Cosmology Inflation Spectral Index From Jcost Ns Rs Lt One

The spectral index bound

In cosmology, the spectral index n_s measures how the strength of density fluctuations in the early universe varied with scale. A value below 1 means the fluctuations were slightly stronger on large scales than on small ones, the tilt that seeded the clumpiness of galaxies. The Planck satellite's 2018 measurements put n_s at 0.965 with an uncertainty of about 0.004.

Recognition Science (RS) is a framework that derives physical structure from a single forced cost function. Its cosmology module defines a candidate spectral index as n_s = 1 − 2/45, using a number called gap45 as a stand-in for the number of e-folds of inflation. The machine-checked library of formal theorems proves three basic facts about this candidate: it is less than 1, it is greater than 0, and it falls in the narrow band between 0.955 and 0.957. The theorem nsRS_lt_one is the first of those, a simple arithmetic consequence of the definition.

The library also proves that this RS value lies within 0.015 of the Planck measurement. That is a true statement, but it is a weak one: the gap between 0.956 and 0.965 is about 0.009, while the measurement's own error bar is only 0.004. The RS candidate sits more than two error bars away from the central measured value. The formal certificate bundles the band and nearness claims into one structure, but it does not assert that the RS value matches the sky within measurement uncertainty.

What the declaration does not claim matters as much as what it proves. It does not derive the spectral index from first principles; gap45 is a defined constant, not a derived one. It does not show that RS predicts the measured value within its error bars. And it says nothing about whether the physical bridge from recognition events to inflationary cosmology is sound, a step the framework itself marks as open.

theorem nsRS_lt_one : nsRS < 1 := by
  unfold nsRS gap45; 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

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

What this page does not claim

The RS candidate does not match the measured Planck value within the measurement's own error bars. The number 45 is a defined constant, not a derived prediction from the framework's axioms. The physical bridge from recognition events to inflationary cosmology is not established by these theorems.

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)

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