VIDraft/riemann-zeta-sonification

A Numerical Spectral Portrait of Suzuki's Weil-Form Operator

  

This repository contains the data, code, and audio behind the interactive demo [Hear the Riemann Zeta Zeros](https://huggingface.co/spaces/VIDraft/zeta-zeros). Its scientific core is, to the best of our knowledge, the first numerical realization of the self-adjoint operator constructed by M. Suzuki in Weil's quadratic form via the screw function (arXiv:2606.09096, 2026) — a theory-only paper that contains no numerics — together with a closed-form characterization of that operator's small-scale spectrum.

Everything here is fully reproducible with standard open-source tools (numpy, scipy, mpmath); no special hardware is required.

⚠️ Honest scope (read first)

This repository is a numerical spectral characterization of a 2026 operator. It is not a proof of, or progress toward, the Riemann Hypothesis (RH), and it makes no such claim.

More precisely:

• —All spectral results below are archimedean / universal: they are governed by the ½·t·log|t| (Gamma-factor) singularity of the screw function, they are the same for every L-function, and they are not specific to the Riemann zeta zeros. The zeta zeros live in the large-a regime and require the full prime sum; the clean small-a results here are, by construction, prime-free.
• —"First realization" is stated to the best of our knowledge.
• —The closed form for the constants (Result 1) is a strong 1σ candidate, not a proof: the additive constant matches −2 ln 2 to one standard error, and a full confirmation would require ~1e-6 accuracy.
• —The GUE / critical-line data reproduce established facts (Montgomery 1972; Odlyzko 1987); they are toolchain certification on the real object, not new mathematics.
• —The sonification is an educational / artistic mapping, not a mathematical claim.

Corrections and independent reproduction are welcome.

Summary of results

We discretize the Weil quadratic form Q_W^a(v) = ∬_{(-a,a)²} g(x−y) v'(x) v'(y) dx dy on H₀¹(−a,a) with P1 finite elements, solve the generalized eigenproblem Q v = λ M v, and write the eigenvalues as λ_k(a) = log(1/a) + μ_k + O(a). Suzuki's Theorem 1.4 gives the leading log(1/a) and asserts the existence of a lowest constant μ₁ > 0 but leaves it undetermined.

The half-integer offset in Result 2 was anticipated by VIDRAFT's circulation-phase hypothesis — from the conservation law Σ δλ = 0 and the even/odd (parity) alternation of the ladder, we conjectured that −½ is not a static constant but a "once-around" winding phase, then confirmed it numerically by the boundary sweep.

Files

Suzuki Weil-form operator (the scientific core)

Riemann zeta zeros and GUE statistics

Sonification ("music of the primes")

Documentation

Key numbers

• —Critical line: all 100 zeros have Re(s) = ½; max|ζ| at those heights ≈ 1e-24.
• —GUE level repulsion: KS distance to GUE ≈ 0.068, to Poisson ≈ 0.349. P(gap < 0.3): empirical ≈ 0 vs Poisson ≈ 0.26.
• —Closed-form constant: B₀ = −1.38612 ± 1.7e-4 vs −2 ln 2 = −1.38629 (1σ agreement), so μ_k = ln(k − ½) + γ + ln(π/2) with R² = 1.000000 over 40 modes.
• —Boundary phase: c(Dirichlet) = −0.4946, c(Neumann) = +0.4971, Δc = +0.99 ≈ +1.
• —Weight readout: von Mangoldt Λ(2)/√2 recovered to 0.08%, Λ(3)/√3 to 1.4%.

Methods (brief)

Operator (Results 1–5). We discretize the Weil quadratic form Q_W^a(v) = ∬ g(x−y) v'(x) v'(y) with g the screw function of ζ (eq. 1.3 of arXiv:2606.09096), using P1 hat functions on a uniform mesh of H₀¹(−a,a). The stiffness-like matrix Q is assembled from the double antiderivative G₂ of g (a Toeplitz construction), and the mass matrix M is the standard P1 mass matrix; eigenvalues come from scipy.linalg.eigh(Q, M). N-convergence of reported eigenvalues is ~1e-4 (Richardson from N = 1000, 2000). For 2a < log 2 the von Mangoldt sum in g is empty, so the small-a spectrum is prime-free and well conditioned; the constants μ_k are extracted by an a-quadratic fit over a ∈ [0.02, 0.05]. The prime-onset study (Result 3) adds a single ramp term g_n(t) = (Λ(n)/√n)·(2a − |t|)_+ that switches on at a = log(n)/2, and reads its first-order effect via perturbation theory δλ_k ∝ v_k'(a) v_k'(−a).

Zeros / GUE (certification). Zeros from mpmath.zetazero; unfolding by the density ρ(t) = (1/2π) log(t/2π); KS distances against the GUE Wigner surmise p(s) = (32/π²) s² e^{−4s²/π} and against Poisson e^{−s}. This reproduces the classic Montgomery (1972) / Odlyzko (1987) result at low height (the razor-sharp match is at height ~10²⁰).

Reproduce

pip install numpy scipy mpmath

# Suzuki operator — one script per result
python suzuki_closedform.py        # 1) μ_k = ln(k−½) + γ + ln(π/2)
python suzuki_boundary_maslov.py   # 2) −½ = boundary Maslov phase
python suzuki_onset_lemma.py       # 3) prime-onset sign law
python suzuki_comb_weights.py      # 3) von Mangoldt weight readout
python suzuki_sumrule.py           # 4) Σ δλ = 0 conservation
python suzuki_spectrum.py          # 5) Weyl counting / Landau ladder

# Zeta zeros, GUE statistics, and the audio
python gen_dataset.py              # zeta_zeros.json, gue_spacings.json, suzuki_mu1.json
python render_song.py              # song.wav

Relation to Suzuki (arXiv:2606.09096)

Suzuki's paper is purely analytic: it constructs the operator, proves self-adjointness and the leading small-a asymptotic (Theorem 1.4), and connects the Weil quadratic form to the explicit formula — but it reports no numerical spectrum and leaves the constants μ_k undetermined. This repository supplies (i) the first numerical spectrum, (ii) a closed-form candidate for the μ_k, (iii) the geometric identification of the −½ offset, and (iv) the prime-onset sign law and its weight readout. Two questions we would ask the author: is the closed form what the construction predicts (can `γ + ln(π/2)` and the `−½` be derived analytically)? and is there a clean asymptotic law for the onset magnitude (whose exponent we find to be mode-dependent, `ν ≈ 1–3.6`, so a naive `ε³` guess is refuted)?

Citation / lineage

The "music of the primes" idea follows Michael Berry (quantum chaos) and Marcus du Sautoy (The Music of the Primes). The operator whose spectrum we characterize is:

M. Suzuki, Weil's quadratic form via the screw function, arXiv:2606.09096 (2026).

References

• —M. Suzuki, Weil's quadratic form via the screw function, arXiv:2606.09096 (2026).
• —H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math. 24 (1973).
• —A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987).
• —H. Widom, spectral asymptotics of integral operators with logarithmic kernels.
• —Chen–Weth, spectral properties of the logarithmic Laplacian (context for the archimedean constants).

Produced by VIDRAFT. Data is honest and reproducible; corrections welcome.