Astrophysics White Dwarf Radius Rs Wdradius Cert

The radius certificate

A white dwarf is the dense, cooling remnant of a Sun-like star, typically about the size of Earth. A typical white dwarf radius is roughly 0.01 times the Sun's radius, or about 7,000 kilometers. The Recognition Science framework defines a cost function, J(x) = (x + 1/x)/2 - 1, which measures the price of a mismatch between two quantities. In the module WDRadiusCert, the framework's library proves three general facts about this cost function when it is applied to a ratio of two numbers, m and e.

The first fact is that the cost is zero when the two numbers are equal. The second is that the cost is never negative when both numbers are positive. The third is a small technical fact: the golden ratio minus 1.5 is greater than zero. These three statements are assembled into a structure called WDRadiusCert, and the library proves that this structure is inhabited, meaning the certificate exists and its claims hold. The proof is machine-checked, meaning a computer program has verified each logical step.

The name WDRadiusCert suggests a connection to white dwarf radii, and a research note in the source file records an idea: phi^19 multiplied by 0.75 kilometers gives about 7,012 kilometers, close to the typical 7,000 kilometer radius. However, the formal proof itself makes no reference to white dwarfs, stars, or astronomy. The definition of the cost function uses the ratio m/e without specifying what m and e represent. The three proven facts are universal statements about the cost function, true for any positive numbers, not just for stellar masses and radii.

In Recognition Science, the framework models the cost of recognition as a forced quantity, and this certificate is an example of the framework's general machinery. The certificate establishes that the cost function has the basic properties one would want for a measure of mismatch: it is zero at agreement, nonnegative, and has a positive threshold related to the golden ratio. These are structural facts about the cost function itself, not empirical claims about white dwarfs.

What the certificate does not claim is that white dwarf radii are derived from the framework. The numerical match between phi^19 times 0.75 and the measured radius is noted in the source as a research idea, not a proved result. To turn this certificate into a theorem about white dwarfs, one would need to define m and e in terms of stellar physics, such as the star's mass and some reference mass. Without such a definition, the certificate remains a piece of pure mathematics about the cost function, not a statement about any astronomical object.

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 WDRadiusCert := ⟨cert⟩

What this page does not claim

No claim that white dwarf radii are derived from the framework. No claim that the numerical match between phi^19 times 0.75 and the measured radius is a proved result. No claim that the certificate makes any statement about astronomy.

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/White_Dwarf_Radius_RS.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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