Cosmology Cosmic Zscale Law Scale Factor Le One

The scale factor bound

In cosmology, the scale factor measures how distances in the universe grow with time. It is conventionally set to 1 today, so a value of 0.5 means distances were half their current size. As we look back in time, we see objects at higher redshift (z), meaning their light has been stretched more by the expansion. The standard relation is a(z) = 1/(1+z), so the scale factor decreases as redshift increases.

The theorem scaleFactor_le_one states a simple consequence of this definition: for any non-negative redshift (z ≥ 0), the scale factor is at most 1. In plain terms, distances in the past were never larger than they are today, given the standard cosmological model. The proof is a direct algebraic manipulation: since z ≥ 0, the denominator 1+z is at least 1, so the fraction 1/(1+z) cannot exceed 1. This is a formal, machine-checked result in the Recognition Science framework's library of theorems.

In Recognition Science, this inequality gains a specific role. The framework models cosmic history through a ledger, a discrete record of recognition events. The scale-affine ledger law states that equal fractions of the scale factor carry equal fractions of the accumulated cosmic record, with no intermediate coordinate preferred. Under this condition, the normalized cosmic-Z history (the accumulated record) is forced to be exactly the scale factor itself: Z/Z_today = a. This forces the dark-energy deviation to take the form δw(z) = δw₀/(1+z), a specific prediction for how dark energy's equation of state evolves.

The theorem does not claim that the scale-affine ledger law is true. It is a conditional result: if that law holds, then the specific redshift history follows. The law itself is an admissibility condition, a definitional choice about what counts as a uniform ledger, not a derived physical fact. The framework proves the implication, not the premise. The inequality scaleFactor_le_one is likewise a property of the standard definition, not an independent physical discovery.

/-- For `z ≥ 0`, the scale factor is at most `1`. -/
theorem scaleFactor_le_one {z : ℝ} (hz : 0 ≤ z) : scaleFactor z ≤ 1 := by
  unfold scaleFactor
  have h : (0 : ℝ) < 1 + z := by linarith
  rw [div_le_one h]
  linarith

/-- 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

/-- 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

What this page does not claim

The scale-affine ledger law is physically true; it is a conditional premise. The inequality scaleFactor_le_one applies to negative redshifts, which would represent the future. The framework derives the value of δw₀; it only fixes the 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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