Cosmology Dark Energy Wof Zstructural W Rs Linear Abs Deviation Eq Threshold

The threshold theorem

In cosmology, the dark-energy equation of state w(z) describes how the pressure of dark energy relates to its density as the universe expands. The standard model, ΛCDM, treats this as a constant, w = -1, meaning dark energy behaves like a fixed cosmological constant. The Recognition Science framework proposes a different form, w_RS_linear(z) = -1 + φ⁻⁴⁴·z, where φ is the golden ratio and φ⁻⁴⁴ is an extremely small positive number, about 6.38 × 10⁻¹⁰. This is a structural placeholder: a deliberately simple linear form chosen to show that the framework's prediction can differ from ΛCDM, not a fully derived dynamical law.

The theorem in question, w_RS_linear_abs_deviation_eq_threshold, proves a precise algebraic fact: for any non-negative redshift z, the absolute difference between the RS value and the ΛCDM value equals a threshold quantity, |w_RS_linear(z) - w_LCDM| = φ⁻⁴⁴·z. The proof is a short chain of rewrites: it expands the definitions, uses the positivity of φ⁻⁴⁴, and applies the fact that the absolute value of a non-negative number is itself. The theorem is axiom-clean, meaning it relies only on the standard logical axioms of the machine-checked library of formal theorems, with no additional assumptions specific to the framework.

The theorem's consequence is a falsification criterion. Because the deviation grows linearly with redshift, at any positive redshift the RS prediction is strictly greater than -1, and the gap is exactly φ⁻⁴⁴·z. A companion theorem proves that any measurement closer to -1 than this threshold cannot equal the RS prediction. This gives a concrete, testable band: at redshift 1, the threshold is φ⁻⁴⁴ itself, about 6.38 × 10⁻¹⁰. If future surveys measure w(z) at that redshift with precision better than this and find it consistent with -1, the linear placeholder is ruled out.

What the theorem does not claim is just as important. It does not derive the specific functional form of w(z); the linear placeholder is documented as one of many possible forms, and the true time-evolution of dark energy remains future work. It does not assert that the framework's prediction is correct, only that it is distinct from ΛCDM in a precisely quantified way. And it does not claim any measurement has been made or any data analyzed. The theorem is purely algebraic: it establishes the size of the gap, not whether the gap exists in nature.

/-- The absolute RS/LCDM separation is exactly the falsifier threshold
at every nonnegative redshift. -/
theorem w_RS_linear_abs_deviation_eq_threshold (z : ℝ) (hz : 0 ≤ z) :
    |w_RS_linear z - w_LCDM_value| = falsifierThreshold z := by
  rw [w_RS_linear_deviation_magnitude]
  unfold falsifierThreshold
  exact abs_of_nonneg (mul_nonneg (le_of_lt phi_neg_44_pos) hz)

/-- The structural discriminator: at positive redshift, the RS w(z)
value strictly exceeds the ΛCDM constant `-1` by the positive
amount `φ^{-44} · z`. -/
theorem w_RS_linear_distinct_from_LCDM_at_positive_z (z : ℝ) (h : 0 < z) :
    w_RS_linear z > w_LCDM_value := by
  unfold w_RS_linear w_LCDM_value
  have hphi : 0 < phi_neg_44 := phi_neg_44_pos
  have : 0 < phi_neg_44 * z := mul_pos hphi h
  linarith

/-- A measurement closer to ΛCDM than the RS structural separation
cannot equal the RS structural prediction. This is the formal falsifier
band used by the dataset row. -/
theorem measured_near_LCDM_not_RS_linear
    (z : ℝ) (h : 0 < z) {w_measured : ℝ}
    (hclose : |w_measured - w_LCDM_value| < falsifierThreshold z) :
    w_measured ≠ w_RS_linear z := by
  intro h_eq
  have hdist := w_RS_linear_abs_deviation_eq_threshold z (le_of_lt h)
  rw [h_eq] at hclose
  rw [hdist] at hclose
  exact (lt_irrefl (falsifierThreshold z)) hclose

/-- A generic non-ΛCDM witness profile (NOT the RS dark-energy prediction).

`w_RS_linear z := -1 + φ^{-44} · z`

At `z = 0`, `w_RS_linear(0) = -1` (matches ΛCDM exactly). At positive
redshift, the deviation is `φ^{-44} · z`, positive.

HONESTY WARNING: this is **not** the RS dark-energy equation of state. Its slope
`φ^{-44} ≈ 6×10⁻¹⁰` is the baryogenesis `η_B` scale, not the dark-energy amplitude;
it is zero today and grows without bound into the past, which is the wrong sign and
scale for the cosmic-aging mechanism (whose deviation is maximal today and decays as
`1/(1+z)`). The correct RS shape is the antitone cosmic-aging kernel
`w(z) = -1 + δw₀/(1+z)`; its un-attenuated ceiling normalization `δw₀ = J(φ) ≈ 0.118`
lives in `Cosmology.DeltaWKernel.canonicalDeltaW` and
`Foundation.MaximalForcing.w_RS_kernel`, while the protocol-carried prediction of
record uses the attenuated `δw₀ = A_RS = φ⁻⁴·J(φ) ≈ 0.017`
(`Cosmology.DarkEnergyCPLPointForced`). `w_RS_linear` is retained only as a generic
witness for structural discriminator and carrier-independence arguments: it shows that
*some* upward-deviating profile distinct from exact ΛCDM exists. -/
noncomputable def w_RS_linear (z : ℝ) : ℝ :=
  -1 + phi_neg_44 * z

What this page does not claim

The theorem does not derive the functional form of w(z); it only establishes a property of a placeholder. The theorem does not claim any observational data supports the RS prediction. The theorem does not prove that dark energy evolves; it only proves a distinction from a constant.

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/DarkEnergyWofZStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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