Cosmology Bitkernel Families Inv One Plus Z Pos

The 1/(1+z) kernel

In cosmology, a kernel is a simple mathematical shape that scales an effect across time or distance. The declaration inv_one_plus_z_pos concerns the kernel K(z) = 1 / (1 + z), where z is the cosmological redshift, a number that grows as we look further back in time. The theorem states that this kernel is positive whenever z is greater than -1, which covers all physically meaningful redshifts, since the universe expands and z is never below zero in practice.

This kernel is one of three candidate families used in the Recognition Science framework's BIT cosmology model to describe how the dark energy equation-of-state parameter w might deviate from -1 as a function of redshift. The other two are a constant kernel K(z) = 1 and an exponential kernel K(z) = exp(-z/z0). All three are designed to equal 1 at z = 0, meaning the deviation is normalized to its present-day value. The 1/(1+z) shape is called the canonical arc-11 kernel in the framework's documentation.

The positivity theorem is a small but necessary piece of a larger structure. It guarantees that the kernel does not cross zero or become negative in the redshift range of interest, which keeps the amplitude of the deviation well-behaved. The framework also proves that the maximum allowed amplitude, δw0, lies between 0 and approximately 0.118, a bound derived from the golden ratio φ. This bound, combined with the positivity of the kernel, keeps the effective equation of state w_eff = -1 + δw0 · K(z) within a physically sensible range.

In Recognition Science, this theorem is a formal result, checked by a machine-checked library of formal theorems. It is not a claim about what the universe actually does. The framework models dark energy with these kernels, but it does not prove that any one kernel is the correct description. The choice between the constant, the 1/(1+z), and the exponential shape is a modeling decision, not a derived law. The positivity theorem only establishes a mathematical property of one candidate function, not its physical reality.

/-- The 1/(1+z) kernel is positive for `z > -1`. -/
theorem inv_one_plus_z_pos (z : ℝ) (h : -1 < z) (z0 : ℝ) :
    0 < kernel KernelFamily.inv_one_plus_z z z0 := by
  unfold kernel
  exact div_pos one_pos (by 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]

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

What this page does not claim

The 1/(1+z) kernel is the correct or observed shape for dark energy evolution. The positivity theorem applies to the exponential kernel or any other function. The BIT model's predictions have been confirmed by 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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: