Cosmology Inflation Spectral Index From Jcost Ns Rs Gt Zero

The spectral index bound

In cosmology, the spectral index n_s describes how density fluctuations in the early universe varied with scale, with 1 meaning perfectly uniform variation. The Planck 2018 satellite data measured n_s = 0.965 ± 0.004. The declaration nsRS_gt_zero is a small formal theorem: it proves that a proposed value for this index, defined as 1 - 2/45, is greater than zero. The proof is a direct calculation: since 2/45 is about 0.0444, subtracting it from 1 leaves roughly 0.956, which is clearly positive.

The value itself comes from the Recognition Science framework, which models reality as a ledger, a discrete record of recognition events. Within that framework, the number 45 is called gap45, described as a body-plan ceiling. The proposed spectral index is then n_s = 1 - 2/gap45, giving 0.956. A companion theorem, nsRS_band, proves this value lies strictly between 0.955 and 0.957. Another theorem, nsRS_near_planck, proves this is within 0.015 of the Planck value, a numerical agreement the framework's documentation describes as close but not exact.

What nsRS_gt_zero does not claim is equally important. It does not prove the spectral index is 0.956. It only proves that a particular defined number is positive. The framework's own notes acknowledge the Planck central value of 0.965 sits above the proposed band, and a simple alternative formula using 60 e-folds gives 0.967, closer to the data. The definition of gap45 as 45 is a modeling choice, not a derived result. The theorem's strength is narrow: it certifies the arithmetic of a definition, not the physical correctness of that definition.

For a reader, the practical takeaway is that the framework offers a concrete, checkable number for the spectral index, and the machine-checked library of formal theorems confirms the basic facts about that number: it is positive, less than one, and within a stated band. Whether that number matches the universe is a separate question, one the formal theorems do not settle.

theorem nsRS_gt_zero : nsRS > 0 := 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

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

What this page does not claim

The theorem does not prove the spectral index equals 0.956. The theorem does not establish that the framework's model is physically correct. The value 45 is defined, not derived from the framework's axioms.

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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