Astrophysics Cosmic Void Size From Phi Ladder

The void-size estimate

Cosmic voids are the vast, nearly empty regions between galaxy clusters and filaments, spanning tens of millions of light-years. A common rule of thumb in observational cosmology links a void's radius to the typical cluster spacing: voids appear a few times larger than the clusters that bound them. One such estimate, recorded as a research note in the framework's library, sets R_void ≈ φ^5 × R_cluster, where φ is the golden ratio, about 1.618. For clusters spaced 2 to 4 megaparsecs apart, this gives voids of roughly 20 to 30 megaparsecs, a scale consistent with observed cosmic structure.

The golden ratio enters through the framework's recognition cost, a measure of how far a measured value sits from an expected one. The framework proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 − 1, and from that function the golden ratio emerges as the unique self-similar scaling. The void-size note applies that scaling to cluster spacing, but the note itself is a research idea, not a derived result. The formal library file that carries the note proves only three general facts about the cost function, none of which mention voids.

What the machine-checked theorems actually establish is modest and universal. For any positive measured and expected values, the cost function J(m/e) is zero when the two match, and nonnegative otherwise. A threshold constant, φ − 3/2, is positive. These facts are true for any pair of positive numbers, not just for cluster radii. The file defines a certificate bundling these three properties and proves it is inhabited, meaning the properties are consistent. None of this depends on what m and e stand for; the void-size formula would become a theorem only if m and e were defined in terms of cluster and void geometry.

In plain language: the framework's library contains a placeholder for a cosmic void result, and the placeholder is honest about its limits. The golden-ratio scaling is a plausible guess, but the formal proof does not yet connect the cost function to actual void sizes. The gap is a missing definition, not a contradiction. A future module could close it by defining m and e from cluster spacing and void radius, then proving the ratio φ^5 follows from the cost function's forcing properties. Until then, the 20 to 30 megaparsec figure remains a research note awaiting its formal grounding.

theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty VoidSizeCert := ⟨cert⟩

What this page does not claim

The void size of 20 to 30 megaparsecs is not a proved result, only a research note. The golden ratio φ^5 scaling is not derived from the cost function in this module. The cost function's properties apply to any positive numbers, not specifically to cosmic voids.

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

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