Cosmology Dark Energy Scale Affinity Derivation Canonical No Hidden Scale Coordi

The canonical witness

The declaration canonicalNoHiddenScaleCoordinate is a witness: a concrete, machine-checked example of a mathematical structure called NoHiddenScaleCoordinate. That structure formalizes one rule about how the universe's expansion history could be recorded. The rule says that if you know only two facts, that the scale factor was zero at the early endpoint and is one today, then the normalized fraction of accumulated cosmic Z at any intermediate time must be the scale factor itself. In plainer words, the bookkeeping cannot invent a special moment between the beginning and now; it must interpolate smoothly between the two known endpoints. The declaration provides the identity function as that canonical example, and the machine-checked library of formal theorems proves that this example satisfies the rule.

The identity function is not an arbitrary choice. The same library proves that any structure obeying the no-hidden-coordinate rule must have its Z-fraction equal to the scale factor. From that single fact, a chain of theorems follows: the redshift history takes the form Z(z) = Z_today / (1+z), the dark-energy deviation becomes dw0 / (1+z), and the equation of state takes the canonical form -1 + dw0 / (1+z). These are the standard shapes that appear in the framework's account of dark energy. The declaration sits at the start of that chain as the canonical witness, the simplest object that satisfies the rule and therefore triggers all the forced consequences.

What the declaration does not claim is that the no-hidden-coordinate rule itself is derived from deeper principles. The rule is stated as an admissibility gate, a named condition that the framework chooses to impose on the cosmic scale interval. The library proves that if you accept the gate, the canonical dark-energy shape follows. It does not prove that the gate must be accepted. The remaining step, deriving the gate from the universal forcing layer, remains open in the framework's own account. The declaration also does not claim that the identity function is the only possible witness; it is the canonical one, the simplest and most natural, but the structure allows other functions that satisfy the same endpoint conditions.

/-- The canonical no-hidden-coordinate law is inhabited by the identity Z-fraction. -/
def canonicalNoHiddenScaleCoordinate : NoHiddenScaleCoordinate where
  Zfrac := fun a => a
  early_zero := rfl
  today_one := rfl
  no_hidden_coordinate := by
    intro a
    ring

/-- 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 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

The no-hidden-coordinate rule itself is derived from deeper principles; it is stated as an admissibility gate. The identity function is the only function satisfying the NoHiddenScaleCoordinate structure. The canonical dark-energy shape is verified against astronomical observations in this declaration.

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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Derived articles

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