Cosmology Bitkernel Shape Forcing Power Kernel Scale Free

The scale-free kernel

In cosmology, a kernel is a function that describes how some influence or quantity fades with distance or redshift. The Recognition Science framework considers kernels that are scale-free, meaning the function's behavior does not change when you zoom in or out on the scale factor. The declaration powerKernel_scaleFree establishes that any such scale-free kernel must be a power law: it has the form (1 + z)^(-s), where z is redshift and s is a real number exponent.

This result is a theorem in the framework's machine-checked library of formal theorems. It proves that the family of scale-free kernels is exactly the set of power laws. The theorem does not, by itself, determine the value of the exponent s. It only establishes the functional form. The value of s is pinned down by a separate condition, the rung condition, which requires the kernel to attenuate by a factor of the golden ratio conjugate when the scale factor increases by the golden ratio. That condition forces s = 1, giving the familiar 1/(1+z) form.

The theorem is a statement about a mathematical class of functions. It does not claim that the universe's actual kernel is scale-free, nor does it claim that the power-law form is the only possible kernel. It also does not determine the amplitude of the dark energy equation of state, which remains an open problem in the framework. The power-law form is a mathematical consequence of the scale-free assumption, not an empirical measurement.

/-- Every power kernel is scale-free. -/
theorem powerKernel_scaleFree (s : ℝ) : ScaleFree (powerKernel s) := by
  intro z w hz hw
  unfold powerKernel
  have hz1 : (0 : ℝ) ≤ 1 + z := by linarith
  have hw1 : (0 : ℝ) ≤ 1 + w := by linarith
  have harg : 1 + ((1 + z) * (1 + w) - 1) = (1 + z) * (1 + w) := by ring
  rw [harg, Real.mul_rpow hz1 hw1]

What this page does not claim

It does not claim that the universe's actual kernel is scale-free. It does not determine the amplitude of the dark energy equation of state. It does not claim that the power-law form is the only possible kernel.

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

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