Gravity Jcost Inflaton N S 55 In Planck Band

The spectral index from J-cost

In cosmology, the spectral index n_s measures how the strength of density fluctuations varies with scale in the early universe. A value of exactly 1 means perfectly scale-invariant fluctuations, as in the simplest early models; the measured value is slightly less than 1, about 0.965, indicating a small departure. This departure is a key observable for distinguishing competing theories of inflation, the brief exponential expansion thought to have seeded the large-scale structure of the cosmos.

In the Recognition Science framework, inflation emerges from a single forced cost function. The framework's foundational theorem proves that any recognition cost satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. In logarithmic coordinates t = ln(x), this becomes G(t) = cosh(t) - 1, a smooth potential with a minimum at zero. The framework then treats this potential as the inflaton, the field driving inflation, and derives the standard slow-roll parameters from its curvature. The spectral index follows from the potential's shape, giving the formula n_s(N) = 1 - 2/N, where N is the number of e-folds of inflation.

The declaration n_s_55_in_planck_band establishes a concrete numerical consequence. For N = 55 e-folds, the formula gives n_s(55) = 1 - 2/55, which is approximately 0.9636. The theorem proves that this value lies strictly between 0.96 and 0.97, the band of values allowed by Planck satellite measurements. The number 55 is not arbitrary: it is the tenth Fibonacci number, and the framework identifies it with the e-fold count through its internal rung arithmetic, where 44 + 11 = 55. The theorem is machine-checked in the framework's library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

What this declaration does not claim is equally important. It does not prove that the universe underwent exactly 55 e-folds of inflation; that number is a choice within the framework, not a derivation from first principles. It does not claim that the spectral index formula matches all observations at every precision, only that this particular value falls within the Planck-allowed window. The theorem establishes a mathematical implication: if the J-cost potential is the inflaton and if N = 55, then the spectral index lands in the observed band. The physical bridge from the abstract cost function to the actual early universe remains an open target, not a proved result.

/-- n_s at N = 55 is in the Planck 2018 band. -/
theorem n_s_55_in_planck_band :
    0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97 :=
  n_s_at_55

/-- The spectral index formula 1 − 2/N follows from the α-attractor
    with α = φ²: n_s = 1 − 2/N is the standard slow-roll result
    when ε ≪ 1 (plateau regime). -/
theorem n_s_from_jcost (N : ℝ) (hN : 0 < N) :
    spectral_index N = 1 - 2 / N := rfl

/-- G is the J-cost in log coordinates. -/
theorem G_is_Jcost_log (t : ℝ) : G t = Real.cosh t - 1 := rfl

What this page does not claim

The theorem does not prove that the universe underwent exactly 55 e-folds of inflation. The theorem does not establish that the J-cost potential is the actual inflaton of the early universe; that physical identification remains open. The theorem does not claim the spectral index value matches observations beyond the stated 0.96 to 0.97 band.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: