Pulse Core Lexicon

Recursive Depth Indicator

Definition

The Recursive Depth Indicator is a weighted average of harmonic index that shows at which recursion levels a harmonic system's energy, and so its computational load, is concentrated. It is one of the book's three spectral complexity measures, used with the Spectral Complexity Index, an information entropy of how energy is spread across harmonics, and the Harmonic Clustering Coefficient, which measures departures from uniform spacing. Each harmonic is weighted by its share of the total energy, and a higher value means the load sits at deeper computational layers. Together the three distinguish recursive substrate architectures from classical uniform harmonic systems. The formal statement is Dℜ = sum_n n² × p_n / sum_n n × p_n, where Dℜ is the Recursive Depth Indicator, n the harmonic index and p_n the normalized energy distribution, equal to E_n/E⊚. All quantities are dimensionless [∅]; the book's check is consistent. It connects to the Spectral Complexity Index, Harmonic Clustering Coefficient, Recursive Depth and Computational Load Accumulation.

Mathematical Expression

From the book · Chapter 7 · Harmonic Complexity and the Recursive Spectrum

Dℜ = sum_n n² × p_n / sum_n n × p_n [∅]

Where:

  • Dℜ [∅] - Recursive Depth Indicator (G)
  • sum_n - summation operator over all harmonic indices n
  • n [∅] - harmonic index
  • p_n [∅] - normalized energy distribution, equal to E_n/E⊚
  • [∅] - quantities without physical units, pure numerical ratios or mathematical constants

Technical Details

Category Concept

Chapter Chapter 7

Importance

Related Terms