Astrophysics Supernova Kickvelocity2

Pulsar kicks and the cost ledger

When a massive star explodes as a supernova, the leftover neutron star often gets a random kick, a natal velocity of roughly 100 to 1000 kilometers per second. Astronomers have measured these pulsar kicks for decades, and the spread is real: some pulsars drift slowly, others race across the galaxy. The cause is thought to be slight asymmetries in the explosion, but the exact mechanism remains an open question in astrophysics.

One way to think about the kick is as a ratio. The observed speed sits between a low reference and a high one, and the ratio of those two numbers carries the information. In the Recognition Science framework, the central object is the cost, a measure of how far a ratio is from being balanced. The framework defines cost as J(x) = (x + 1/x)/2 - 1, which is zero when x equals 1 and grows as x moves away from 1. For a pulsar, the module sets the cost as J(m/e), where m is the measured kick and e is a reference speed.

The module astrophysics-supernova-kickvelocity2 proves three general facts about this cost function. First, the cost is zero when the measured kick equals the reference. Second, the cost is never negative for positive inputs, so the ledger never rewards a mismatch. Third, the golden ratio phi minus 1.5 is positive, a small arithmetic fact about the constant that appears throughout the framework. These three statements are proved in the machine-checked library of formal theorems, but the module itself does not prove anything specific to pulsars. The definition of m and e in astrophysical terms is a research note, not a theorem.

What the module does establish is a template. It shows that the cost function has the right shape to serve as a distance-like measure for any ratio, including a pulsar kick. The research note attached to the module proposes that the kick velocity follows a phi-power ladder: phi^0 to phi^5 times a base speed of 100 km/s gives 100 to 1100 km/s, which overlaps the observed 100 to 1000 km/s range. That is a hypothesis, not a proof, and it names its falsifier: a measured pulsar kick far outside the phi-power ladder would break the pattern.

In plain terms, the module is a small but honest piece of the framework. It takes a general cost function, proves its basic properties, and leaves the astrophysical connection as a clearly labeled idea. The reader can see exactly what is proved and what is proposed, which is the point of the exercise.

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]

What this page does not claim

The module proves nothing specific to pulsars; the kick-ladder idea is a research note, not a theorem. The phi-power ladder is not a derived prediction with a named falsifier beyond the range itself. The framework does not explain the physical cause of pulsar kicks.

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.

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