Cosmology Primordial Spectrum Tensor To Scalar Upper Bound

The tensor-to-scalar upper bound

In cosmology, the tensor-to-scalar ratio r compares the strength of primordial gravitational waves (tensor perturbations) to the density fluctuations (scalar perturbations) that seeded galaxies. A value of r = 0 would mean no gravitational waves from inflation; a larger r means stronger waves. The current observational upper bound, set by the BICEP/Keck and Planck collaborations, is r < 0.06 at 95% confidence. This number is a measurement, not a prediction.

The machine-checked library of formal statements, called the framework's library, records this bound as a definition: tensor_to_scalar_upper_bound is set to 0.06. This is a bookkeeping choice, a way to store the observed limit inside the formal system. It does not derive the number from deeper principles. The library also defines the ratio itself: given a scalar power spectrum amplitude and a tensor amplitude, the ratio is simply the tensor amplitude divided by the scalar amplitude, provided the scalar amplitude is positive.

In Recognition Science, the framework models primordial fluctuations as arising from quantum fluctuations in a cost function, a ledger of recognition events. The framework's library contains a prediction for r: it suggests r ≈ 0.02, based on the golden ratio conjugate raised to the fourth power, (φ - 1)^4. However, the formal proof for this prediction only establishes a weaker bound, that the value lies between 0.1 and 0.2. The comment in the source code notes that the intended value 0.021 would come from (2 - φ)^4, but this is not what the definition computes.

What the declaration does not claim is important. The upper bound of 0.06 is not a theorem derived from the framework's axioms; it is a recorded observation. The framework's own prediction of r ≈ 0.02 remains a hypothesis with a named falsifier: the structure SpectrumFalsifier contains a proposition r_contradicts, which would mark the prediction as falsified if observations contradicted it. The exact value of r, whether it is 0.02 or something else, remains an open question for observation to settle.

/-- The tensor-to-scalar ratio r < 0.06 (Planck + BICEP/Keck). -/
noncomputable def tensor_to_scalar_upper_bound : ℝ := 0.06

/-- RS prediction for r:

    r may be φ-related. Possible predictions:
    - r = (φ - 1)⁴ = 0.021 (testable by CMB-S4!)
    - r = 1/(8φ⁵) = 0.011
    - r = 1/φ⁸ = 0.021

    All these are in the observable range! -/
noncomputable def rs_prediction_r : ℝ := (phi - 1)^4

theorem r_prediction :
    -- r ≈ 0.02 is a testable RS prediction
    -- NOTE: The comment "(φ-1)⁴ = 0.382⁴" is incorrect.
    -- φ - 1 ≈ 0.618 (the golden ratio conjugate), so (φ-1)⁴ ≈ 0.146.
    -- The correct value 0.021 would be (2-φ)⁴ = 0.382⁴.
    -- For now, we prove a weaker bound: 0.1 < (φ-1)⁴ < 0.2
    0.1 < rs_prediction_r ∧ rs_prediction_r < 0.2 := by
  unfold rs_prediction_r
  -- φ - 1 ≈ 0.618, so (φ-1)⁴ ≈ 0.146
  -- Using bounds: 1.61 < φ < 1.62, so 0.61 < φ-1 < 0.62
  have h_phi_gt : phi - 1 > 0.61 := by
    have h := phi_gt_onePointSixOne
    linarith
  have h_phi_lt : phi - 1 < 0.62 := by
    have h := phi_lt_onePointSixTwo
    linarith
  -- 0.61^4 ≈ 0.138 > 0.1, 0.62^4 ≈ 0.148 < 0.2
  have h_low : (0.61 : ℝ)^4 > 0.1 := by norm_num
  have h_high : (0.62 : ℝ)^4 < 0.2 := by norm_num
  have h_phi_pos : phi - 1 > 0 := by linarith [one_lt_phi]
  constructor
  · calc 0.1 < (0.61 : ℝ)^4 := h_low
       _ < (phi - 1)^4 := by
           apply pow_lt_pow_left₀ h_phi_gt (by norm_num) (by norm_num)
  · calc (phi - 1)^4 < (0.62 : ℝ)^4 := by
           apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_phi_pos) (by norm_num)
       _ < 0.2 := h_high

/-- The derivation would be falsified if:
    1. n_s has no φ-connection
    2. r contradicts (φ-1)⁴ prediction
    3. Large non-Gaussianity found -/
structure SpectrumFalsifier where
  ns_no_phi : Prop
  r_contradicts : Prop
  large_nongaussianity : Prop
  falsified : ns_no_phi ∧ r_contradicts → False

What this page does not claim

The upper bound 0.06 is not a theorem derived from the framework's axioms; it is a recorded observation. The framework does not prove that r equals 0.02; the formal proof only establishes a range between 0.1 and 0.2. The declaration does not claim that the golden ratio directly sets the tensor-to-scalar ratio; the connection remains a hypothesis.

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/PrimordialSpectrum.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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