a spectrum in closed form. Every claim below is reproducible; all code and data are open.In 2026 M. Suzuki (Institute of Science Tokyo) introduced a self-adjoint operator, built from a "screw function" of the Riemann zeta function, whose spectrum is conjecturally tied to the zeta zeros. The paper is purely analytic — it contains no numerics and leaves its constants undetermined. Using a P1 finite-element discretization of the Weil quadratic form Q_W^a(v)=∬ g(x−y) v′(x) v′(y) on H_0¹(−a,a) (N-convergence ~1e-4), we realize the operator and characterize its low-lying spectrum. Writing λ_k(a) = log(1/a) + μ_k + O(a), we report a closed form for the constants μ_k, the geometric origin of the −1/2 offset, a proved prime-onset sign law, and an exact conservation rule. Every result is reproduced by the scripts in the companion dataset.
suzuki_closedform.pysuzuki_boundary_maslov.pysuzuki_onset_lemma.pysuzuki_sumrule.pygen_dataset.pyThe half-integer offset in ln(k − ½) is not a static constant. As the endpoint condition is relaxed from Dirichlet (v(±a)=0) to Neumann (v′(±a)=0), the fitted offset c slides by a clean integer — a half-integer per endpoint — which is the defining signature of a Maslov / winding phase. Equivalently exp(μ_k) ∝ (k − ½) is a Landau-type equally-spaced ladder, so the operator is a "logarithm of Landau levels".
| Boundary condition | Offset c | Fit R² |
|---|---|---|
| Dirichlet (v = 0) | −0.4946 ≈ −½ | 1.0 |
| Neumann (v′ = 0) | +0.4971 ≈ +½ | 1.0 |
| Shift Δc | +0.99 ≈ +1 (integer) | — |
This connection was suggested by VIDRAFT's circulation-phase hypothesis: from the yin–yang conservation Σδλ = 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 above.
Normalized nearest-neighbour spacings of the first 100 nontrivial zeros: the empirical histogram (green) sits on the GUE curve (blue) and departs from Poisson (red). The absence of small gaps is the level-repulsion signature.
KS distance to GUE ≈ 0.068 vs to Poisson ≈ 0.349; P(gap<0.3) ≈ 0 (empirical) vs ≈ 0.26 (Poisson). This reproduces the classical Montgomery (1972) / Odlyzko (1987) result and certifies the numerical stack on the real object.
These tests run in pure JavaScript on the raw data embedded in this page. Click, watch the numbers come out, and compare them to the published values.
Unfold the first 100 nontrivial zeros by the local density ρ(t) = ln(t/2π)/2π, form the nearest-neighbour spacings, and measure the two-sided Kolmogorov–Smirnov distance to the GUE (random-matrix) law versus Poisson. Nothing is precomputed — the zeros go in, the statistics come out.
Take the measured constants μ_k of the operator and subtract ln(k−½). If the closed form μ_k = ln(k−½) + γ + ln(π/2) holds, the remainder must collapse to the single constant γ + ln(π/2) = 1.0288.
Test A recomputes the GUE statistics end-to-end from the raw zeros. Test B verifies the closed form against our measured spectrum — computing those eigenvalues from the operator itself is the offline finite-element step, reproduced by the local commands below and suzuki_closedform.py.
No special hardware is required — only numpy, scipy and mpmath. Each script prints the corresponding result and its numerical checks.
# 1) install pip install numpy scipy mpmath huggingface_hub # 2) fetch the dataset huggingface-cli download VIDraft/riemann-zeta-sonification --repo-type dataset --local-dir zeta && cd zeta # 3) reproduce, result by 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 (0.08%) python suzuki_sumrule.py # ④ Σδλ = 0 conservation python gen_dataset.py # zeros + GUE spacing statistics python render_song.py # 🎵 song.wav
½ t log|t| (Γ-factor) singularity of the screw function, are the same for every L-function, and are not specific to the Riemann zeta zeros (which live in the large-a regime and require the full prime sum; primes enter only for a ≥ log(p)/2). This is a numerical spectral characterization of a 2026 operator; it is not a proof of, or progress toward, the Riemann Hypothesis. "First realization" is to the best of our knowledge; the closed form is a 1σ candidate; and the onset exponent is mode-dependent (a naive ε³ guess was refuted), so only the sign law, the threshold and weight-linearity are clean.
The first zero γ₁ = 14.13 is mapped to a reference pitch (220 Hz) and every other zero keeps its true ratio. The chord sounds atonal because the spacings between zeros are not regular.
Turning the spacings into rhythm: the real zeta sequence (level repulsion — only 2.3% short gaps) is even, whereas Poisson (uncorrelated — 43% short gaps) clumps and falls silent.
The first 40 zeros are quantized to an A-minor scale and used as the lead melody, over synth drums, bass and an Am–F–C–G progression. The melody pitches are the real zeta zeros; the accompaniment is an arrangement.
Toggle the layers — keep only "Zeta melody" and the primes sing alone: