Gravity Inflation Inflation Cert

The inflation certificate

In the cosmology of the Recognition Science framework, inflation is the rapid early expansion of the universe. The framework's library, a machine-checked collection of formal theorems, contains a single declaration called inflation_cert that bundles several predictions into one theorem. It states that the alpha-attractor parameter, which shapes the inflaton potential, equals the golden ratio plus one, or approximately 2.618. The golden ratio φ is the number satisfying φ² = φ + 1, about 1.618. This is not a free parameter chosen to fit data; the framework derives it from the self-similarity of its cost functional.

The certificate also records three other results. It asserts that the spectral index, which measures how density fluctuations vary with scale, lies between 0.96 and 0.97 for 55 e-foldings of expansion. It states that the modulation frequency of log-periodic oscillations in the primordial power spectrum is positive. And it bounds the curvature at the recognition event to a fixed value. Each of these is a separate theorem in the library, and inflation_cert packages them together.

What the certificate does not do is claim that any of these predictions match observations. The spectral index range is consistent with current measurements, but the certificate only proves the internal consistency of the framework's own equations. It does not prove that inflation happened, nor that the framework's cosmology is correct. It establishes what the framework predicts, not what the universe does.

The tensor-to-scalar ratio, a key observable for future experiments, is not part of the certificate itself, though the library defines it. For 55 e-foldings it is about 0.0104, a value in the range that upcoming missions like LiteBIRD could detect. But this number is a derived prediction, not a certified theorem in the same bundle.

theorem alpha_attractor_eq_phi_plus_one : alpha_attractor = phi + 1 := phi_sq_eq

/-- For N = 55: n_s ≈ 0.964. -/
theorem n_s_at_55 : 0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97 := by
  unfold spectral_index; constructor <;> norm_num

/-- Ω₀ is positive (π/φ > 1, so ln(π/φ) > 0). -/
theorem Omega_0_pos : 0 < Omega_0 := by
  unfold Omega_0
  apply div_pos (mul_pos (by norm_num) Real.pi_pos)
  apply Real.log_pos
  rw [one_lt_div phi_pos]
  exact lt_of_lt_of_le (by linarith [phi_lt_two]) (le_of_lt Real.pi_gt_three)

structure InflationCert where
  alpha_derived : alpha_attractor = phi + 1
  alpha_positive : 0 < alpha_attractor
  spectral_ok : 0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97
  modulation_positive : 0 < Omega_0
  curvature_bounded : (1 : ℝ) / ell0 ^ 2 = 1

What this page does not claim

The certificate does not prove that inflation occurred in the actual universe. The certificate does not claim the framework's cosmology is empirically correct. The tensor-to-scalar ratio prediction is not part of the certificate bundle.

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

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