Gravity Inflation R At 55 Bounds

The tensor ratio bound

In cosmology, the tensor-to-scalar ratio r measures the relative strength of primordial gravitational waves compared to density fluctuations in the early universe. A positive value means gravitational waves were actually generated during inflation, the brief period of exponential expansion right after the Big Bang. A zero value would mean no such waves were produced, which would rule out many inflationary models. The ratio is a key target for next-generation observatories like LiteBIRD and CMB-S4.

The Recognition Science framework, which derives physical structure from a forced cost function, makes a specific prediction for this ratio. Its model of inflation sets the α-attractor parameter to φ², where φ is the golden ratio, approximately 1.618. This choice is not a free parameter; it follows from the self-similarity condition of the framework's cost functional. The standard slow-roll formula then gives r ≈ 12φ²/N², where N is the number of e-foldings of expansion.

At N = 55 e-foldings, a common reference point, the framework predicts r ≈ 12 × 2.618 / 3025 ≈ 0.0104. The theorem r_at_55_bounds in the framework's machine-checked library of formal theorems proves a weaker but rigorous statement: that the ratio at 55 e-foldings is strictly greater than zero. This is a positivity bound, not a precise numerical match. The proof relies only on the definition of the ratio and the fact that both the numerator and denominator are positive, which follows from the positivity of the golden ratio and the natural numbers.

In Recognition Science, this positivity result is part of a larger picture. The framework derives the golden ratio from its core cost function, then uses that ratio to fix the inflationary parameters. The theorem establishes that the model's prediction is physically meaningful: it predicts a detectable signal, not a null result. The framework's library also proves that the ratio remains positive for N = 50 and N = 60, and that the spectral index, another inflationary observable, falls in the narrow range 0.96 to 0.97 at N = 55.

What the theorem does not claim is equally important. It does not prove that r equals 0.0104 exactly; that numerical value is a separate, unformalized estimate from the docstring. It does not prove that the framework's inflationary model is correct, or that gravitational waves from inflation exist. It only proves a mathematical consequence of the framework's definitions: a specific ratio, as defined, is positive. The physical relevance depends entirely on whether the framework's assumptions about inflation are valid, which is a separate empirical question.

/-- For N = 55 e-foldings: r ≈ 12 * 2.618 / 3025 ≈ 0.0104. -/
theorem r_at_55_bounds : tensor_to_scalar 55 > 0 := by
  unfold tensor_to_scalar
  apply div_pos
  · exact mul_pos (by norm_num) alpha_attractor_pos
  · positivity

/-- The α-attractor parameter: α = φ².
    In RS, this arises from the self-similarity condition of the
    cost functional: the inflaton potential inherits the quadratic
    character of J(x) near x = 1, with the φ² = φ + 1 identity
    setting the curvature scale. -/
noncomputable def alpha_attractor : ℝ := phi ^ 2

/-- The tensor ratio r is in the range detectable by LiteBIRD/CMB-S4.
    For α = φ² and N ∈ [50, 60]: r ∈ (0.005, 0.02). -/
theorem r_in_detectable_range :
    tensor_to_scalar 60 > 0 ∧ tensor_to_scalar 50 > 0 := by
  unfold tensor_to_scalar
  constructor <;> (apply div_pos (mul_pos (by norm_num : (0:ℝ) < 12) alpha_attractor_pos)
                    (by positivity))

/-- 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

/-- Tensor-to-scalar ratio: r ≈ 12α/N² = 12φ²/N².
    This is the RS-SPECIFIC prediction: the standard α-attractor formula
    with α = φ² (not a free parameter). -/
noncomputable def tensor_to_scalar (N : ℝ) : ℝ := 12 * alpha_attractor / N ^ 2

What this page does not claim

The theorem does not prove that the tensor-to-scalar ratio equals approximately 0.0104 at 55 e-foldings. The theorem does not establish that gravitational waves from inflation exist in the actual universe. The theorem does not validate the Recognition Science framework's assumptions about the early universe.

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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