Cosmology Inflation Spectral Index From Jcost Spectral Index Cert

A certified spectral index

The spectral index, written n_s, is a number cosmologists use to describe the primordial density variations that seeded galaxies. A value of exactly 1 means those variations were perfectly uniform in scale; the measured value from the Planck satellite's 2018 data is 0.965, with an uncertainty of about 0.004. In the Recognition Science framework, the index is predicted from a discrete count of recognition events, the framework's term for the universe storing a record of its own states.

The framework's library, a machine-checked collection of formal theorems, defines the prediction as n_s = 1 - 2/45, which works out to about 0.956. The declaration SpectralIndexCert bundles two formal results into a single package: first, that this predicted value lies strictly between 0.955 and 0.957, and second, that it differs from the Planck value by less than 0.015. Both statements are proved with no gaps in the formal proof; the certificate is the conjunction of those two facts.

The number 45 enters as a fixed constant called gap45, which the framework identifies with a structural ceiling on its recognition cycle. The prediction is not fitted to the Planck data; the constant comes from the framework's internal logic. The formal proof shows the arithmetic is correct, not that the framework's identification of 45 is physically required. That identification is a definitional choice within the framework, not a theorem about the cosmos.

What the certificate does not claim is worth stating plainly. It does not prove that the spectral index is 0.956; it proves that the framework's formula yields that number. It does not establish that the framework's model of inflation is correct, nor that the gap-45 identification is uniquely forced. The agreement with Planck, about 0.009 away, is a numerical proximity that the certificate records, not a physical derivation it establishes.

/-- RS spectral index: n_s = 1 - 2/gap45. -/
noncomputable def nsRS : ℝ := 1 - 2 / (gap45 : ℝ)

structure SpectralIndexCert where
  nsRS_band : (0.955 : ℝ) < nsRS ∧ nsRS < 0.957
  nsRS_near_planck : |nsRS - nsPlanck| < 0.015

/-- Planck observed value 0.965 is close to RS prediction within 0.01. -/
def nsPlanck : ℝ := 0.965

What this page does not claim

This does not claim the framework proves the spectral index is 0.956; it proves its formula yields that number. This does not claim the gap-45 constant is physically required by the framework's axioms. This does not claim the proximity to Planck data establishes the framework's inflationary model as correct.

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/InflationSpectralIndexFromJCost.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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