Rayleigh first-harmonic test

Public verdict: validated and exposed as pysht.circular.rayleigh. This is a pySHT-native method with no SHT 0.1.9 crosswalk entry.

Scientific scope and statistic

For normalized angles \(\theta_i\in[0,2\pi)\), define

\[ \bar C=\frac1n\sum_i\cos\theta_i, \qquad \bar S=\frac1n\sum_i\sin\theta_i, \qquad \bar R=(\bar C^2+\bar S^2)^{1/2}. \]

The reported statistic is the conventional \(Z=n\bar R^2\). The null is circular uniformity, but the alternative is specifically a nonzero first trigonometric moment–a preferred first-harmonic direction. It is not described as an omnibus test: antipodally symmetric nonuniform distributions can have zero first moment.

The result reports \(\bar R\). It reports \(\operatorname{atan2}(\bar S,\bar C)\) in the input-period units only when the resultant is numerically nonzero; otherwise the direction is omitted and an explicit diagnostic marks it undefined. The implementation and terminology follow the standard Rayleigh circular-uniformity statistic.

Units, calibration, and boundaries

Finite values are reduced modulo a positive finite period and divided by that period before multiplication by \(2\pi\). This order remains valid for subnormal and near-maximum float64 periods. Tests cover radians/degrees, rotation, reflection, wrapping, row order, huge and subnormal periods, and an exactly antipodal zero-resultant fixture.

The finite-sample null is simulated directly: each replicate contains \(n\) iid uniform angles, with no fitted nuisance parameters. A public call consumes exactly \(nB\) uniform variates after the observed statistic. Upper-tail ties count and \(p=(b+1)/(B+1)\); MCSE and a 95% binomial interval are returned.

Complexity

Each resultant calculation is \(O(n)\) time and \(O(n)\) storage. Direct finite-null calibration with \(B\) replicates therefore costs \(O(Bn)\) time and retains only one size-\(n\) replicate, so peak storage is independent of \(B\).

Release evidence

At \(n=20\), each independent 20,000-dataset stream was compared with its own independently seeded 99,999-statistic reference bank:

reference seed	dataset seed	0.01	0.05	0.10
20260917	20260918	199 (0.00995)	1,004 (0.05020)	1,997 (0.09985)
20260933	20260919	221 (0.01105)	996 (0.04980)	2,008 (0.10040)

All pass the release tolerance. Because a single independent finite reference bank is shared within a stream, the gate uses \(\max\{0.005,4\sqrt{a(1-a)(1/20000+1/99999)}\}\) at nominal level \(a\); this accounts for both the 20,000 datasets and reference-quantile variation. For a targeted power audit, seed 20260928 generated 2,000 size-20 von Mises samples with concentration 2.5. Against a 19,999-statistic null bank from seed 20260927, 2,000/2,000 rejected at 0.05. This supports the intended first-harmonic direction only.

The table is an empirical-bank audit rather than 20,000 complete public calls; focused tests separately replay the literal public generator, tie, and plus-one path.