Physics Gravitational Wave Strain From Jcost

The strain formula

Gravitational wave strain, usually written h, is the fractional amount by which a passing wave stretches and squeezes space. A wave that changes a kilometer-long detector by one thousandth of the width of a proton produces a strain of about 10^-21. The standard formula for the strain from two orbiting masses is h = 4G/(c^4 R) times a term built from the chirp mass and the orbital frequency, where G is Newton's constant, c is the speed of light, and R is the distance to the source.

In Recognition Science, the framework models a discrete ledger, a record of recognition events, and defines a cost that measures how expensive a recognition is. The cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions, and it vanishes when x equals 1. The framework's library, a machine-checked collection of formal theorems, applies this cost to the ratio of two masses in orbit, defining the cost as J(m/e) for masses m and e.

The module proves three general facts about this cost. It vanishes when the two masses are equal, it is never negative for positive masses, and the number phi - 3/2 is positive, where phi is the golden ratio. The golden ratio, about 1.618, is the unique number that satisfies r^2 = r + 1. These three facts are assembled into a certificate, a small package of guarantees that the cost behaves as expected.

What the module does not do is connect this cost to actual gravitational wave strain. The definition of the cost uses the ratio m/e without saying what m and e mean physically. The docstring records a research note that the strain at LIGO's detection threshold might be approximately J(phi) times 10^-21, but this is an idea for where the work could go, not a result the module proves. The module itself establishes only the general properties of the cost function, not any specific claim about gravitational waves.

In plain language, the module shows that a particular mathematical cost function has the right basic shape for a physical application: it is zero when things match, never negative, and has a positive threshold value. The step from that shape to a real prediction about LIGO's sensitivity remains open. The framework's contribution here is a clean starting point, not a finished measurement.

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_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 does not establish any specific value for gravitational wave strain at LIGO's threshold. The research note about J(phi) times 10^-21 is not a theorem and is not proved in the module. The module does not define what the masses m and e represent in a gravitational wave context.

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

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