Astrophysics Coronal Temperature From Jcost

The coronal temperature ratio

The Sun's corona, its outer atmosphere, reaches about 1 to 3 million kelvin, while the visible surface, the photosphere, sits near 5778 K. That is a temperature ratio of roughly 200 to 500. For decades this has been the coronal heating paradox: the corona is farther from the Sun's energy source, yet it is dramatically hotter. The standard explanation invokes magnetic fields and plasma waves, but the puzzle of why the ratio lands in that specific range remains open.

In Recognition Science, the framework models this ratio as a step on a discrete ladder. The central object is the cost, a forced penalty for any mismatch between two quantities, defined as J(x) = (x + 1/x)/2 - 1. The framework's library proves three general facts about this cost when it is applied to a ratio of two positive numbers: it is zero when the numbers are equal, it is never negative, and a particular threshold involving the golden ratio is positive. These are the only machine-checked results in the module; they are universal statements about the cost function, not specific to the Sun.

The astrophysical claim is a research note attached to the module, not a proved theorem. It observes that the observed coronal-to-photospheric temperature ratio, roughly 200 to 500, brackets the eleventh power of the golden ratio, phi^11, which is about 199. In the framework's language, the corona would sit at rung 11 above the photosphere on a phi-power ladder of temperatures. This is a hypothesis with a clear falsifier: a measured coronal temperature ratio far from the phi^11 band, say below 150 or above 600, would break the pattern.

What the module itself establishes is narrower and fully rigorous. The machine-checked library proves that the cost function vanishes at equality, stays nonnegative for positive inputs, and that phi - 3/2 is positive. A certificate structure bundles these three facts together. The module proves nothing about the Sun because the definition of the cost does not yet connect the abstract ratio to any physical temperature. The bridge from the general cost theorem to the coronal temperature is the missing step.

For a reader, the distinction matters. The framework offers a concrete, testable prediction: the coronal temperature ratio should cluster near phi^11, roughly 199, not at some arbitrary value. That is a number a solar physicist could check against new observations. The formal library, meanwhile, guarantees only the abstract properties of the cost function. The physics enters when someone defines the mass and energy variables in the corona's own terms, a step the module itself does not take.

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]

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The module proves the coronal temperature ratio equals phi^11; it only records the observed bracket as a research note. The coronal heating paradox is resolved by this module; the physical mechanism remains unmodeled. The cost function's properties are proved for any positive ratio, not specifically for 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.

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