Cosmology Inflaton Potential Structural Spectral Index Band

The spectral index band

In cosmology, the spectral index n_s measures how the strength of primordial density fluctuations varies with scale. A value of exactly 1 means fluctuations are scale-invariant, identical at every size. The observed value is slightly less than 1, meaning fluctuations are a bit stronger on large scales than on small ones. The Planck satellite measurements put n_s at approximately 0.965, with an uncertainty of about 0.004.

The Recognition Science framework, a research program that derives physical structure from a forced cost function, contains a machine-checked library of formal theorems. One theorem, called spectralIndex_band, proves a simple numerical fact: the expression 1 - 2/45, which equals about 0.9556, lies strictly between 0.955 and 0.957. The theorem is a statement about real numbers, verified by direct computation. It does not connect this number to any measurement or to the framework's deeper structure.

In the framework's cosmology module, this expression appears as the predicted value of n_s - 1. The module also defines five structural regimes for the inflaton potential, a scalar field thought to drive cosmic inflation: slow-roll plateau, slow-roll slope, hilltop decline, reheating, and the post-reheating radiation era. The framework sets the number of e-folds, the measure of how much the universe expanded during inflation, to 44. These are definitional choices, not derived results.

What the theorem actually establishes is narrow: it proves that a particular rational number falls inside a specified interval. The interval 0.955 to 0.957 sits below the measured value of about 0.965, so the framework's number does not match observation. The theorem makes no claim about the physical universe, only about arithmetic. The framework's own documentation labels the slow-roll parameters and the spectral index as structural, not as measured quantities.

In Recognition Science, this band is presented as a structural consequence of the framework's assumptions. The framework models the inflaton potential with five regimes and derives slow-roll parameters from the golden ratio phi, which appears throughout the framework. The spectral index expression follows from those parameters. But the step from the framework's internal structure to an observational prediction remains a hypothesis, not a theorem. The machine-checked part ends at the arithmetic.

/-- n_s - 1 = -2/45 gives n_s ∈ (0.955, 0.957). -/
theorem spectralIndex_band :
    ((0.955 : ℝ) < 1 - 2/45) ∧ (1 - 2/45 < (0.957 : ℝ)) := by
  refine ⟨?_, ?_⟩ <;> norm_num

/-- e-fold count N_e = 44 (gap-45 ladder). -/
def efoldCount : ℕ := 44

inductive InflatonRegime where
  | slowRollPlateau
  | slowRollSlope
  | hilltopDecline
  | reheating
  | radiationEra
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The spectral index band does not match the measured Planck value of approximately 0.965. The theorem does not connect the numerical band to any physical observation or measurement. The framework does not derive the spectral index value from its foundational cost function; it defines it structurally.

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/InflatonPotentialStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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