Cosmology Dark Energy Wof Zstructural Falsifier Threshold At Redshift Half

The falsifier threshold

In cosmology, the dark energy equation of state is a number, usually written w, that describes how the density of dark energy responds to the expansion of the universe. The standard cosmological model, ΛCDM, treats dark energy as a constant with w = -1. The Recognition Science framework models dark energy as evolving slightly with redshift, the measure of how much the universe has expanded since light left a distant object. Its placeholder form is w(z) = -1 + φ⁻⁴⁴ · z, where φ is the golden ratio and φ⁻⁴⁴ is about 6.38 × 10⁻¹⁰.

The declaration falsifierThreshold_at_redshift_half is a proved theorem in the framework's machine-checked library of formal theorems. It computes the threshold at redshift 0.5, meaning the universe was half as large as it is today. The theorem establishes that the threshold equals φ⁻⁴⁴ / 2, roughly 3.19 × 10⁻¹⁰. A companion theorem proves that if a measurement of w at any positive redshift falls within this threshold of -1, then that measurement cannot equal the framework's predicted value. In plain terms, the framework has drawn a line: observe w closer to -1 than this tiny amount, and the structural form of its prediction is falsified.

The threshold is not a claim about what w actually is. It is a claim about what w would have to be for the framework to survive. The framework's own placeholder form is explicitly not a derived prediction; the specific functional dependence of w on redshift remains future work. The threshold only tests the algebraic discriminator: that the framework's w is strictly greater than -1 at any positive redshift, while ΛCDM's is exactly -1. The framework proves this separation is real, but it does not prove that the separation is large enough to be measured by current or planned instruments.

What the threshold changes is the nature of the test. Instead of a vague hope that dark energy might deviate from -1, the framework has committed to a precise, checkable boundary. Any future measurement of w at redshift 0.5 that lands within 3.19 × 10⁻¹⁰ of -1 would eliminate the framework's structural form. A measurement outside that band would be consistent with the framework, though it would not confirm the placeholder's specific shape. The declaration turns a philosophical stance into a numerical contract.

/-- Falsifier threshold at `z = 0.5`. -/
theorem falsifierThreshold_at_redshift_half :
    falsifierThreshold redshift_half = phi_neg_44 / 2 := by
  unfold falsifierThreshold redshift_half
  ring

/-- 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

/-- 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

What this page does not claim

The declaration does not claim that the framework's placeholder form w(z) = -1 + φ⁻⁴⁴ · z is the true evolution of dark energy. The declaration does not claim that any measurement within the threshold is impossible, only that it would falsify the structural form. The declaration does not claim that the threshold is observable with present instruments.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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