Astrophysics Supernova Kickvelocity2 Pulsar Kick Cert2

What the certificate proves

Pulsar kicks are the sudden velocities, often 100 to 1000 kilometers per second, that neutron stars receive at birth in a supernova. Astronomers have measured these speeds for decades, but no single theory explains them. In the Recognition Science framework, a file called PulsarKickCert2 was meant to connect those observed speeds to the framework's cost function. The file's own research note states the intended range: kick velocity roughly phi^k times 100 km/s, with phi the golden ratio, covering 100 to 1100 km/s. That is the plan. The formal content, however, proves something narrower.

What the machine-checked library of formal theorems actually establishes is three general facts about the cost function J(x) = (x + 1/x)/2 - 1, evaluated at a ratio m/e. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the number phi - 3/2 is positive. These are collected in a structure called PulsarKickCert2, which bundles the three statements together, and a proof that such a bundle exists. The proofs use only the definition of the cost and standard properties of the golden ratio. Nothing in the file defines what m or e mean for a pulsar, a supernova, or a kick velocity.

In Recognition Science, the cost function is a measure of recognition effort, and its uniqueness is a proved theorem elsewhere. The certificate here is a template: it shows that if you identify m and e with physical quantities, the cost has basic sanity properties. But the identification itself is absent. The docstring is explicit: the module proves nothing specific to pulsars because domainCost is defined as Jcost (m / e) without reference to the subject. The same body is shared verbatim with 2383 sibling modules, each with a different intended topic but the same three generic lemmas.

So the honest takeaway is that PulsarKickCert2 is a formal skeleton, not a physical result. It guarantees that a certain cost function behaves well at equality and stays nonnegative, and that a threshold constant is positive. It does not predict a kick velocity, does not derive the 100 to 1000 km/s range, and does not connect phi to pulsar data. The research note records where the idea was meant to go; the Lean code records only what was actually proved. A reader should not mistake the certificate for evidence about neutron stars.

structure PulsarKickCert2 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 cert_inhabited : Nonempty PulsarKickCert2 := ⟨cert⟩

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

PulsarKickCert2 does not predict any pulsar kick velocity. The certificate does not derive the 100 to 1000 km/s range from the framework. The file does not establish any connection between the golden ratio and measured neutron star speeds.

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/Supernova_Kickvelocity2.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

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