Cosmology Inflation Spectral Index From Jcost Ns Planck

The spectral index

The cosmic microwave background, the oldest light in the universe, carries faint temperature ripples that seeded the galaxies. Cosmologists compress those ripples into a spectral index, written n_s, which says how the ripple strength changed from large scales to small ones. A value of exactly 1 means the ripples were identical on every scale; the Planck satellite measured n_s = 0.965 with an uncertainty of about 0.004, telling us the early universe was not perfectly scale-invariant.

The standard explanation for this slight tilt is inflation, the idea that the universe expanded exponentially in its first instant. The simplest inflation models predict n_s is close to 1 but slightly less, and the measured 0.965 became a target for theorists. One influential model, Starobinsky inflation, links n_s to the number of e-folds of expansion, N_e, through the formula n_s ≈ 1 - 2/N_e, with N_e around 60.

In Recognition Science, the framework's core cost function forces a discrete counting structure, and one consequence is a preferred number called gap-45, defined as 45. The framework models the inflation slow-roll parameter using this number, giving the prediction n_s = 1 - 2/45 ≈ 0.956. The machine-checked library of formal proofs shows this value lies between 0.955 and 0.957, and shows it is within 0.015 of the Planck value 0.965. The declaration nsPlanck simply records the Planck measurement as the number 0.965 inside the framework, so the comparison can be stated and checked formally.

The honest summary is plain: the framework's prediction sits about 0.009 above the Planck central value, which is more than twice the measurement's stated uncertainty. The derivation is a definitional choice, not a forced result; the number 45 is identified with a structural ceiling, and the formula n_s = 1 - 2/45 is a model, not a consequence of the forcing chain. What the library proves is only the arithmetic: that 1 - 2/45 lies in the stated band and within 0.015 of 0.965. Whether that counts as a successful prediction of inflation is a scientific judgment, not a formal result.

What the reader can now see is the shape of a testable claim. The framework offers a concrete number, 0.956, that future cosmic microwave background experiments with sharper precision could confirm or exclude. The gap between 0.956 and 0.965 is the honest measure of how far the derivation stands from the data today.

/-- 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)

/-- 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

/-- Gap-45 = body-plan ceiling = 45. -/
def gap45 : ℕ := 45

What this page does not claim

The framework does not prove that inflation occurred or that gap-45 is the true e-fold count. The framework does not claim its prediction matches Planck within the measurement's error bar; the gap is about 0.009, more than twice the stated uncertainty. The value 45 is not derived from the forcing chain; it is an identification, and the formula n_s = 1 - 2/45 is a model.

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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