe's menu of possibilities. The threshold J(1) = 0 is a mathematical identity about the cost function, not a physical law about when inflation must end in the actual cosmos.

The practical consequence is a clean bookkeeping statement: the framework has fixed its list of inflation models and specified a formal condition for their endpoint. This gives cosmologists working inside the framework a precise, checkable object to build on, while leaving the physical question of which model is correct entirely open.

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

inductive InflationModel where
  | chaotic | natural | starobinsky | higgsInflation | axionMonodromy
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not prove that these five models are the only possible models of inflation in physics. The declaration does not derive the physical mechanism of reheating. The threshold J(1) = 0 is a mathematical identity, not a physical law about the actual end of cosmic inflation.

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: