Cosmology Dark Energy Scale Affinity Derivation No Hidden Forces Linear Z

The redshift law

In cosmology, the redshift of light from distant galaxies is a measure of cosmic expansion. The standard model of dark energy predicts a specific form for how this redshift evolves over time. The Recognition Science framework, a mathematical system that derives physical laws from a ledger of recognition events, has produced a machine-checked theorem that forces this standard form from a single, simple assumption.

The theorem, noHidden_forces_linearZ, states that if a certain condition holds, then the redshift history must follow the linear law: Z(z) = Z_today / (1+z). Here, Z(z) is the redshift at a cosmic scale factor z, and Z_today is its value today. The condition, called NoHiddenScaleCoordinate, is a mathematical principle of parsimony: it forbids the universe's history from inserting any extra preferred moment between the early endpoint (a=0, the Big Bang) and today (a=1). The only allowed behavior is a straight-line interpolation between these two fixed points.

This is a theorem in the framework's machine-checked library of formal theorems, meaning the logical derivation is verified by a computer. The theorem is conditional: it holds if the NoHiddenScaleCoordinate principle is accepted as an axiom. The framework does not claim to derive this principle from its deeper forcing chain; it is stated as an admissibility gate for cosmic redshift. The theorem also does not claim to explain the physical origin of dark energy itself, nor does it predict the numerical value of Z_today.

What the theorem does establish is a clean logical link: a minimal, natural assumption about cosmic time leads directly to the standard redshift law. This is a significant result because it shows the standard law is not arbitrary but follows from a single, defensible principle. The framework's library proves this equivalence, and the result is a step toward showing that the entire standard model of cosmology may be a consequence of the framework's foundational axioms.

/-- 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 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

What this page does not claim

The NoHiddenScaleCoordinate principle is not derived from the framework's universal forcing layer; it is stated as an admissibility gate. The theorem does not predict the numerical value of Z_today or any other cosmological parameter. The theorem does not explain the physical nature or origin of dark energy itself.

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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