Cosmology Inflation Spectral Index From Jcost Ns Rs Near Planck

The near-match certificate

The spectral index of the cosmic microwave background, written n_s, is a number cosmologists use to describe how the density fluctuations in the early universe varied with scale. A value of exactly 1 would mean perfectly scale-invariant fluctuations. The Planck satellite's 2018 measurements gave n_s = 0.965 with an uncertainty of 0.004. That measurement is a standard result of observational cosmology, not something the Recognition Science framework produced.

Recognition Science (RS) is a framework that derives physical constants from a single cost function, a mathematical rule that assigns a price to every recognition event, where a recognition event is a discrete record of some comparison. Within RS, the spectral index is predicted to follow the formula n_s = 1 - 2/N, where N is a number called the gap. In the framework, the gap is set to 45, a value that comes from an internal structural ceiling. The prediction is therefore n_s = 1 - 2/45, which is about 0.9556.

The machine-checked library of formal theorems contains a declaration called nsRS_near_planck. It proves a precise inequality: the absolute difference between the RS prediction (0.9556) and the Planck value (0.965) is less than 0.015. The same library also proves that the RS value lies strictly between 0.955 and 0.957. These are exact statements in the formal system, checked with no gaps and no extra assumptions.

What this certificate does not claim is equally important. It does not say the RS prediction matches the Planck measurement within the measurement's own uncertainty. The Planck error is 0.004, while the RS value is about 0.0094 away, so the difference is larger than the error bar. The theorem only establishes a coarse nearness, a band of 0.015. It also does not derive the Planck value from first principles; the Planck number is entered as a definition in the library, not proved. The framework's formula for n_s itself is a model choice, not a derived theorem.

In plain terms, the declaration is a sanity check: it shows that a number coming out of the framework's internal machinery is not absurdly far from what telescopes actually see. It is a certificate of rough agreement, not a derivation of the cosmic measurement. A reader should treat it as evidence that the framework's machinery produces a plausible cosmological number, while noting that the precise match to 0.965 remains an open target.

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

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

theorem nsRS_near_planck : |nsRS - nsPlanck| < 0.015 := by
  rw [nsRS_val]
  unfold nsPlanck
  rw [abs_lt]
  constructor <;> norm_num

theorem nsRS_near_planck : |nsRS - nsPlanck| < 0.015 := by
  rw [nsRS_val]
  unfold nsPlanck
  rw [abs_lt]
  constructor <;> norm_num

/-- n_s_RS ∈ (0.955, 0.957). -/
theorem nsRS_band : (0.955 : ℝ) < nsRS ∧ nsRS < 0.957 := by
  rw [nsRS_val]; norm_num

What this page does not claim

The RS prediction matches Planck within the measurement's uncertainty of 0.004. The Planck value 0.965 is derived from the framework; it is entered as a definition. The formula n_s = 1 - 2/45 is a proved theorem; it is a model choice.

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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