Cosmology Cosmic Zhistory Linear Accumulation Forces Canonical Kernel

The dark-energy shape

Dark energy is the name cosmologists give to the observed acceleration of the expansion of the universe. In the standard model, it is described by an equation of state, a number w that relates its pressure to its density. The simplest possibility, the cosmological constant, has w = -1 exactly, meaning the energy density stays constant as the universe expands. Observations allow small deviations from this value, and how those deviations change with cosmic time is called the dark-energy shape.

In the Recognition Science framework, this shape is tied to a quantity called cosmic Z-complexity, a measure of accumulated structure in the universe. The framework's Bosonic Identity Theorem (BIT) provides a dark-energy kernel, an equation of state written as w(z) = -1 + δw₀ · Z(z)/Z_today, where z is redshift (a measure of how much the universe has expanded since the light was emitted), Z(z) is the integrated complexity at that redshift, and Z_today is its value today. A theorem in the machine-checked library proves that the normalized deviation from w = -1 is exactly the normalized cosmic-Z history. This is a precise reduction: the dark-energy shape problem is the problem of deriving Z(z).

The declaration linear_accumulation_forces_canonical_kernel establishes a conditional result. If cosmic Z-complexity accumulates linearly with the scale factor a(z) = 1/(1+z), so that Z(z) = Z_today/(1+z), then the BIT kernel produces exactly the canonical deviation δw(z) = δw₀/(1+z). In other words, a single stated premise about how complexity grows forces the familiar 1/(1+z) shape. The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions in the derivation itself.

What the declaration does not claim is that complexity actually does accumulate linearly. That premise, called LinearScaleFactorAccumulation, is a hypothesis, not a proved result. The framework's own certificate notes that the accumulation law is the only remaining freedom in the problem. A power-law history, for instance, would give δw(z) = δw₀/(1+z)^p, where p is a shape index. The theorem shows what follows if the premise holds; it does not establish the premise itself.

/-- **SHAPE REDUCTION.** The normalized dark-energy deviation equals the normalized
cosmic-Z history. Deriving the dark-energy shape is exactly deriving `Z(z)`. -/
theorem shape_reduction (dw0 Zt : ℝ) (Zhist : ℝ → ℝ) (z : ℝ) (hdw : dw0 ≠ 0) :
    bitDeviation dw0 Zt Zhist z / bitDeviation dw0 Zt Zhist 0
      = (Zhist z / Zt) / (Zhist 0 / Zt) := by
  rw [bitDeviation_eq, bitDeviation_eq, mul_div_mul_left _ _ hdw]

/-- **LINEAR ACCUMULATION FORCES THE CANONICAL KERNEL.** With the linear-`a` cosmic-Z
history, the BIT kernel produces exactly the canonical `δw(z) = δw₀/(1+z)` deviation. The
`1/(1+z)` shape is derived from the accumulation premise, not posited. -/
theorem linear_accumulation_forces_canonical_kernel (dw0 Zt z : ℝ)
    (hZt : Zt ≠ 0) (_hz : (1 : ℝ) + z ≠ 0) :
    bitDeviation dw0 Zt (linearZ Zt) z = dw0 / (1 + z) := by
  rw [bitDeviation_eq]
  unfold linearZ
  rw [div_div, mul_comm (1 + z) Zt, ← div_div, div_self hZt, mul_one_div]

What this page does not claim

The linear-accumulation premise is not proved; it is a hypothesis. The theorem does not derive the value of δw₀, the present-day deviation. The framework does not claim dark energy is a cosmological constant; it allows for a time-varying shape.

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

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