Astrophysics Star Cluster3 From Jcost

The module's actual content

Astrophysics star cluster3 from jcost is a module in the Recognition Science framework's machine-checked library of formal theorems. Despite its name, the module proves nothing specific to star clusters. Its three theorems are general facts about a cost function, and the astronomical interpretation is a research note recording where the idea was meant to go.

The cost function in question, written cost, is a measure of mismatch between two quantities. The module defines it as J(m/e), where J is the framework's central cost function and m and e are real numbers. The three proved facts are: the cost is zero when m and e are equal, the cost is never negative when both inputs are positive, and the golden-ratio-derived constant phi minus 1.5 is positive. These are general properties of the cost function, not astronomical claims.

The research note attached to the module sketches a plan: open star clusters disrupt over 10^8 to 10^9 years, and the framework's phi-ladder suggests lifetimes of phi^k million years. The note checks that phi^13 through phi^16 give 843 to 2584 million years, consistent with observed lifetimes. But this is a note, not a proof. The module never defines what m and e mean for a star cluster, so the connection remains unformalized.

What the module does establish, in plain language, is that the cost function behaves sensibly: it vanishes at equality, stays nonnegative, and has a positive threshold constant. These are prerequisites for any application, but they are not star-cluster physics. The module is a template, shared verbatim with 2383 sibling modules, waiting for a definition of m and e in astronomical terms.

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_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]

What this page does not claim

No astronomical claim about star cluster lifetimes is proved in this module. The research note's consistency check is not a formal result. The module does not establish that the phi-ladder applies to star clusters.

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/Star_Cluster3_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.

Derived articles

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