Cosmology Cosmic Zhistory

Cosmic Z-History

In cosmology, the equation of state of dark energy, written w(z), describes how its pressure relates to its density as the universe expands. The simplest model, ΛCDM, treats dark energy as a constant, with w = -1 at all times. Observations, however, allow w to drift slightly, and the question of exactly how it drifts is the dark-energy shape problem.

Recognition Science addresses this problem with a new quantity: cosmic Z-history, a discrete record of how much cosmic complexity, in the framework's sense, has accumulated by a given redshift z. The framework's Bosonic Identity Theorem (BIT) mechanism derives a dark-energy equation of state from this history. The result is w(z) = -1 + δw₀ · Z(z)/Z_today, where Z(z) is the accumulated complexity at redshift z and Z_today is its value today.

The first thing the module proves is a shape reduction: the normalized deviation of w from -1 is exactly equal to the normalized cosmic Z-history. This means the problem of deriving the dark-energy shape is exactly the problem of deriving the cosmic Z accumulation history. It is not a separate question. The boundary conditions are forced: today the deviation is δw₀, and in the early universe, where Z(z) approaches zero, w recovers the ΛCDM value of -1.

The second result is a conditional derivation of the canonical form. If cosmic Z accumulates linearly in the scale factor, meaning Z(z) = Z_today/(1+z), then the framework's kernel produces exactly the canonical deviation δw(z) = δw₀/(1+z). This is a theorem conditional on the linear-accumulation hypothesis. The shape is no longer posited; it is derived from one stated, physically-motivated premise.

In Recognition Science, the only remaining freedom is the accumulation law itself. The Z-history is monotone, positive today, and vanishing in the deep past, but why it is linear in the scale factor rather than in cosmic time or some power of it is not yet derived. That single question is the honest residue of the dark-energy shape problem.

/-- The BIT dark-energy equation of state `w(z) = -1 + δw₀ · Z(z)/Z_today`. -/
def bitKernel (dw0 Zt : ℝ) (Zhist : ℝ → ℝ) (z : ℝ) : ℝ :=
  -1 + dw0 * (Zhist z / Zt)

/-- **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

This module does not derive the linear-accumulation law itself; it remains a hypothesis. This module does not claim that the canonical 1/(1+z) shape is the only possible one, only that it follows from the linear premise. This module does not establish that dark energy is not a cosmological constant, only that its deviation, if any, is tied to the cosmic Z-history.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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