Astrophysics Pulsar Period From Rung Normal Median Rung Eq

The median rung

Pulsars are rapidly spinning neutron stars that beam radio waves like lighthouses. Astronomers have long known their spin periods fall into two distinct families: normal pulsars, with periods from about 16 milliseconds to 10 seconds and a peak near 0.5 to 1 second, and millisecond pulsars, recycled by accreting matter from a companion, with periods from about 1 to 30 milliseconds and a peak near 3 to 5 milliseconds. The gap between the families sits at roughly 30 to 100 milliseconds, a separation confirmed by major catalogs such as the ATNF catalog and the EPTA and NANOGrav projects.

In Recognition Science, the framework models these periods as lying on a ladder of discrete steps, where each step multiplies the period by the golden ratio φ, about 1.618. The normal pulsar family has a base time τ_neutron, and its periods are τ_neutron · φ^k for integer rungs k. The declaration normal_median_rung_eq establishes that the median rung of the normal pulsar family is k = 4. Because adjacent rungs differ by exactly φ, this means the median period is φ^4 · τ_neutron, roughly 0.7 seconds, which sits squarely in the observed normal-pulsar peak.

The theorem itself is a simple identity: it states that the defined constant normal_median_rung equals 4. It is a structural claim inside the framework, not a measurement. The framework does not claim that this theorem, by itself, proves the observed pulsar distribution. The empirical bimodality, the gap at 30 to 100 milliseconds, and the ratio between the two families are addressed by other declarations in the same module, such as bimodal_ratio_gt_thirty, which shows the ratio of normal to millisecond median periods is φ^8, greater than 30.

What normal_median_rung_eq does not claim is just as important. It does not claim that the number 4 is derived from first principles; it is a definitional choice for the median rung. It does not claim that the framework has proved the observed pulsar period distribution; that remains an empirical comparison. And it does not claim that the framework has explained why the gap at 30 to 100 milliseconds is empty; that is a separate structural hypothesis with a named falsifier. The theorem is a precise, machine-checked statement about a constant, and its value lies in being one rung of a larger ladder, not in standing alone.

theorem normal_median_rung_eq : normal_median_rung = 4 := rfl

/-- Adjacent rungs differ by exactly `φ`. -/
theorem period_geometric (P_base : ℝ) (k : ℕ) :
    period_at_rung P_base (k + 1) = period_at_rung P_base k * phi := by
  unfold period_at_rung
  rw [pow_succ]
  ring

/-- The bimodal ratio is strictly greater than 30 (sharply distinguishable
from a continuous distribution). -/
theorem bimodal_ratio_gt_thirty : 30 < bimodal_ratio := by
  unfold bimodal_ratio recycling_rung_shift
  -- phi^8 ≥ (1.61)^8 = ?
  have h_phi : 1.61 < phi := phi_gt_onePointSixOne
  have h_pow : (1.61 : ℝ)^8 ≤ phi^8 := by
    have h_pos : (0 : ℝ) ≤ 1.61 := by norm_num
    exact pow_le_pow_left₀ h_pos (le_of_lt h_phi) 8
  -- (1.61)^8 = 45.39... > 30
  have h_compute : (30 : ℝ) < (1.61 : ℝ)^8 := by norm_num
  linarith

What this page does not claim

The theorem does not prove the observed pulsar period distribution. The median rung value 4 is a definitional choice, not a derived outcome. The framework does not claim that the 30 to 100 millisecond gap is explained by this theorem alone.

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

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