Cosmology Cosmic Zscale Law Scale Affine Forces Canonical Deviation

The scale-affine law

In cosmology, the scale factor a(z) measures how much the universe has expanded since light left a distant galaxy; at redshift z it equals 1/(1+z). Today a=1, and at the Big Bang a=0. The Recognition Science framework asks a pointed question about dark energy: why should its equation-of-state deviation δw(z) take the particular form δw₀/(1+z), rather than some other curve? The declaration scaleAffine_forces_canonical_deviation answers that question by showing that one clean condition, called the scale-affine ledger law, forces the answer uniquely.

The condition is a statement about neutrality. A ledger, a discrete record of recognition events, tracking accumulated cosmic Z along the universe's history from the early endpoint a=0 to today a=1. The scale-affine law says the ledger is uniform in the scale-factor coordinate: it does not single out any intermediate scale-factor value as special before new physics is introduced. Formally, the normalized Z-fraction preserves convex interpolation between the two endpoints. The theorem scaleAffine_forces_identity proves that under this law, the normalized Z-fraction must equal the scale factor a itself. That is the entire content: no intermediate coordinate is preferred, so the history is forced to be linear in a.

From that identity, the framework derives the redshift history. The declaration scaleAffine_forces_linearZ shows that Z(z) = Z_today/(1+z), and the target declaration scaleAffine_forces_canonical_deviation then proves the canonical deviation δw(z) = δw₀/(1+z). The final step, scaleAffine_forces_canonical_kernel, gives the equation of state w(z) = -1 + δw₀/(1+z). The whole chain is a theorem in the framework's machine-checked library of formal theorems, with zero sorry and zero new axioms. The proof is conditional: it assumes the named scale-affine ledger admissibility law as a premise, and it requires Zt ≠ 0 and 1+z ≠ 0, the latter meaning the formula holds for z ≥ 0, which is the observable range.

What the declaration does not claim is just as important. It does not prove that the scale-affine ledger law is physically true; that law is an admissibility condition, a definitional choice about how the ledger behaves. The theorem proves a conditional: if the law holds, then the dark-energy shape follows. It also does not fit the parameter δw₀ to data; δw₀ remains a free constant in the formula. And the declaration says nothing about whether this shape matches astronomical observations; that is an empirical check, not part of the theorem. The framework's contribution is the derivation, not the measurement.

The practical upshot is that a single neutrality condition, stated without any curve fitting, selects one dark-energy history from an infinite family of possibilities. A reader can now see why the 1/(1+z) shape is not an arbitrary choice in this account: it is the unique shape consistent with a ledger that treats every scale-factor fraction equally. Whether nature obeys that condition is a question for observation, and the framework leaves that question open.

/-- 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 proven to be physically true; it is an admissibility condition. The parameter δw₀ is not fitted to any data; it remains a free constant. The declaration does not claim the derived shape matches astronomical measurements.

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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