Astrophysics Pulsar Emission Regimes From Rs Period Ratio

The period ratio

A pulsar is a rotating neutron star that emits beams of radiation, appearing to pulse as it spins. Astronomers group pulsars into five broad emission classes: normal pulsars, millisecond pulsars, magnetars, rotating radio transients, and fast radio burst sources. These classes differ in spin period, magnetic field strength, and emission mechanism.

In Recognition Science, the framework's library of formal theorems defines a toy model of these classes. The model assigns each class a period on a ladder of powers of the golden ratio, φ ≈ 1.618. The theorem period_ratio proves that the ratio of any period to the next lower period is exactly φ. In plain words: if you climb from one rung of this ladder to the next, the period multiplies by the golden ratio each time.

The proof is short and mechanical. The period is defined as φ raised to an integer power. Dividing φ^(k+1) by φ^k cancels the common factor and leaves φ. The library also verifies that every such period is positive, since φ is positive and any power of a positive number is positive.

This is a statement about a defined model, not about measured pulsars. The theorem does not claim that real pulsar periods actually follow this ladder. It establishes a property of a mathematical construction: within the framework's five-regime model, adjacent periods are related by the golden ratio. Whether real pulsars match this pattern is an empirical question the theorem does not address.

theorem period_ratio (k : ℕ) : period (k + 1) / period k = phi := by
  unfold period
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

noncomputable def period (k : ℕ) : ℝ := phi ^ k

theorem period_pos (k : ℕ) : 0 < period k := pow_pos phi_pos k

noncomputable def period (k : ℕ) : ℝ := phi ^ k

What this page does not claim

The theorem does not claim that real pulsar periods follow the golden-ratio ladder. The theorem does not identify which observed pulsar belongs to which regime. The model does not specify the physical origin of the golden-ratio scaling.

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/PulsarEmissionRegimesFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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