Cosmology Bitkernel Families Delta W0 Max Lt One

The amplitude bound

In cosmology, the parameter w describes how dark energy responds as the universe expands. A value of exactly -1 means a constant energy density, the cosmological constant. Any deviation, written δw, would signal something more dynamic. The Recognition Science framework's BIT model proposes that this deviation follows a simple shape: δw(z) = δw₀ · K(z), where K(z) is one of three kernel families: constant, 1/(1+z), or exponential. Each kernel is normalized to equal 1 at the present day, z = 0, and each stays within [0, 1] for all future times, z ≥ 0.

The declaration delta_w0_max_lt_one proves a bound on the present-day amplitude δw₀. It shows that δw₀ must be strictly less than 1. More precisely, the maximum allowed value is J(φ) = φ - 3/2, where φ is the golden ratio, approximately 1.618. This gives δw₀ < 0.118. The proof is a short argument from the definition: since φ is known to be less than 2, subtracting 3/2 gives a number less than 1/2, hence certainly less than 1. The companion theorem delta_w0_max_pos establishes that this maximum is positive, so the allowed range is 0 < δw₀ < 0.118.

This bound matters because it constrains the entire family of possible dark energy histories. No matter which kernel shape is chosen, the amplitude at the present day cannot exceed this value. The effective equation of state, w_eff = -1 + δw₀ · K(z), therefore always lies between -1 and about -0.882 at z = 0. The framework's machine-checked library of formal theorems certifies this bound as a theorem, not a conjecture. The master certificate bundles all four key facts: kernels equal 1 at z = 0, the maximum amplitude is positive, it is less than 1, and w_eff at z = 0 equals -1 + δw₀.

What this does not claim is equally important. The bound does not say which kernel is correct, nor does it predict a specific measured value for δw₀. It only sets an upper limit. It also does not assert that dark energy actually deviates from -1; the framework allows δw₀ = 0, which recovers the cosmological constant. The theorem is purely a constraint on the model's parameters, not a measurement of them.

theorem delta_w0_max_lt_one : delta_w0_max < 1 := by
  unfold delta_w0_max
  have := phi_lt_two
  linarith

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

theorem delta_w0_max_pos : 0 < delta_w0_max := by
  unfold delta_w0_max
  have := phi_gt_onePointFive
  linarith

What this page does not claim

The theorem does not identify which kernel (constant, 1/(1+z), or exponential) is physically realized. It does not predict a specific measured value for δw₀, only an upper limit. It does not assert that dark energy deviates from -1; δw₀ = 0 remains allowed.

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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