Cosmology Dark Energy Scale Affinity Derivation Canonical No Hidden Maps To Cano

The canonical map

In the Recognition Science framework, the declaration canonicalNoHidden_maps_to_canonical is a machine-checked theorem that establishes a precise identity: the canonical witness for the no-hidden-scale-coordinate principle is exactly the same object as the canonical scale-affine law. The witness is simply the identity function, which maps every scale factor to itself. The theorem proves that when this identity function is used as the normalized Z-fraction, it satisfies the no-hidden-coordinate condition and, through the framework's conversion, lands precisely on the canonical scale-affine law.

The no-hidden-scale-coordinate principle is an admissibility condition about how the cosmic scale interval is treated. It states that once the early endpoint (scale factor 0) and today endpoint (scale factor 1) are fixed, the framework's recognition ledger may not insert an extra preferred coordinate inside the interval. The normalized Z-fraction must preserve endpoint convex interpolation, which means it must be a straight line between the two endpoints. The identity function is the unique function that does this, and the theorem confirms that this canonical choice maps to the canonical scale-affine law.

The theorem is a consistency and closure result. It shows that the canonical witness for the no-hidden-scale-coordinate condition is not an arbitrary choice but is exactly the function that the scale-affine law expects. The proof is by reflexivity, meaning the two objects are definitionally equal in the framework's formal system. This establishes that the no-hidden-scale-coordinate gate, when instantiated with its canonical witness, produces the canonical dark-energy shape without any additional structure or hidden assumptions.

What this theorem does not claim is equally important. It does not prove that the no-hidden-scale-coordinate principle itself is the fundamental origin of the scale-affine law. The principle is stated as an admissibility gate, not derived from the deeper universal forcing layer. The theorem also does not establish that the canonical dark-energy shape is the only possible one; it only shows that the canonical witness maps to the canonical law. Finally, it makes no empirical claim about the actual value of the dark-energy equation-of-state parameter; it is a formal identity about the framework's internal structures.

/-- 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 canonical no-hidden-coordinate law maps to the canonical scale-affine law. -/
theorem canonicalNoHidden_maps_to_canonical :
    (noHidden_to_scaleAffine canonicalNoHiddenScaleCoordinate).Zfrac =
      CosmicZScaleLaw.canonicalScaleAffineZLaw.Zfrac := by
  rfl

What this page does not claim

The no-hidden-scale-coordinate principle is not derived from the universal forcing layer. The theorem does not establish the empirical value of the dark-energy equation-of-state parameter. The theorem does not prove uniqueness of the canonical dark-energy shape among all possible witnesses.

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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