Volume III · In preparation · Expected late 2026

The Multi-Level Gap Hierarchy and Structural Extensions

The third volume of the research programme extends the framework with theorems on the multi-level gap hierarchy of the prime sequence, scale invariance at very large column widths, and perfect number classification within the Tesfa Grid.

Tesfaye Dereje In preparation Preprint expected late 2026
Volume III: In preparation

Volume III is currently under private preparation and will be released as a preprint following completion. This page summarizes the scope and the principal results in preliminary form. Specific theorems, proofs, and quantitative findings will be made available with the full preprint.

Scope

Where Volume I establishes the algebraic and spectral foundation of the Tesfa Grid, and Volume II presents empirical evidence for a connection between Tesfa Grid residuals and the Riemann zeta zero spectrum, Volume III turns the lens inward on the prime sequence itself examining the fine structure of prime gaps and their higher-order derivatives.

The volume develops what we call the multi-level gap hierarchy - the sequence of first, second, third, and fourth differences of the prime sequence and establishes several structural properties of this hierarchy.

Principal results (preliminary)

  1. X
    Scale Invariance Cross-scale correlations of the Tesfa structural signal remain essentially unity across a very wide range of integer scales. The construction exposes a scale-free property of the prime sequence invisible to the Tesfa Grid alone at moderate widths.
  2. XI
    Perfect Number Classification Perfect numbers are confined to specific rows and a small set of columns within the Tesfa Grid. The classification is complete for all currently-known perfect numbers and has an arithmetic proof grounded in the Euclid–Euler form 2^(n−1)(2^n − 1) together with the grid's two-phase structure.
  3. -
    Mean Convergence Law The means of the second, third, and fourth gap sequences converge to zero as the prime range grows, while the mean of the first gap sequence scales as ln(p). The prime sequence is in this precise sense a self-regulating steady-state signal at every higher derivative level.
  4. -
    Prime Oscillation Law (conjecture, with strong empirical support) The gap-of-gaps sequence alternates in sign with a characteristic frequency that is significantly different from random expectation. Prime gaps display a rhythmic grow-and-shrink behavior rather than Gaussian drift. Full numerical characterization with permutation controls is provided in Volume III.
On disclosure

Volume III contains applied material beyond the pure-mathematical scope of Volumes I and II. Specifically, the unified prediction system derived from the gap hierarchy will be released separately from the theoretical content, subject to intellectual-property protections appropriate for applied technology. The research community will receive the full theoretical content; the applied prediction system will be presented in a form that supports reproducibility without compromising those protections.

To be notified when Volume III is released as a preprint:

Subscribe to programme updates →