Cosmology Dark Energy Wof Zstructural Lcdm Abs Deviation From W Rs Linear Eq Thr

The deviation threshold

In cosmology, dark energy is often summarized by a single number, the equation-of-state parameter w, which describes how the pressure of dark energy relates to its density. The standard cosmological model, ΛCDM, treats this number as a constant, pinned to w = -1. Recognition Science (RS), a framework that derives physical structure from the cost of recognition events, proposes a different form: a value that grows slightly with redshift, the measure of how much the universe has expanded since light left a distant object.

The declaration LCDM_abs_deviation_from_w_RS_linear_eq_threshold is a theorem in the framework's machine-checked library of formal theorems. It states that for any non-negative redshift z, the absolute difference between the ΛCDM constant value and the RS linear proposal equals a specific threshold. That threshold is the product of z and a tiny number, φ⁻⁴⁴, roughly 6.38 × 10⁻¹⁰. In symbols: |w_ΛCDM - w_RS_linear(z)| = φ⁻⁴⁴ · z. The theorem also proves this threshold is positive for any positive redshift, meaning the two descriptions are never identical once the universe has expanded.

The practical consequence is a falsifier band. The library proves that if a measurement of w at some redshift lands closer to the ΛCDM value than this threshold, then that measurement cannot equal the RS linear prediction. The threshold therefore names the precision required to distinguish the two models. At redshift one, the required precision is about 6.38 × 10⁻¹⁰; at redshift one-half, half that. This is a structural result about the algebraic form, not a claim about actual data.

What the theorem does not claim is just as important. The linear form w_RS_linear(z) = -1 + φ⁻⁴⁴ · z is explicitly a placeholder, a non-vacuous witness to show the discriminator inequality can hold. The specific functional dependence of the RS deviation on redshift, the dynamics of cosmic aging that would produce it, remains future work. The theorem also does not assert that ΛCDM is wrong, nor that any measurement has yet reached the required precision. It only establishes the algebraic gap and the precision needed to test it.

/-- Symmetric form: the ΛCDM value is separated from the RS structural
prediction by exactly the falsifier threshold. -/
theorem LCDM_abs_deviation_from_w_RS_linear_eq_threshold (z : ℝ) (hz : 0 ≤ z) :
    |w_LCDM_value - w_RS_linear z| = falsifierThreshold z := by
  have h := w_RS_linear_abs_deviation_eq_threshold z hz
  have hswap :
      w_LCDM_value - w_RS_linear z = -(w_RS_linear z - w_LCDM_value) := by
    ring
  rw [hswap, abs_neg, h]

/-- The falsifier threshold is positive at every positive redshift. -/
theorem falsifierThreshold_pos (z : ℝ) (h : 0 < z) :
    0 < falsifierThreshold z := by
  unfold falsifierThreshold
  exact mul_pos phi_neg_44_pos h

/-- 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 claim that the RS linear form is the true physical law. The theorem does not claim any measurement has reached the required precision. The theorem does not claim ΛCDM is falsified.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: