Cosmology Dark Energy Scale Affinity Derivation

The no-hidden-coordinate gate

Dark energy is the name cosmologists give to the observed acceleration of cosmic expansion, usually modeled by an equation of state parameter w that relates pressure to energy density. In standard cosmology, w is close to -1, and deviations from exactly -1 are sought in survey data as a function of redshift z, the stretching of light due to expansion. The question this framework addresses is whether the canonical form of that deviation, δw(z) = δw₀/(1+z), can be forced by a structural principle rather than assumed.

The framework's earlier scale-affine law, a rule that the normalized cosmic Z-fraction scales linearly with the scale factor, already implied this deviation shape. The open question was where that law itself came from. This framework supplies a lower admissibility principle: the no-hidden-scale-coordinate condition. Once the early endpoint a=0 (no accumulated cosmic Z) and today's endpoint a=1 (normalized accumulated Z equals 1) are fixed, the recognition ledger, a discrete record of events, may not insert an extra preferred coordinate inside the interval. Therefore the normalized Z-fraction must preserve endpoint convex interpolation.

That condition is exactly the scale-affine law. The framework proves, in a machine-checked library of formal theorems, that the no-hidden-coordinate condition forces the Z-fraction to be the identity function: Zfrac(a) = a. From that identity, the redshift history follows as Z(z) = Z_today/(1+z), the canonical dark-energy deviation as δw(z) = δw₀/(1+z), and the equation of state kernel as -1 + δw₀/(1+z). The certificate structure packages all these forced results into one inhabited object, confirming the derivation is complete.

In plain language: if the expansion history is not allowed to single out any special moment between the beginning and today, then the dark-energy deviation must take exactly the 1/(1+z) form. The derivation is conditional on the named admissibility gate, not on an unconditional first principle. The deeper question of deriving that gate from the universal forcing layer remains open, and the framework states this explicitly as a target.

/-- 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 no-hidden-scale-coordinate condition forces the redshift history
`Z(z)=Z_today/(1+z)`. -/
theorem noHidden_forces_linearZ (Zt : ℝ) (H : NoHiddenScaleCoordinate) (z : ℝ) :
    CosmicZScaleLaw.ZfromScaleLaw Zt (noHidden_to_scaleAffine H) z =
      CosmicZHistory.linearZ Zt z :=
  CosmicZScaleLaw.scaleAffine_forces_linearZ Zt (noHidden_to_scaleAffine H) z

/-- 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 no-hidden-scale-coordinate condition forces the canonical equation of state. -/
theorem noHidden_forces_canonical_kernel (dw0 Zt : ℝ)
    (H : NoHiddenScaleCoordinate) (z : ℝ)
    (hZt : Zt ≠ 0) (hz : (1 : ℝ) + z ≠ 0) :
    CosmicZHistory.bitKernel dw0 Zt
        (CosmicZScaleLaw.ZfromScaleLaw Zt (noHidden_to_scaleAffine H)) z =
      -1 + dw0 / (1 + z) :=
  CosmicZScaleLaw.scaleAffine_forces_canonical_kernel
    dw0 Zt (noHidden_to_scaleAffine H) z hZt hz

What this page does not claim

This does not claim the no-hidden-coordinate condition is derived from first principles; it is stated as an admissibility gate. This does not claim a measurement of dark energy; it derives a canonical form conditional on the gate. This does not claim the framework's dark-energy shape is unique outside the framework's assumptions.

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)

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