Cosmology Rs Cosmo Module 011

A template for cosmic dust

Interstellar dust grains absorb and scatter starlight, shaping how we see the galaxy. A research note in the Recognition Science framework's module 011 proposes that the fraction of mass locked in dust follows a specific formula: the square of the framework's cost function J evaluated at the golden ratio φ, which gives about 1.39 percent, matching a rough empirical estimate of one percent. The note calls this a match.

But the machine-checked library of formal theorems behind the framework proves something more general and less specific. The module defines a quantity called domainCost, a discrete record of the cost of recognition, as J(m/e), where m and e are real numbers. The formal theorems establish three facts about this quantity: it equals zero when m equals e, it is never negative when both inputs are positive, and the number φ − 3/2 is positive. These are properties of the cost function itself, not of dust. The module contains no definition of m or e in terms of any physical property of interstellar matter.

In plain language, the module proves a template. The same three facts would hold if m and e were masses of any two objects, or any other pair of positive real numbers. The docstring is explicit: the research note records where the idea was meant to go, not a result about dust. The formal content is shared verbatim across 2383 sibling modules, each with a different subject heading but the same universal template.

What the module does establish is a small piece of the framework's foundation. It shows that the cost function J, which the framework derives from first principles, has the basic regularity one would expect: zero cost when the two quantities are equal, nonnegative cost otherwise, and a threshold value tied to the golden ratio. These are stepping stones, not conclusions about cosmology.

To make module 011 a theorem about dust, one would need to define m and e in terms of dust itself: for instance, m as the mass of dust and e as some reference mass. That definition does not appear in the module. The gap is acknowledged in the source, and it means the empirical match in the research note remains a hypothesis, not a proved result.

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

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 does not prove that the interstellar dust fraction is 1.39 percent. The module does not provide a physical definition of the mass variables m and e. The empirical match to one percent is a research note, not a formal result.

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/RS_Cosmo_Module_011.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

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