Astrophysics Neutron Star Cooling From Jcost Nsthermal Cert

A certificate for cooling

A neutron star is the crushed core left after a massive star explodes. Born with surface temperatures near a billion kelvin, it cools over millennia; at an age of ten thousand years, typical surface temperatures sit around a million kelvin. The Recognition Science framework approaches this through its ledger, a discrete record of recognition events, and the cost of recognition, a forced penalty for mismatch. The declaration NSThermalCert packages three proved facts about that cost into a single structure.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 - 1, applied here to a ratio m/e of two positive real numbers. First, the cost vanishes when the two numbers are equal: domainCost r r = 0 for any nonzero r. Second, the cost is never negative for positive inputs: 0 ≤ domainCost m e whenever m and e are both positive. Third, the constant phi - 3/2, called the canonical threshold, is positive; phi is the golden ratio, about 1.618, so the threshold is about 0.118. The declaration assembles these three statements into a structure called NSThermalCert, and the library proves such a certificate exists.

What the declaration does not do is as important as what it does. The documentation is explicit: these three facts are proved for the cost function in general, not for neutron stars specifically. The cooling formula T_NS(t) = T_0 * phi^(-n), with n = t / t_cooling, appears in the research note as an intended direction, not as a theorem. Nothing in NSThermalCert defines what m and e mean for a neutron star, so the certificate establishes no physical cooling law. It records that the cost function has three useful properties, and that a certificate of those properties exists.

The value of the declaration is structural. It shows the framework can package general cost properties into a named, reusable certificate, ready for a future definition that ties m and e to neutron star mass and energy. Until that definition exists, the cooling formula remains a research note, not a result. The certificate is a foundation stone with the building still unbuilt.

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 NSThermalCert := ⟨cert⟩

What this page does not claim

The cooling formula T_NS(t) = T_0 * phi^(-n) is not proved by NSThermalCert. The declaration does not define m or e in neutron star terms. The certificate does not predict a specific surface temperature at any age.

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)

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.

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