Cosmology Bitkernel Families Delta W0 Max Pos

The maximum amplitude

In cosmology, the parameter w describes the pressure of dark energy relative to its density. A value of exactly -1 means dark energy behaves like a constant energy density, the cosmological constant. Observations allow w to drift slightly with redshift z, which measures cosmic time. The framework writes this drift as δw(z) = δw₀ · K(z), where δw₀ is the amplitude at the present day and K(z) is a shape function, called a kernel, that describes how the drift evolves.

The declaration delta_w0_max_pos proves that the maximum allowed amplitude, δw₀ = φ - 3/2, is positive. Here φ is the golden ratio, approximately 1.618, so the number is about 0.118. The proof is direct: since φ exceeds 1.5, subtracting 1.5 leaves a positive remainder. A companion theorem shows the same number is less than 1. Together they place the amplitude in the interval (0, 1), a range the framework's BIT theorem derives for the maximum of the cost function J.

The framework's library, a machine-checked collection of formal theorems, also defines three concrete kernels for forecasting. The constant kernel K(z) = 1 models a drift that never changes. The canonical kernel K(z) = 1/(1+z) decays slowly with redshift. The exponential kernel K(z) = exp(-z/z₀) decays faster, controlled by a scale z₀. All three equal 1 at z = 0, so at the present day each reduces to δw₀ itself. The effective equation of state is w_eff = -1 + δw₀ · K(z), which at z = 0 becomes -1 + δw₀ for any kernel.

In Recognition Science, this result is one small link in a longer chain. The BIT theorem forces the maximum amplitude to sit at J(φ) = φ - 3/2 ≈ 0.118, and delta_w0_max_pos certifies that this value is a valid positive bound. The declaration does not assert that any particular kernel is correct, nor that the amplitude actually reaches its maximum. It only establishes that the ceiling is positive and therefore a meaningful constraint for forecasts such as the DESI Y3 analysis that uses these kernels.

theorem delta_w0_max_pos : 0 < delta_w0_max := by
  unfold delta_w0_max
  have := phi_gt_onePointFive
  linarith

theorem delta_w0_max_lt_one : delta_w0_max < 1 := by
  unfold delta_w0_max
  have := phi_lt_two
  linarith

/-- All three kernels equal 1 at `z = 0`. -/
theorem kernel_at_zero (k : KernelFamily) (z0 : ℝ) :
    kernel k 0 z0 = 1 := by
  cases k <;> simp [kernel]

/-- At `z = 0`, `w_eff = -1 + δw_0` for any kernel. -/
theorem w_eff_at_zero (k : KernelFamily) (delta_w0 z0 : ℝ) :
    w_eff k 0 delta_w0 z0 = -1 + delta_w0 := by
  unfold w_eff
  rw [kernel_at_zero]
  ring

What this page does not claim

The declaration does not prove that any particular kernel shape is the correct one. It does not assert that the dark energy drift actually reaches its maximum value. It does not establish a prediction for the measured value of δw₀.

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

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