Astrophysics Gravitational Wave From Jcost Strain At Rung

The phi-decay law for wave strain

Gravitational wave strain is the fractional stretching of space that a passing wave produces, measured as h = ΔL/L, where L is the distance between two test masses and ΔL is the change that the wave causes. For binary mergers, the peak strain scales with the masses and distance as h ~ M^(5/3) / D. The Recognition Science library contains a small formal theory that arranges the five canonical source categories, neutron star pairs, black hole with neutron star, black hole pairs, supermassive black holes, and stochastic background, into a discrete ladder of strain amplitudes.

The central definition, strainAtRung, assigns to each integer rung k the value phi^k, where phi is the golden ratio (1 + √5)/2 ≈ 1.618. The library proves that the ratio of strain at one rung to the strain at the next lower rung is exactly phi. In plain terms, adjacent source classes differ in peak strain by a factor of the golden ratio, a phi-decay law. This is a theorem inside the framework: the machine-checked library of formal theorems verifies the ratio statement with zero axioms beyond the standard logical ones.

The framework's claim is narrow. It does not say that any particular binary merger produces a strain that equals phi^k in absolute terms. It does not derive the overall amplitude scale h_max from first principles, and it does not predict the measured waveform of any specific event such as GW150914. The definition is a relative statement: given a ladder of categories, the ratios follow the golden ratio. The absolute scale, the mapping of categories to real masses and distances, and the connection to observed signals are all left open.

What the definition does establish is a structural claim about the framework's own models. If the five categories are accepted as the right partition of gravitational wave sources, then the strain ratios are forced by the same phi-ladder that appears elsewhere in the framework. This is a consistency result, not a detection claim. A reader should take it as a formal property of a mathematical object, one that could in principle be compared to catalog data, but the comparison itself is not part of the theorem.

theorem strainRatio (k : ℕ) :
    strainAtRung (k + 1) / strainAtRung k = phi := by
  unfold strainAtRung
  have hpos := pow_pos phi_pos k
  rw [pow_succ]; field_simp [hpos.ne']

inductive GWSourceCategory where
  | nsNS | bhNS | bhBH | smbh | stochastic
  deriving DecidableEq, Repr, BEq, Fintype

theorem gwSourceCount : Fintype.card GWSourceCategory = 5 := by decide

What this page does not claim

The absolute strain amplitude h_max is not derived from the phi-ladder. No specific observed event, such as GW150914, is predicted by this definition. The mapping from rung numbers to real masses and distances is not established.

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

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