Astrophysics Coronal Temperature From Jcost Coronal Temp Cert

The certificate

The Sun's corona, its outer atmosphere, is a few million kelvin while the visible surface below sits near 5,778 K. That contrast, a factor of roughly 200 to 500, has long puzzled physicists, because heat normally flows from hot to cold, not outward into a thinner, cooler region. The puzzle is called the coronal heating problem.

In the Recognition Science framework, the cost function J(x) = (x + 1/x)/2 - 1 measures the forced expense of a recognition event, a discrete record of comparison between two quantities. The framework's library of machine-checked theorems proves that any cost meeting five plain conditions must equal this J. The declaration CoronalTempCert packages three general facts about J applied to a ratio of two real numbers m and e: the cost is zero when m equals e, it is never negative when both inputs are positive, and the golden-ratio constant phi minus 3/2 is positive.

The certificate proves these three facts for any positive m and e. It does not define what m and e mean. The name suggests a solar application, and a research note attached to the file records an intended identification: the corona sits at rung 11 above the photosphere on the framework's phi-power ladder, since phi^11 is about 199, matching the observed temperature ratio. That note is a plan, not a result. The Lean code itself defines domainCost as J(m/e) without any reference to the Sun, and the certificate inherits that emptiness.

What the certificate does establish is reusable. Any future definition that sets m and e to physical quantities, such as coronal and photospheric temperatures, can cite the certificate's three facts as lemmas. The structure guarantees the cost vanishes at equality and stays nonnegative, and it records that the golden-ratio threshold is positive. Those are the load-bearing pieces; the subject-specific meaning must come from a separate definition that the current file does not supply.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not prove the corona is at rung 11 above the photosphere. The certificate does not resolve the coronal heating paradox. The certificate does not establish any empirical agreement with measured solar temperatures.

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/CoronalTemperatureFromJCost.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: