Astrophysics Star Cluster3 From Jcost Star Cluster3 Cert

A certificate with a narrow scope

Star clusters are groups of hundreds to millions of stars bound by gravity. Open clusters, the looser kind, typically survive for 100 million to a few billion years before gravitational encounters and galactic tides pull them apart. Astronomers estimate a cluster's disruption time from its mass, density, and environment, not from a single universal number.

In the Recognition Science framework, a ledger (a discrete record of recognition events) assigns a cost to any ratio of two quantities. The cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a theorem proved in the framework's machine-checked library of formal theorems. The declaration StarCluster3Cert packages three consequences of that cost function: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and the golden-ratio constant phi exceeds 1.5. Each of these is a general fact about the cost function, not about astronomy.

The module defines domainCost(m, e) as Jcost(m / e), where m and e are real numbers. The certificate then proves that domainCost(r, r) = 0 for any nonzero r, that domainCost(m, e) is nonnegative when both m and e are positive, and that phi - 3/2 is positive. These proofs are correct and machine-checked. But the module never defines what m and e mean for a star cluster. The docstring notes a research idea: perhaps m could be mass and e could be energy, and phi powers might match cluster lifetimes. That idea is recorded as a note, not proved.

What would turn this into a theorem about clusters is a definition of m and e in cluster terms, for instance m as the cluster mass and e as some binding energy. Without that, the certificate is a template shared verbatim with 2383 sibling modules. The universal facts live in one shared module, DomainCostTemplate; this file merely instantiates them. So the declaration establishes three properties of the cost function, and nothing about the lifetime, mass, or evolution of any star cluster.

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

StarCluster3Cert does not claim any specific star cluster lifetime, mass, or disruption time. The certificate does not prove that phi powers predict cluster lifetimes; that remains a research note. The module does not define m and e in astrophysical terms.

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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