Astrophysics Hertzsprung Russell Rs

The H-R diagram and its cost

The Hertzsprung-Russell diagram, introduced independently by Ejnar Hertzsprung in 1911 and Henry Norris Russell in 1913, plots stars by luminosity against surface temperature. Most stars fall along a diagonal band called the main sequence, where hotter stars shine brighter. The band spans a vast range: the brightest main-sequence stars are roughly ten billion times more luminous than the faintest, a spread of about ten orders of magnitude in luminosity.

In Recognition Science, this astrophysical pattern becomes a test case for a general idea. The framework models any domain as a ledger, a discrete record of recognition events with a forced cost. The cost function J(x) = (x + 1/x)/2 - 1, proved unique in the framework's machine-checked library of formal theorems, measures the price of relating two quantities. The module under discussion defines a domain cost for the H-R diagram by applying J to the ratio of a star's mass m to some reference value e.

The module proves three general facts about this cost. It vanishes when m equals e, so matching the reference costs nothing. It stays nonnegative for positive inputs, so no ratio produces a negative price. And the threshold phi - 3/2, where phi is the golden ratio, is positive. These are properties of the cost function itself, not of any particular star.

The research note attached to the module suggests a structural ambition: that the main sequence width in luminosity might equal phi^7 times 10, about 2900, against an empirical range near 10^10. That comparison is not a theorem. The machine-checked code proves nothing specific to the H-R diagram, because the module defines m and e without reference to stellar physics. The note records where the idea was meant to go, not a result.

What the module does establish is a template. It shows how the framework's universal cost function can be pointed at an astrophysical ratio, and it certifies the basic properties any such application inherits. The step from template to astronomy requires defining mass and reference in stellar terms, which remains open.

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]

theorem cert_inhabited : Nonempty HRDiagramCert := ⟨cert⟩

What this page does not claim

The module does not prove that the main sequence width equals phi^7 times 10. The module does not identify m or e with any measured stellar quantity. The framework does not derive the H-R diagram from first principles; it only supplies a cost template.

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

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