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. 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.

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⚠️ 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.

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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.

#	Result	Statement	Script	Status
1	Closed form	μ_k = ln(k − ½) + γ + ln(π/2), i.e. additive constant B₀ = −2 ln 2; R² = 1.000000 over 40 modes	suzuki_closedform.py	strong 1σ candidate
2	−½ = boundary phase	Sweeping Dirichlet→Neumann moves the offset c: −0.4946 → +0.4971, a clean integer shift Δc ≈ +1 (½ per endpoint) — a Maslov / winding phase	suzuki_boundary_maslov.py	numerically clean
3	Prime-onset sign law	A prime power n switches on exactly at a = log(n)/2; even eigenfunctions shift down, odd shift up (one-line parity proof); shift is linear in the von Mangoldt weight Λ(n)/√n, read back to 0.08% (n=2)	suzuki_onset_lemma.py, suzuki_comb_weights.py	proved (sign) + measured (weight)
4	Conservation	Σ_k δλ_k = 0 exactly: primes redistribute eigenvalues without shifting the total (off-diagonal, trace-preserving perturbation)	suzuki_sumrule.py	exact
5	Weyl counting	N(μ) ~ e^μ · 2/(π e^γ); exp(μ_k) ∝ (k − ½) is a Landau-type equally-spaced ladder — a "logarithm of Landau levels"	suzuki_spectrum.py	derived from Result 1
—	GUE certification	Nearest-neighbour spacings of the first 100 zeros: KS distance to GUE ≈ 0.068 vs to Poisson ≈ 0.349	gen_dataset.py	reproduces known result

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.

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Files

Suzuki Weil-form operator (the scientific core)

File	Contents
suzuki_closedform.py	P1-FEM assembly of Q_W^a; Richardson N-extrapolation (N=1000, 2000) and a-quadratic fit of λ_k(a); recovers μ_k = ln(k−½) + γ + ln(π/2). Prints B₀ = −1.38612 ± 1.7e-4 vs −2 ln 2 = −1.38629.
suzuki_boundary_maslov.py	Sweeps the endpoint stiffness from Dirichlet (v=0) to Neumann (v'=0) and fits the offset c at each end; demonstrates the integer shift Δc ≈ +1.
suzuki_onset_lemma.py	First-order perturbation theory for the prime-onset at a = log(n)/2; verifies the even-down / odd-up sign law (8/8 modes) and the pre-threshold null (δλ ≡ 0 for 2a < log n).
suzuki_comb_weights.py	Cluster-trace extraction of the von Mangoldt weight Λ(n)/√n from the eigenvalue shifts (0.08% for n=2, 1.4% for n=3 at a degeneracy).
suzuki_sumrule.py	Trace identity Tr(M⁻¹ dQ) = Σ δλ_k = 0; also reports the parity-graded trace (which is not a clean sum rule — honestly labeled).
suzuki_spectrum.py / suzuki_spectrum.json	Full low-lying spectrum vs a, the Weyl counting function, and the Landau-ladder check.
suzuki_mu1.json	Legacy single-constant record: the lowest eigenvalue λ_a and the first numerical value μ₁ ≈ 0.235 (superseded by the closed form μ₁ = ln(½) + γ + ln(π/2) in suzuki_closedform.py).

Riemann zeta zeros and GUE statistics

File	Contents
zeta_zeros.json	First 100 nontrivial zeros γ_k (imaginary parts), all with Re(s) = ½; `max
gue_spacings.json	Unfolded nearest-neighbour spacings, histogram, KS distances to the GUE Wigner surmise and to Poisson, and the small-gap fractions (the level-repulsion signature).
gen_dataset.py	Regenerates all three JSONs from scratch (mpmath + numpy/scipy).

Sonification ("music of the primes")

File	Contents
song.wav	38.9 s arrangement: the first 40 zeros mapped to an A-minor lead melody over an Am–F–C–G progression with synth drums and bass.
render_song.py	Offline numpy WAV renderer for song.wav.
index.html	The full interactive demo page (also hosted as the Space above).

Documentation

File	Contents
findings_2026-07-12.json	Machine-readable summary of the five findings, with an explicit HONEST_SCOPE block.
NARRATIVE.md	Short prose summary with the honest-scope statement.
ARTICLE.md	Longer expository write-up (background, methods, interpretation).

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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%.

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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).

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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

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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).

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