Astrophysics Pulsar Period From Rung Period At Rung Pos

The period ladder

Pulsars are rapidly spinning neutron stars that emit beams of radiation, and their rotation periods are famously bimodal: most fall into a normal group with periods from about 16 milliseconds to 10 seconds, peaking near 0.5 to 1 second, while a smaller recycled group spins much faster, with periods from about 1 to 30 milliseconds, peaking near 3 to 5 milliseconds. The gap between these populations, roughly 30 to 100 milliseconds, is empirically clear in catalogs such as the ATNF Pulsar Catalogue. In Recognition Science (RS), this structure is modeled by placing periods on a ladder: each family has a base period, and the period at rung k is the base period multiplied by the golden ratio φ raised to the k-th power, so that adjacent rungs differ by exactly φ.

The declaration period_at_rung_pos establishes a simple but load-bearing fact: if the base period is positive, then the period at every rung is also positive. This is a theorem proved in the framework's machine-checked library of formal theorems, with no unproved axioms beyond the standard logical ones. The proof is short: a positive base times a positive power of φ is positive, since φ itself is positive. This positivity result is what makes the ladder physically meaningful, ensuring that no rung can correspond to a zero or negative period, which would be unphysical for a spinning star.

The theorem does not claim that any particular pulsar actually sits on any particular rung. The framework's model chooses the rung structure and the base periods, but the assignment of observed pulsars to specific rungs is an empirical check, not a proved result. The declaration also does not claim that the bimodal distribution itself is proved; the framework exposes the structural gap of seven unstable intermediate rungs as a sub-cert, but the empirical bimodality is a measured fact, not a theorem. What is proved is the arithmetic: the ratio between the median periods of the two families is φ^8, which is strictly greater than 30, a sharp separation.

The consequence for the reader is a precise, checkable relation: if the normal family's median rung is 4 and the recycled family's median rung is also 4 but with a base period smaller by φ^8, then the period ratio is φ^8, about 47. This is a concrete prediction that can be compared against the measured pulsar catalog. The framework's contribution is not a claim that pulsars must exist, but a structural account of why their periods cluster where they do, with one clean arithmetic fact as the central point.

theorem period_at_rung_pos {P_base : ℝ} (h : 0 < P_base) (k : ℕ) :
    0 < period_at_rung P_base k := by
  unfold period_at_rung
  exact mul_pos h (pow_pos phi_pos k)

/-- 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 claim any observed pulsar sits on a specific rung. The theorem does not prove the empirical bimodal distribution of pulsar periods. The theorem does not derive the value of the base period τ_neutron.

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)

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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