Volume II · Preprint · March 2026 · Headline result

Empirical Evidence for the Tesfa-Zeta Conjecture

Three independent spectral methods consistently identify peaks in the Tesfa Grid residual spectrum at frequencies coinciding with the imaginary parts of the non-trivial zeros of the Riemann zeta function. Sub-0.1 accuracy on multiple zeros. All signals absent in shuffled, noise, and fake-prime controls.

Tesfaye Dereje March 2026 18+ pages (V6 edition with beat-cipher discovery and AI benchmark) Preprint Not a proof of the Riemann Hypothesis
Full PDF → Read Volume I

Abstract

We present the first direct experimental test of the Tesfa-Zeta Conjecture - the proposition that the Fourier transform of the Tesfa Grid residuals ε(j) = ρ(j) − 1/ln(6j), computed in logarithmic column-index space, produces spectral peaks at frequencies coinciding with the imaginary parts γ_n of the non-trivial zeros of the Riemann zeta function.

Three independent methods were applied across grid widths C = 10,000 to 500,000 (seven grid widths), with 300 randomized controls per C value. All 50 zeta zeros (γ₁ through γ₅₀) are recovered consecutively at C = 200,000 and C = 500,000. Tightest alignment: δ(γ₄) = 0.00242 at C = 300,000. Beat frequency discovery reveals four zero pairs producing deterministic binary key sequences. AI benchmark (Tesfa Bench): Claude Opus scores 0.478, failing on spectral computation tasks.

All signals are absent in shuffled-residual, white-noise, and fake-prime controls. These results constitute strong empirical support for the Tesfa-Zeta Conjecture. The Riemann Hypothesis remains unproved.

Headline result

δ = 0.00242 for γ₄ = 30.425 at C = 300,000 (0.008% accuracy). δ = 0.00376 for γ₂₇ = 94.651 at C = 300,000 (0.004% accuracy). δ = 0.00465 for γ₁₃ = 59.347 at C = 50,000 (0.008% accuracy). Three sub-0.005 matches. All 50 zeros recovered consecutively at C = 200,000.

Three independent methods, converging

Each method was applied at seven grid widths (C = 10,000 to 500,000) with 300-trial randomization controls. V6 sliding window construction with adaptive spectral windows. All three methods agree on the central finding.

Method 1: Log-space DFT
50/50 consecutive
Direct discrete Fourier transform with ln(6j) as time variable. All 50 zeros recovered consecutively at C = 200,000 and C = 500,000.
Method 2: Lomb-Scargle
δ < 0.005
Optimal for logarithmically-sampled data. Three sub-0.005 matches across C values. Sharpest: δ(γ₄) = 0.00242 at C = 300,000 (0.008% accuracy).
Method 3: Matched Filter
24/30 significant (C=10K)
z-score versus 300 shuffled controls. Expected by chance: 1.5. Observed at C = 10,000: 24. p < 0.001.
Control specificity
Zero
Shuffled residuals, white noise, and fake-prime sequences all fail to reproduce the signal. Unique to the real prime sequence.

What this establishes

The Tesfa Grid residuals, analyzed in log(j) frequency space, carry power elevated above random baseline at frequencies coinciding with the imaginary parts of Riemann zeta zeros. This is confirmed by three independent methods at seven grid widths (C = 10,000 to 500,000) with 300-trial randomization controls.

What this does not establish

This result does not prove the Riemann Hypothesis. It does not prove the Tesfa-Zeta Conjecture (which requires convergence as C → ∞). The analytic connection between the Tesfa Grid operator and the zeta function remains open.

Correct statement

The Tesfa-Zeta Conjecture has passed its first direct experimental test. The evidence is strong enough to report. It is not strong enough to claim proof of anything beyond what the data directly shows.

Beat Frequency Discovery (V6)

The V6 analysis reveals oscillating amplitude ratios between adjacent zeta zeros as C varies. Four beat pairs tracked across seven C values produce deterministic binary key sequences:

γ₄/γ₅: key = 1100111  |  γ₃₄/γ₃₅: key = 0100110
γ₄₅/γ₄₆: key = 0110101  |  γ₁/γ₂: key = 0111111

Tesfa Bench: AI Evaluation

36 challenge problems with provably correct answers. Claude Opus scores 1.000 on knowledge lookup, 0.000 on spectral computation. Current LLMs cannot compute what the Tesfa spectral engine computes.

0.478
Claude Opus overall score
1.000
Knowledge lookup
0.000
Spectral computation

Open problems

OP 1TZC Convergence

Does δ_n(C) → 0 as C → ∞ for all n? Prove or disprove that Lomb-Scargle peak positions converge to the imaginary parts γ_n of the zeta zeros. Empirical fit at current scale: δ_n(C) ~ C^(−α) with α ≈ 0.12.

OP 2Null-to-Gap Alignment

Prove that the Tesfa Harmonic Sieve nulls at f_k = k/C are positioned between consecutive γ_n values, making the Tesfa operator a natural band-pass filter for zeta-zero frequencies.

OP 3Amplitude Formula

Derive an analytic expression for the z-score z(γ_n) as a function of C, γ_n, and the Tesfa operator transfer function. If z(γ_n) → ∞ as C → ∞, this implies convergence.

OP 4Critical Line

Can the Tesfa spectral framework constrain the real part of the zeta zeros to Re(ρ) = 1/2? This would constitute a new approach to the Riemann Hypothesis, one not rooted in contour integration or random matrix methods, but in discrete geometric sieving.
See all open problems across the programme →