Cosmology Cosmic Zscale Law

The cosmic z-scale law

In cosmology, the scale factor measures how much the universe has expanded since light at a given redshift was emitted. At redshift zero, the scale factor equals one. At higher redshifts, looking further back in time, the universe was smaller, so the scale factor is less than one. This simple function is the backbone of how astronomers compare distances and times across cosmic history.

The cosmic z-scale law in Recognition Science asks a precise question: if the universe keeps a ledger, a discrete record of recognition events, what shape must the dark energy equation of state take? The answer comes from a single condition called the scale-affine ledger law. This law states that along the cosmic interval from the early universe to today, equal fractions of the scale factor carry equal fractions of the accumulated ledger. No intermediate scale-factor coordinate is special or selected before a new physical input is supplied.

This condition is not a curve fit. It is a symmetry statement. The theorem, proved in the machine-checked library of formal theorems, shows that this scale-affine ledger law uniquely forces the normalized ledger history to be the identity map on the scale factor. From this, the redshift history becomes linear in the scale factor, and the dark energy deviation takes the canonical form δw(z) = δw₀/(1+z). The equation of state is therefore w(z) = -1 + δw₀/(1+z).

The result is a theorem conditional on the named scale-affine ledger admissibility law. The proof contains zero sorry and zero new axioms. What this establishes in plain language is that a single symmetry principle, the uniformity of the ledger in the scale-factor coordinate, completely determines the dark energy shape. The framework does not fit this shape to data; it derives it from the admissibility condition.

The consequence is that the dark energy shape is not arbitrary. If the universe's ledger is uniform in scale-factor fractions, then the canonical form δw₀/(1+z) is the only possibility. This gives a concrete, testable prediction that distinguishes this framework from models where the dark energy shape is a free function.

/-- The scale-affine ledger law uniquely forces the normalized Z-fraction to be the identity
map on scale factor. -/
theorem scaleAffine_forces_identity (law : ScaleAffineZLaw) (a : ℝ) :
    law.Zfrac a = a := by
  have h := law.scale_affine_from_early_to_today a
  simpa [law.early_zero, law.today_one] using h

/-- Scale-affinity forces the redshift history to be the linear-in-scale-factor history
`Z(z)=Z_today/(1+z)`. -/
theorem scaleAffine_forces_linearZ (Zt : ℝ) (law : ScaleAffineZLaw) (z : ℝ) :
    ZfromScaleLaw Zt law z = CosmicZHistory.linearZ Zt z := by
  unfold ZfromScaleLaw CosmicZHistory.linearZ scaleFactor
  rw [scaleAffine_forces_identity law]
  rw [mul_one_div]

/-- Therefore scale-affinity forces the canonical BIT deviation
`δw(z)=δw₀/(1+z)`. -/
theorem scaleAffine_forces_canonical_deviation (dw0 Zt : ℝ) (law : ScaleAffineZLaw)
    (z : ℝ) (hZt : Zt ≠ 0) (hz : (1 : ℝ) + z ≠ 0) :
    CosmicZHistory.bitDeviation dw0 Zt (ZfromScaleLaw Zt law) z = dw0 / (1 + z) := by
  have hfun : ZfromScaleLaw Zt law = CosmicZHistory.linearZ Zt := by
    funext x
    exact scaleAffine_forces_linearZ Zt law x
  rw [hfun]
  exact CosmicZHistory.linear_accumulation_forces_canonical_kernel dw0 Zt z hZt hz

/-- The induced equation of state is the canonical kernel
`w(z)=-1+δw₀/(1+z)`. -/
theorem scaleAffine_forces_canonical_kernel (dw0 Zt : ℝ) (law : ScaleAffineZLaw)
    (z : ℝ) (hZt : Zt ≠ 0) (hz : (1 : ℝ) + z ≠ 0) :
    CosmicZHistory.bitKernel dw0 Zt (ZfromScaleLaw Zt law) z = -1 + dw0 / (1 + z) := by
  have h := scaleAffine_forces_canonical_deviation dw0 Zt law z hZt hz
  unfold CosmicZHistory.bitDeviation at h
  linarith [h]

What this page does not claim

The scale-affine ledger law is not derived from more basic principles; it is a named admissibility condition. This module does not prove that the dark energy shape matches astronomical measurements. The result does not determine the value of δw₀, only the functional form of its redshift dependence.

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

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