---
type: Paper Section
title: Research Hypotheses — Abstract
description: >-
  Abstract of the Part V preregistration: five primary hypotheses (optimal filling,
  scale invariance, zeta gap distribution, universal repulsion, biotic decoupling) and
  four fractal-string spatial hypotheses for old-growth canopies, with the LiDAR, DBC
  and statistical methods to test them.
resource: https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/abstract/
tags: [paper, part-v, preregistration, forest-ecology, canopy-height-model, spectral-geometry]
paper:
  title: "Spectral Geometry and Metabolic Scaling in Forest Canopy Architecture: A Preregistration Framework"
  part_title: Research Framework and Testable Hypotheses
  landing: https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/
  stage: preregistration
sources:
  - id: lapidus-vanfrankenhuijsen2013
    resource: https://doi.org/10.1007/978-1-4614-2176-4
    title: "Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Lapidus and van Frankenhuijsen, 2nd ed. 2013, Springer)"
  - id: sarkar-chaudhuri1994
    resource: https://doi.org/10.1109/21.259692
    title: An efficient differential box-counting approach to compute fractal dimension of image (Sarkar and Chaudhuri 1994, IEEE Transactions on Systems, Man, and Cybernetics)
  - id: odlyzko1987
    resource: https://doi.org/10.1090/S0025-5718-1987-0866115-0
    title: On the distribution of spacings between zeros of the zeta function (Odlyzko 1987, Mathematics of Computation)
  - id: bak1987
    resource: https://doi.org/10.1103/PhysRevLett.59.381
    title: "Self-organized criticality: an explanation of 1/f noise (Bak, Tang and Wiesenfeld 1987, Physical Review Letters)"
generated:
  by: human:tswetnam
  at: "2025-12-14T00:00:00Z"
status: draft
---

# Abstract

<a rel="license" href="http://creativecommons.org/licenses/by/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>.

**Spectral Geometry and Metabolic Scaling in Forest Canopy Architecture: A Preregistration Framework**

This chapter presents a formal preregistration framework for investigating whether old-growth forests at "Self-Organized Criticality" (SOC) exhibit distinct geometric and spectral signatures compared to disturbed or managed forests. By treating the Canopy Height Model (CHM) as a stochastic surface, we propose that optimal space packing and metabolic scaling result in specific Fractal Dimensions (\(D\)) and Lacunarity (\(\Lambda\)) curves. Furthermore, we test the hypothesis that the spatial repulsion of dominant tree crowns in steady-state systems follows "Universal Laws of Repulsion" (random matrix statistics) akin to the spacing of Riemann Zeta zeros, rather than simple Poissonian randomness.

We develop five primary testable hypotheses:

1. **The "Optimal Filling" Hypothesis:** Old-growth forests maximize light interception while minimizing self-shading, resulting in fractal dimensions significantly higher than disturbed or managed stands.

2. **The "Scale Invariance" Hypothesis:** Disturbance creates characteristic gap sizes (breaks in scaling), whereas steady-state forests exhibit scale invariance (gaps of all sizes following strict power-law decay).

3. **The "Zeta Distribution" Hypothesis:** The size-frequency distribution of canopy gaps in old-growth forests follows a Power Law distribution where the exponent \(\alpha\) approximates the Zeta function parameter associated with 2D packing (\(\alpha \approx 2.0\), related to \(\zeta(2)\)).

4. **The "Universal Repulsion" Hypothesis:** The Nearest Neighbor Spacing distribution of dominant tree apices in old-growth forests fits the Wigner-Dyson distribution (GUE/GOE statistics), indicating rigid repulsion rather than random Poisson placement.

5. **The "Biotic Decoupling" Hypothesis:** Mature ecosystems buffer environmental constraints through niche construction, resulting in lower correlation between local fractal dimension and topographic variables in old-growth versus young or disturbed stands.

Additionally, we present four spatial distribution hypotheses derived from Fractal String theory: the Fractal String Gap Hypothesis, the Prime Number Repulsion (GUE) Hypothesis, the Complex Dimension Oscillation Hypothesis, and the Riemann Gas Density Hypothesis. These provide testable predictions for how old-growth forests should distribute their largest trees in space.

For each hypothesis, we specify null and alternative formulations amenable to rigorous statistical testing using high-resolution LiDAR point clouds, Canopy Height Models, and Differential Box Counting (DBC) algorithms. The methodology includes ANOVA, regression analysis, Likelihood Ratio Tests, and Chi-Squared goodness-of-fit tests comparing empirical distributions to theoretical predictions.

This framework connects biological systems (forests) at equilibrium to the same mathematical "universality classes" found in Number Theory (Riemann Zeta function) and Quantum Chaos (Random Matrix Theory), establishing a non-destructive, remote-sensing method for identifying forests that have reached "Optimal Packing" (Old Growth) versus those that are biologically immature or degraded.

**Keywords:** spectral geometry, fractal dimension, lacunarity, Riemann Zeta distribution, self-organized criticality, GUE statistics, canopy structure, metabolic scaling, complex dimensions, old-growth forest