Cosmology Cosmic Zscale Law Scale Affine Forces Linear Z

The scale-affine law

In cosmology, redshift z measures how much the universe has stretched since light left a distant galaxy. The scale factor a(z) = 1/(1+z) is the standard coordinate for that stretch, equal to 1 today and approaching 0 at the Big Bang. The Recognition Science framework asks what shape the dark-energy equation of state must take, and its machine-checked library of formal theorems answers with a conditional result: if one admissibility law holds, the shape is forced to be linear in redshift.

The law is called scale-affine, meaning that equal fractions of the scale factor carry equal fractions of accumulated cosmic recognition, a discrete record of events. The framework models the normalized cosmic-Z history as a function Zfrac(a) with two endpoints: Zfrac(0)=0 at the early zero-complexity limit, and Zfrac(1)=1 today. The admissibility condition states that Zfrac preserves convex interpolation between these endpoints. In plain terms, no intermediate scale-factor coordinate is singled out before new physics is supplied.

Under that condition, the theorem scaleAffine_forces_linearZ proves that Z(z) = Z_today / (1+z). The proof is short: the interpolation condition forces Zfrac(a) to be the identity map on the scale factor, and substituting a(z)=1/(1+z) gives the linear history. The declaration then derives the canonical deviation δw(z) = δw₀/(1+z) and the equation of state w(z) = -1 + δw₀/(1+z), conditional on the named law.

The result is a theorem conditional on the scale-affine law, not an unconditional derivation from first principles. The law itself is a structure with fields, not a proved fact. The theorem shows that if the law holds, the dark-energy shape follows; it does not prove the law holds in the actual universe. The framework's library includes a canonical witness that the admissibility class is inhabited, but that witness is a definitional choice, not evidence from observation.

What the declaration establishes is a tight logical link: scale-affinity forces linear-Z. It does not claim the scale-affine law is true of nature, nor that the linear shape matches any particular supernova or CMB dataset. Those comparisons would be empirical checks, not theorems. The certificate structure assembles the forced results, but the certificate is inhabited only under the same conditional law.

/-- 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]

/-- 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

/-- 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]

/-- The canonical scale-affine law, included as the witness that the admissibility class is
inhabited. -/
def canonicalScaleAffineZLaw : ScaleAffineZLaw where
  Zfrac := fun a => a
  early_zero := rfl
  today_one := rfl
  scale_affine_from_early_to_today := by
    intro a
    ring

What this page does not claim

The scale-affine law is not proved to hold in the actual universe; it is an admissibility condition, not an observed fact. The declaration does not compare the linear-Z prediction against any specific astronomical dataset. The theorem does not derive the scale-affine law from the BIT kernel or from the cost function J; it takes the law as a named premise.

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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