Cosmology Cosmic Inflation From Jcost Inflation Ends At Threshold

The threshold

Cosmic inflation is the brief exponential expansion thought to have occurred in the first fraction of a second after the Big Bang. In the standard picture, a field called the inflaton drives this expansion; when the field settles, inflation ends and the universe reheats. The Recognition Science framework models this same process with a recognition ratio, a discrete record of events comparing two quantities. When the ratio is much larger than 1, the framework's cost function is large, and inflation proceeds; when the ratio approaches 1, the cost drops to zero, and inflation ends.

The formal declaration inflation_ends_at_threshold establishes exactly this endpoint. It states that the cost function J evaluated at 1 equals 0, written as J(1) = 0. This is a proved theorem in the machine-checked library of formal theorems, with no unproved assumptions. The threshold is not an arbitrary choice; it follows from the same cost function that governs all recognition dynamics in the framework. The declaration also certifies that the five canonical inflation models, chaotic, natural, Starobinsky, Higgs inflation, and axion monodromy, are represented as a finite set of exactly five possibilities.

What the declaration does not claim is equally important. It does not assert that any particular inflation model is correct, nor that the framework's description matches observed cosmic data. It establishes a formal condition, the endpoint of inflation in the framework's own terms, not an empirical prediction. The framework's library proves the mathematics of the threshold; whether the universe follows this exact path remains a question for observation, not for the formal theorem.

In plain terms, the declaration pins down when inflation stops in the framework: at the point where the recognition ratio equals 1, the cost vanishes, and the exponential expansion ceases. This is a clean, provable statement about the framework's internal logic. It gives cosmologists working within Recognition Science a precise condition to test, and it leaves the physical question of whether the real universe obeys this threshold open to empirical check.

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

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

What this page does not claim

The declaration does not claim any empirical match with observed cosmic data. The declaration does not assert that any particular inflation model is physically correct. The declaration does not establish that the framework's threshold is the actual reheating condition in the real universe.

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)

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