Cosmology Inflation Spectral Index From Jcost

The spectral index from a counting rule

The spectral index n_s is a number cosmologists use to describe the earliest moments of the universe. In the standard picture of cosmic inflation, a period of explosive expansion stretched quantum fluctuations into the seeds of galaxies. The spectral index measures how the strength of those fluctuations varies with scale. A value of exactly 1 means the fluctuations are perfectly scale-invariant, the same on all sizes. The Planck satellite measured n_s = 0.965 ± 0.004, meaning the fluctuations are slightly stronger on large scales than on small ones, a result announced in 2018.

Inflation theories predict n_s through a parameter called the slow-roll parameter, which depends on the shape of the inflaton potential. The simplest models give n_s = 1 - 2/N_e, where N_e is the number of e-folds, the amount by which the universe expanded during inflation. A typical value is N_e ≈ 60, which gives n_s ≈ 0.967, close to the Planck measurement. The Starobinsky model, a particular inflation theory based on a modified gravity action, produces this same formula and is considered one of the best fits to the data.

In Recognition Science, the framework derives a specific value for N_e from its own principles. The framework's central result is a forced cost function J(x) = (x + 1/x)/2 - 1, which emerges from five plain conditions on how recognition events are recorded. From this cost function, the framework derives a chain of results including the golden ratio φ and an eight-tick recognition cycle. The framework models the number of e-folds as a count derived from this structure, arriving at N_e = 45, a value it calls the body-plan ceiling.

Using this value in the standard slow-roll formula gives the framework's prediction for the spectral index: n_s = 1 - 2/45 ≈ 0.956. This is the result formalized in the module InflationSpectralIndexFromJCost. The machine-checked library of formal theorems proves that this value lies strictly between 0.955 and 0.957, and that it is within 0.015 of the Planck measured value of 0.965. The formal proof contains no unproven assumptions and no axioms beyond the standard logical ones.

The agreement is not a precise match. The framework's central value of 0.956 sits about 0.009 below the Planck central value of 0.965, a difference of roughly two to three times the measurement's stated uncertainty. The module does not claim to reproduce the measurement exactly; it establishes a nearby prediction from a different starting point. The value of 45 is a definition within the framework, not a derived constant, and the step from the framework's cost function to this specific count is not yet formalized in the library.

What the module does establish is a concrete, checkable bridge. It takes a number from the framework's internal structure, plugs it into the standard, well-tested formula of inflation, and produces a number that lands in the right neighborhood of a precise cosmological observation. That is a meaningful test for a framework that aims to derive physical constants from a single principle. The gap between 0.956 and 0.965 is a target for future work, a place where the framework's assumptions could be refined or shown to be incomplete.

/-- Planck observed value 0.965 is close to RS prediction within 0.01. -/
def nsPlanck : ℝ := 0.965

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

theorem nsRS_val : nsRS = 1 - 2 / 45 := by
  unfold nsRS gap45; norm_cast

/-- 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

What this page does not claim

The framework's prediction of 0.956 is not an exact match to the Planck value of 0.965; the difference is about 0.009. The value of 45 for the e-fold count is a definition within the framework, not a derived theorem. The module does not prove that the framework's cost function itself implies the spectral index; it formalizes the arithmetic of a specific proposed value.

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)

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.

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