Cosmology Bitkernel Families Kernel At Zero

The kernel at zero

In cosmology, a kernel is a mathematical filter that shapes how a physical effect grows or fades with distance or time. The Recognition Science framework uses one to model a possible slow drift in the dark energy equation of state, written δw(z) = δw₀ · K(z). The factor K(z) is the kernel: it says how much of the maximum amplitude δw₀ survives at a given redshift z. Three candidate kernels are on the table: a constant K(z) = 1, a simple decline K(z) = 1/(1+z), and an exponential falloff K(z) = exp(−z/z₀).

The theorem kernel_at_zero settles a common ground for all three. It proves that at redshift z = 0, today, every one of these kernels evaluates to exactly 1. The constant kernel is 1 everywhere, by definition. The 1/(1+z) kernel at z = 0 is 1/1 = 1. The exponential kernel at z = 0 is exp(0) = 1. The proof in the machine-checked library of formal theorems is a short case analysis: check each of the three possibilities and simplify. The result is a recognition event, a discrete record of a fact, that anchors the whole family of models at the present epoch.

This single value carries a consequence. The effective equation of state in these models is w_eff(z) = −1 + δw₀ · K(z). At z = 0, since K(0) = 1, the equation simplifies to w_eff(0) = −1 + δw₀. The framework can therefore state, as a proved theorem, that the present-day value of the dark energy equation of state depends only on the amplitude δw₀, not on which kernel family one chooses. The choice of kernel affects how the drift evolves into the past, but not its value today.

What the theorem does not do is choose among the kernels. It proves a shared starting point, not a preferred shape. The constant, inverse, and exponential forms all pass through the same point at z = 0, and the theorem is silent on which one better describes actual cosmic data. That comparison is an empirical question for surveys like DESI, not a formal result in the library. The theorem also does not fix the amplitude δw₀ itself; it only says the amplitude lies in the range [0, φ − 3/2], where φ is the golden ratio, and that bound is a separate theorem.

In plain terms, kernel_at_zero is a normalization fact. It ensures that all three models agree on the present moment, which lets the framework compare their past behavior without a distracting offset at z = 0. The reader can now see that the kernel choice is purely about evolution away from today, and that any future measurement of w_eff at z = 0 constrains δw₀ directly, independent of the kernel.

/-- 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

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 theorem does not say which kernel family is physically correct. The theorem does not determine the value of the amplitude δw₀. The theorem does not establish that the dark energy equation of state actually drifts.

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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