Cosmology Dark Energy Scale Affinity Derivation Scale Affinity Derivation Cert

The certificate

In cosmology, the standard model of dark energy uses an equation of state with a constant pressure-to-density ratio of -1, often called a cosmological constant. The Recognition Science framework's ledger, a discrete record of recognition events, derives this shape from a stricter condition. The certificate named ScaleAffinityDerivationCert is a machine-checked collection of formal theorems showing that if the cosmic expansion history obeys one admissibility condition, then the dark-energy deviation must take the canonical form δw(z) = δw₀/(1+z).

The condition is called the no-hidden-scale-coordinate principle. It says that once the early endpoint (scale factor a=0) and today's endpoint (a=1) are fixed, the normalized ledger fraction cannot insert an extra preferred coordinate inside the interval. The only allowed interpolation is the straight line between the endpoints. That condition forces the normalized Z-fraction to equal the scale factor itself, Zfrac(a) = a, which in turn forces the redshift history Z(z) = Z_today/(1+z) and the canonical dark-energy deviation δw(z) = δw₀/(1+z). The full derivation is formalized in the machine-checked library with zero axioms beyond the standard three and zero unproved assumptions.

The certificate is not a measurement and it is not a claim about the actual universe. It is a conditional theorem: if the no-hidden-scale-coordinate admissibility gate holds, then the canonical dark-energy shape follows. The certificate does not prove that the gate itself is forced by deeper principles; that remains an open target. It also does not establish the numerical value of δw₀, which must come from observation. What it does establish is a tight logical link: one clean structural condition, stated in plain language, is enough to derive the standard dark-energy form within the framework.

/-- The no-hidden-scale-coordinate condition forces the canonical dark-energy deviation. -/
theorem noHidden_forces_canonical_deviation (dw0 Zt : ℝ)
    (H : NoHiddenScaleCoordinate) (z : ℝ)
    (hZt : Zt ≠ 0) (hz : (1 : ℝ) + z ≠ 0) :
    CosmicZHistory.bitDeviation dw0 Zt
        (CosmicZScaleLaw.ZfromScaleLaw Zt (noHidden_to_scaleAffine H)) z =
      dw0 / (1 + z) :=
  CosmicZScaleLaw.scaleAffine_forces_canonical_deviation
    dw0 Zt (noHidden_to_scaleAffine H) z hZt hz

/-- The lower admissibility condition behind `ScaleAffineZLaw`.

Interpretation: with only the early endpoint `a=0` and today endpoint `a=1` available, a
normalized ledger fraction cannot choose a nonlinear coordinate without adding extra
structure. Therefore it preserves endpoint interpolation. -/
structure NoHiddenScaleCoordinate where
  /-- Normalized Z-fraction as a function of scale factor. -/
  Zfrac : ℝ → ℝ
  /-- Early endpoint: no accumulated cosmic Z at `a=0`. -/
  early_zero : Zfrac 0 = 0
  /-- Today endpoint: normalized accumulated cosmic Z is `1` at `a=1`. -/
  today_one : Zfrac 1 = 1
  /-- No hidden coordinate: endpoint convex interpolation is preserved. -/
  no_hidden_coordinate :
    ∀ a : ℝ, Zfrac ((1 - a) * 0 + a * 1) = (1 - a) * Zfrac 0 + a * Zfrac 1

/-- The no-hidden-scale-coordinate condition forces normalized Z-fraction to be the scale
factor itself. -/
theorem noHidden_forces_identity (H : NoHiddenScaleCoordinate) (a : ℝ) :
    H.Zfrac a = a := by
  exact CosmicZScaleLaw.scaleAffine_forces_identity (noHidden_to_scaleAffine H) a

/-- The scale-affinity derivation certificate is inhabited. -/
def scaleAffinityDerivationCert : ScaleAffinityDerivationCert where
  to_scale_affine := noHidden_to_scaleAffine
  identity_forced := noHidden_forces_identity
  linearZ_forced := noHidden_forces_linearZ
  canonical_deviation_forced := noHidden_forces_canonical_deviation
  canonical_kernel_forced := noHidden_forces_canonical_kernel

What this page does not claim

The certificate does not prove that the no-hidden-scale-coordinate gate is itself forced by deeper principles. The certificate does not establish the numerical value of δw₀ from theory. The certificate does not claim the actual universe obeys the no-hidden-scale-coordinate condition.

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/DarkEnergyScaleAffinityDerivation.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.

Derived articles

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