Data License:
All data and visualizations generated by this explorer may be used only with proper attribution to:
Flamandzki Harmonic Operator – millenniumchecked.org
Note: For performance reasons, only up to 10,000 primes are allowed in this demo.
Additional note: Vertical lines indicating zeta zeros are based on data of the first 10,000 non-trivial zeros of the Riemann zeta function, retrieved from LMFDB. The harmonic operator itself, when generating the signal, has no knowledge of these zeros beyond its inherent structure; the zero data is used purely for visualization reference.

🔬 Harmonic Interference Operator – RH Analysis

This interface computes the harmonic signal from primes using variable σ to detect destructive interference along the critical line ℜ(s) = 0.5.

📋 Output Summary

Interpretation Note: Destructive Interference vs. Depth

Minima of the harmonic amplitude function can appear for different values of σ. However, only at σ = 0.5 do these minima result from true destructive interference – that is, complete phase cancellation of oscillatory components. This phenomenon reflects a harmonic equilibrium where component vectors align in opposite phases, producing structured cancellation that corresponds to Riemann zeta zeros.

In contrast, deeper minima observed at other σ values (e.g., σ = 0.4) arise from asymmetric suppression, not equilibrium. These are not caused by full interference, but by dominance of certain frequency components. The interpreter distinguishes these by labeling only σ = 0.5 as "destructive interference (critical line)", and others as "partial interference" or "constructive accumulation".

Interpret amplitude with respect to phase structure, not numeric depth alone.

Ready.