Gravity Jcost Inflaton N S At 55 From Jcost

A number from the cost of recognition

In cosmology, the spectral index n_s measures how the density fluctuations left over from cosmic inflation vary with scale. A value close to 1 means the fluctuations are nearly scale-invariant, and the Planck satellite's measurements put the observed value at about 0.965, with a band of 0.96 to 0.97. The Recognition Science theorem n_s_at_55_from_jcost proves that when the number of e-folds of inflation is taken to be 55, the spectral index equals 1 minus 2/55, which is approximately 0.9636, squarely inside the Planck band.

The derivation starts from the framework's central object, the cost of recognition, a number that measures how expensive it is for reality to register a distinction between two states. The framework proves this cost must take a specific form, J(x) = (x + 1/x)/2 - 1. In logarithmic coordinates, this cost becomes cosh(t) - 1, which is a plateau-shaped potential of the type used in Starobinsky inflation models. From this potential, the framework derives the slow-roll parameters and then the spectral index formula n_s = 1 - 2/N, where N is the number of e-folds.

The theorem n_s_at_55_from_jcost states that with N = 55, the spectral index lies strictly between 0.96 and 0.97. The number 55 is not arbitrary: it is the tenth Fibonacci number, and the framework identifies it with the number of e-folds through a separate arithmetic theorem. The choice of 55 is a definitional identification, not itself derived from first principles within the framework.

In Recognition Science, the result is a theorem in the machine-checked library of formal theorems, meaning it is proved from the framework's axioms with no gaps. The empirical agreement with Planck is a successful check, not a derivation of the measured value itself. The framework does not claim to derive the fine-structure constant, and it does not claim that 55 e-folds is uniquely forced by the mathematics; the identification of 55 with the e-fold count is a modeling choice.

/-- For N = 55 e-foldings, the spectral index satisfies n_s ∈ (0.96, 0.97). -/
theorem n_s_at_55_from_jcost : 0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97 :=
  n_s_at_55

/-- 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

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

theorem fib_10_eq : fib_10 = 55 := rfl

What this page does not claim

The framework does not derive the fine-structure constant alpha. The framework does not prove that 55 e-folds is uniquely forced by the mathematics; it is a definitional identification. The theorem does not prove the measured Planck value itself, only that the formula lands inside the observed 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: