Cosmology Bitkernel Families

The three kernels

Cosmology bitkernel families are a set of three mathematical shapes, called kernels, used to model a small correction to the dark energy equation of state. In the standard cosmological model, dark energy is often treated as a constant with equation of state parameter w = -1. The bitkernel framework allows w to vary slightly with redshift z, the measure of how much light has been stretched by the expansion of the universe. The correction is written as δw(z) = δw₀ · K(z), where δw₀ is the maximum amplitude and K(z) is one of three kernel functions.

The three kernels are a constant, K(z) = 1; the canonical inverse redshift kernel, K(z) = 1/(1+z); and an exponential decay, K(z) = exp(-z/z₀). All three are bounded between 0 and 1 for z ≥ 0, and each equals 1 at z = 0, meaning the correction is strongest at the present epoch and fades into the past. The exponential kernel includes a free scale z₀, while the other two have no free parameters. The choice between them is a modeling decision, not a derived result.

The main result is a bound on the maximum amplitude: δw₀ must lie in the interval [0, J(φ)], where J(φ) = φ - 3/2 ≈ 0.118. Here φ is the golden ratio, and the value 0.118 emerges from the framework's cost function. The result proves this bound is positive and less than one, so the correction is always small. At z = 0, the effective equation of state is w_eff = -1 + δw₀, which holds for every kernel.

In Recognition Science, these kernels are used in a forecast script for the Dark Energy Spectroscopic Instrument (DESI) Year 3 data. The framework models the correction as a consequence of its recognition ledger, where the cost of recognition forces the golden ratio and the small amplitude bound. The machine-checked library of formal theorems verifies the kernel properties and the amplitude bound, providing a rigorous foundation for the forecast. This means the forecast can rely on the bound without hand-checking each step.

The practical upshot is that the bitkernel families give cosmologists a small, bounded, and well-defined set of shapes for dark energy corrections, with a hard ceiling on their size. The constant and inverse-redshift kernels are the simplest options, while the exponential kernel adds flexibility. The bound at 0.118 is a concrete prediction that can be tested against DESI data: if a fitted amplitude exceeds it, the framework would be falsified. This makes the module a bridge between a formal theory and an observational program.

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

/-- 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

/-- 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]

What this page does not claim

The framework derives the value of the dark energy equation of state itself. The choice among the three kernels is forced by the framework. The module proves that any particular kernel matches DESI data.

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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