Theorem IX · Volume I, Part II

The Harmonic Sieve Theorem

The generalized Tesfa Grid operator has exact spectral nulls at rational frequencies. Oscillatory content is suppressed exactly at harmonic sub-multiples of the grid width; the DC component is preserved.

Theorem IX Proved via the Dirichlet kernel Published: Volume I, Part II

Statement

The generalized Tesfa Grid operator T_{C,w} has exact spectral nulls at frequencies f_k = k/C for k = 1, 2, …, w − 1. The transfer function H(f) satisfies H(f_k) = 0 at every null. The operator preserves the DC component while annihilating exactly w − 1 oscillatory frequencies per period.

Proof outline

The result follows from a Dirichlet-kernel analysis of the operator's spectral response. The full proof, including the explicit transfer function and the kernel cancellation argument, is given in Volume I, Sections 12–13.

Comparison with classical kernels

Three properties distinguish the Tesfa operator from standard smoothing kernels:

  1. Annihilation, not attenuation. The Fejér kernel smoothly attenuates all frequencies by a bell-shaped taper; the Tesfa operator produces exact zeros at w − 1 specific frequencies.
  2. DC preservation. The operator does not affect the mean of the input sequence. The ratio of structured low-frequency power to total power is therefore enhanced, not just preserved.
  3. Structural independence of input. The null locations depend only on the grid parameters (C, w) and are independent of what sequence is passed through. Any input coincidentally carrying energy at these frequencies will have that energy suppressed.

Empirical consequence: the Harmonic Sieve in action

When the prime sequence is passed through the Tesfa operator, the low-frequency power ratio of the output is structured 0.406 versus a random-permutation baseline of 0.094 ± 0.018 (Z ≈ 17σ, p < 0.005; 200 Monte Carlo trials, see Volume I, Section 14). This effect is width-robust across w = 2–5 and scale-strengthening as C grows from 100 to 500.

That the prime sequence, passed through an operator with purely geometric null structure, yields an elevated low-frequency power ratio is evidence that the Harmonic Sieve is interacting with genuine structure in the prime distribution structure that is destroyed by shuffling and absent in random-gap surrogate sequences.

Cite this result

Dereje, T. (2026). The Tesfa Grid, Volume I, Theorem IX (Harmonic Sieve Theorem). Part II, Sections 12–13.