Gravity Inflation R In Detectable Range

A testable window for gravity waves

Cosmic inflation, the leading theory of the universe's first instants, predicts that gravitational waves from that violent expansion should leave a faint imprint on the cosmic microwave background. The strength of that imprint is measured by a number called the tensor-to-scalar ratio, written r. A value of r near 0.01 would confirm the simplest inflation models; a value far below 0.001 would make them very hard to test at all.

The Recognition Science framework, which derives physical structure from a forced cost of recognition, makes a specific prediction for this number. In its library of formal theorems, the declaration r_in_detectable_range proves that for a number of e-foldings N between 50 and 60, the tensor-to-scalar ratio r is positive. The docstring states the intended scope: for α = φ² and N in [50, 60], r lies in the range (0.005, 0.02). This window sits squarely within the sensitivity of the LiteBIRD and CMB-S4 experiments, which are designed to detect r down to about 0.001.

The proof itself is modest. It shows that the ratio is positive at N = 50 and N = 60, given the framework's definition of the ratio as r = 12α/N² with α = φ². Here φ is the golden ratio, about 1.618, so α = φ² ≈ 2.618. The theorem does not prove the ratio falls in the stated range; that claim lives in the docstring as a numerical illustration. The formal result is narrower: it establishes positivity at two endpoints, which is a necessary precondition for the ratio to be in any physically meaningful positive band.

What the declaration does not claim is just as important. It does not prove that the ratio is detectable, that LiteBIRD or CMB-S4 will see it, or that the framework's inflation model is correct. The theorem only asserts a mathematical fact about a defined quantity. Whether the actual universe matches this prediction is an empirical question for the experiments to answer.

/-- 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))

/-- 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))

/-- 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))

What this page does not claim

The theorem does not prove the ratio falls in the stated detectable range (0.005, 0.02). The theorem does not claim that LiteBIRD or CMB-S4 will detect the signal. The theorem does not establish that the framework's inflation model is the correct description of 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)

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

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