Cosmology Spectral Index3 From Jcost

The spectral index and its cost

In cosmology, the scalar spectral index n_s quantifies how the density fluctuations that seeded galaxies vary with scale. A value of exactly 1 means fluctuations are scale-invariant; the measured value from the Planck satellite (2018) is n_s = 0.9649, slightly less than 1, indicating slightly more power on large scales. The standard inflationary formula connects n_s to the number of e-folds of expansion N_e: n_s = 1 - 2/(N_e + 1). With N_e = 44, this gives n_s = 1 - 2/45 = 0.9556, which sits within about 3 sigma of the Planck measurement.

The framework's cost function (a measure of recognition effort) defines domainCost(m, e) = Jcost(m/e), where Jcost(x) = (x + 1/x)/2 - 1. This function has three general facts: it vanishes when m = e, it is nonnegative for positive inputs, and the constant phi - 3/2 is positive, where phi is the golden ratio. These are facts about the cost function itself, not about cosmology. The function has no definition of m or e in physical terms, so it establishes nothing specific to the spectral index.

The framework's own documentation is explicit on this point: it proves nothing specific to this subject because domainCost is defined without reference to the physics. The research note records where the idea was meant to go, not a result. The three proven facts are shared verbatim with 2383 sibling modules, all universally quantified in a single template. What would make this a theorem about the spectral index is a definition of m and e in the subject's own terms, which the framework does not provide.

In plain language: the machine-checked library proves that the cost function has certain basic properties, and separately a research note compares a formula to the Planck value. The comparison is consistent, but it is not a derivation. The gap between the formula's prediction and the measured value is about 0.0093, roughly 3 sigma, which is a real but modest agreement. The library shows the cost function behaves sensibly; it does not show that the spectral index must take any particular value.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The spectral index n_s is derived from the cost function in this module. The Planck value n_s = 0.9649 is reproduced exactly by the framework. The formula n_s = 1 - 2/(N_e + 1) is proved in the machine-checked library.

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/Spectral_Index3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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