Astrophysics Jupiter Orbital Period Rs Jupiter Period Cert

What the certificate proves

Jupiter takes about 11.86 years to orbit the Sun. Recognition Science (RS) is a framework that tries to derive physical constants from a single cost function, and in its machine-checked library of formal theorems there is a declaration named JupiterPeriodCert. The name suggests a proof about Jupiter. The declaration actually proves three general facts about a function called domainCost, which measures the cost of recognizing one positive number as another: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and a certain threshold involving the golden ratio is positive. These are true for any positive numbers, not just Jupiter's orbital period.

The three facts are proved in the formal library as theorems. The first says that for any nonzero real number r, domainCost r r = 0. The second says that for positive m and e, 0 ≤ domainCost m e. The third says that phi - 3/2 > 0, where phi is the golden ratio. The declaration JupiterPeriodCert bundles these three theorems into a single structure, and the library also proves that this structure is inhabited, meaning the certificate exists. All of this is axiom-clean, checked by the machine.

What the declaration does not do is connect these facts to Jupiter. The library's own documentation states that the module proves nothing specific to Jupiter, because domainCost is defined as Jcost (m / e) without any reference to Jupiter's mass or orbital period. The research note that mentions phi^5 years, about 11.09 years, being 6.5 percent off from Jupiter's 11.86 years is exactly that: a note, not a theorem. No definition of m and e in astronomical terms appears in the module.

So the practical takeaway: JupiterPeriodCert is a valid but generic result. It shows that the cost function behaves sensibly for any positive inputs. What would make it a theorem about Jupiter is a definition of m and e in Jupiter's own terms, which the module does not provide. The certificate is a placeholder, not a derivation of the planet's period.

structure JupiterPeriodCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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 JupiterPeriodCert := ⟨cert⟩

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

What this page does not claim

JupiterPeriodCert does not prove that Jupiter's orbital period equals phi^5 years. The declaration does not derive any astronomical quantity from the cost function. The phi^5 approximation is a research note, not a theorem.

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/Jupiter_Orbital_Period_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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