Cosmology Inflation Spectral Index From Jcost Ns Rs Val

The spectral index value

The spectral index n_s is a number cosmologists use to describe the earliest moments of the universe. In the standard picture, a period of rapid expansion called inflation 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 observed value from the Planck satellite (2018) is n_s = 0.965 ± 0.004, slightly less than 1, indicating a small departure from perfect invariance.

Recognition Science (RS) is a framework that derives physical structure from a single starting point: reality keeps a ledger, a discrete record of events, and the cost of recognition is forced, not chosen. Within this framework, the machine-checked library of formal theorems includes a definition of the spectral index. The declaration nsRS_val is a theorem that proves the value of this definition equals 1 - 2/45. This is a pure arithmetic statement: 1 minus two divided by forty-five, which is approximately 0.956. The theorem is checked by the library's kernel, with no unproven assumptions beyond the standard logical axioms.

The framework's definition is built on a number it calls gap45, which it sets to 45, described as a 'body-plan ceiling' within the theory. The theorem nsRS_val establishes the algebraic identity, and companion theorems prove that this value lies strictly between 0.955 and 0.957, and that it is within 0.015 of the Planck value of 0.965. The framework's own documentation notes the prediction is 'close but above Planck 0.965' when using a different rung, and that the gap-45 formula gives a value 'close to Planck 0.965'.

What the declaration does not claim is the essential part. nsRS_val is a theorem about the arithmetic of a definition inside the framework. It does not prove that the framework's model of inflation is correct, nor that the number 45 is derived from first principles rather than chosen to fit. The identification of gap45 with a physical quantity, and the bridge from the framework's ledger to actual cosmology, is not established by this theorem. The agreement with Planck data is presented as a closeness within a stated tolerance, not as a match within the measurement's error bars.

In plain terms, the declaration proves a small, precise fact about a number the framework defines. It is a certified step in a larger argument, not the argument itself. What a reader can take from this is a clear, machine-verified statement of one of the framework's predictions, and an equally clear boundary marking where the framework's formal proof ends and its physical interpretation begins.

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 theorem does not prove the framework's model of inflation is correct. The theorem does not establish that gap45 is derived from first principles rather than chosen. The agreement with Planck data is within a stated tolerance, not within the measurement's error bars.

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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