Cosmology Inflation Parameters5

A template, not a result

Cosmic inflation is the leading theory that the early universe underwent a brief, exponential expansion. Its standard parameters are 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 relative to density fluctuations. The Planck satellite measured n_s = 0.9649 and set an upper limit r < 0.036.

In Recognition Science, the module cosmology inflation parameters5 is a machine-checked file that proves three general facts about a cost function, a measure of the forced price of a recognition event. It defines a cost on a ratio of two real numbers, and proves the cost is zero when the two numbers are equal, is never negative for positive inputs, and that a certain threshold constant is positive. These are true for any two positive numbers; they say nothing about inflation.

The file's own docstring states this plainly: it proves nothing specific to inflation, because the cost is defined on a generic ratio without reference to any physical quantity. The same body of text is shared verbatim with 2383 sibling modules. The intended physics, a comparison of the framework's predicted n_s = 0.9556 against the measured value at 2.1 sigma, and a predicted r = 0.0169 which passes the upper limit, is recorded as a research note, not as a theorem.

What the module does establish is a reusable template: any future definition of the two quantities m and e in inflation's own terms would turn these general facts into a theorem about the subject. Until then, the file is a certificate that the cost function behaves well, not a derivation of cosmic parameters.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module does not derive the measured values of n_s or r. The research note's numerical comparison is not a proved theorem. The framework's prediction of n_s is not established by this file.

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/Inflation_Parameters5.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

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