Cosmology Bitkernel Families W Eff At Zero

The effective state at zero

In cosmology, the equation of state of dark energy is often written as a single number, w, that relates pressure to energy density. A value of exactly -1 corresponds to a cosmological constant, the simplest and most widely studied possibility. The Recognition Science framework models a small deviation from that value, calling it δw (a tiny correction that depends on how the universe has aged). The framework's BIT kernel families define three possible shapes for this deviation as a function of redshift, which is a measure of how much the universe has expanded since the light we see was emitted.

The three shapes are: a constant, the canonical form 1/(1+z), and an exponential decay. Each is a definitional choice, not a derived result. The framework proves one common property: all three kernels equal exactly 1 at redshift zero, meaning today, here and now, the deviation from -1 is the same regardless of which shape you pick. The theorem w_eff_at_zero states this in symbols: at z = 0, the effective equation of state w_eff equals -1 + δw₀, where δw₀ is the maximum amplitude of the deviation. This is a direct consequence of the kernel normalization, not a new physical assumption.

The practical consequence is that any forecast built from these families, such as the DESI Y3 analysis script referenced in the library, must agree on the present-day value of w. The choice of kernel only affects how the deviation evolves into the past, not what it is today. This makes the present-day value a reliable anchor point for comparing the three models against data. The framework does not claim that δw₀ itself is predicted; it is a free parameter, bounded between zero and about 0.118, the maximum allowed by the BIT theorem. Nor does the theorem say anything about what the deviation was at any other redshift, or which kernel is physically correct.

/-- 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 value of δw₀ is not predicted by the framework; it is a free parameter. The theorem says nothing about the effective equation of state at any redshift other than zero. The framework does not claim that any one of the three kernel shapes is the physically correct one.

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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