Astrophysics Intergalactic Voids From Jcost

What the module proves

Astrophysics intergalactic voids are the large, nearly empty regions between galaxy filaments and clusters, occupying roughly 75 to 80 percent of the universe's volume. In Recognition Science, a ledger, a discrete record of recognition events, assigns a cost to comparing two quantities. The module in question defines a domain cost as the function J applied to the ratio of two real numbers, m over e, and proves three general properties of that cost.

The three proved facts are: the cost is zero when the two inputs are equal, the cost is never negative for positive inputs, and the golden ratio phi is greater than 1.5. These are formal theorems in the machine-checked library of formal theorems. The first says that comparing a quantity to itself costs nothing. The second says that comparing two positive quantities never yields a negative cost. The third establishes that a canonical threshold, defined as phi minus 1.5, is positive.

What the module does not do is connect these facts to cosmic voids. The definition of domain cost uses the ratio m/e without specifying what m and e mean in astrophysical terms. The paragraph in the source describing a void fraction formula, void fraction equals 1 minus J(phi) times (1 plus density contrast), is a research note recording where the idea was meant to go, not a result. The text is shared verbatim with 2383 sibling modules, all stating the same three general facts.

In Recognition Science, the framework models this as a template: the general cost properties are proved once, universally quantified, and any subject-specific theorem would require defining m and e in that subject's own terms. For cosmic voids, that definition does not exist in the module. The three proved facts stand on their own as properties of the cost function, but they do not establish anything about the actual void fraction of the universe.

theorem domainCost_at_eq (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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module proves any specific fact about the observed void fraction of the universe. The research note's formula for void fraction is a theorem. The value 0.882 for void fraction is derived or measured in this module.

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/IntergalacticVoids_FromJCost.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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