Astrophysics Neutron Star Cooling From Jcost

Neutron star cooling and the cost function

Neutron stars are the crushed cores left after massive stars explode. A typical neutron star is about 20 kilometers across but holds more mass than the Sun, and its surface starts at roughly a million kelvin. Over thousands of years it cools by radiating neutrinos and photons. The classic picture: at age 10,000 years, the surface temperature is around 10^6 K, and the star fades on a timescale set by its internal heat capacity and neutrino emission.

In Recognition Science, the framework asks whether that cooling curve is not arbitrary but forced by a single cost function, a penalty that reality pays for every recognition event it records. The framework's central result, proved in a machine-checked library of formal theorems, is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The framework models neutron star cooling by writing the cost as a function of a ratio: domainCost(m, e) = J(m / e), where m and e are two quantities in the star's physics. The framework's library proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the number phi - 3/2 is positive, where phi is the golden ratio.

What the module actually establishes is narrower than its title suggests. The Lean file proves the three facts above, but it defines domainCost as J(m / e) without connecting m and e to any specific neutron star property. The docstring records a research note: the idea was to set T_NS(t) = T_0 * phi^(-n) with n = t / t_cooling, so that at t = 10^4 years = phi^k years the temperature drops by a factor phi^k. That is a hypothesis, not a theorem. The file proves nothing specific to neutron stars because the bridge from the cost function to a real cooling law is not yet defined in the framework's own terms.

The honest summary: the framework has a proved cost function with clean properties, and a plausible but unproved guess about how it might apply to neutron star cooling. The three proved facts are real but generic. The specific claim, that a neutron star's temperature follows a phi-power ladder, remains a target. A reader should take the general cost theorem seriously and the cooling application as an open question.

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]

What this page does not claim

The module does not prove any specific neutron star cooling law. The phi-power temperature ladder is a research note, not a theorem. The cost function's three proved properties do not by themselves select the golden ratio as a cooling rate.

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

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