Cosmology Bitkernel Families Constant Kernel Eq One

The constant kernel

In cosmology, the dark energy equation of state is often written w(z) = -1 + δw₀ · K(z), where z is redshift, a measure of how much the universe has expanded since the light we see was emitted. The function K(z) is a kernel, a shape that describes how the deviation from a pure cosmological constant evolves over cosmic time. A constant kernel, K(z) = 1, means the deviation does not evolve at all: dark energy behaves exactly like a cosmological constant at every redshift. The other two kernels in the family, K(z) = 1/(1+z) and K(z) = exp(-z/z₀), allow the deviation to decay or grow with time.

The formal theorem constant_kernel_eq_one states, for any real numbers z and z₀, that the constant kernel evaluates to 1. This is a definitional identity, true by the way the kernel is defined, not a physical discovery. It is one of three kernels in a family used by the framework's DESI Y3 forecast script to project how future measurements might constrain dark energy. The theorem's content is modest: it confirms that the constant shape is literally constant, and it acts as a building block for the larger claim that all three kernels equal 1 at z = 0, where the deviation δw₀ is defined.

In Recognition Science, this theorem sits inside a broader framework. The framework models recognition events in a discrete ledger, and from that starting point it derives a cost function J(x) = (x + 1/x)/2 - 1. The maximum allowed amplitude δw₀ for the dark energy deviation is bounded by J(φ) = φ - 3/2 ≈ 0.118, where φ is the golden ratio. The constant kernel theorem does not derive this bound; it only fixes the shape of one possible evolution. The bound itself is a separate definition, delta_w0_max, with its own proof that it is positive and less than 1.

What the theorem does not claim is worth stating plainly. It does not say that dark energy actually is constant; it only defines one of three candidate shapes. It does not assert that the constant kernel is preferred by data. The framework's forecast script uses all three kernels to explore possibilities, not to declare a winner. The theorem also does not claim anything about the physical mechanism behind dark energy. It is a small, precise piece of a larger formal structure, and its value is in what it enables: a clean statement that the constant shape is exactly what its name says.

/-- The constant kernel is `1` everywhere. -/
theorem constant_kernel_eq_one (z z0 : ℝ) :
    kernel KernelFamily.constant_kernel z z0 = 1 := rfl

/-- 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]

/-- The maximum BIT amplitude is `J(φ) = φ - 3/2 ≈ 0.118`. -/
def delta_w0_max : ℝ := phi - 3 / 2

What this page does not claim

The theorem does not assert that dark energy is constant in the actual universe. The theorem does not claim that the constant kernel is preferred by observational data. The theorem says nothing about the physical origin of dark energy.

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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