Expected Results¶
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This section outlines expected outcomes under null and alternative hypotheses for each of the five primary hypotheses, including predicted measurement ranges and statistical considerations.
Hypothesis 1: The "Optimal Filling" Hypothesis (Fractal Dimension)¶
Expected Outcomes Under \(H_{10}\) (Null)¶
If the null hypothesis is correct---that there is no difference in fractal dimension across forest types---we predict:
- Fractal Dimension: Invariant across all groups at \(D = 2.3 \pm 0.2\)
- ANOVA: \(F\)-statistic with \(p > 0.05\)
- No systematic trend with forest age or disturbance history
Expected Outcomes Under \(H_{1a}\) (Alternative)¶
If the alternative hypothesis is correct---that old-growth forests exhibit higher fractal dimensions due to optimal space packing---we predict:
Group A (Old Growth):
- \(D_{\text{surf}} = 2.6\)--\(2.8\)
- Multi-layered canopy structure
- Maximum "surface area" for photosynthesis
Group B (Late-Successional with Disturbance):
- \(D_{\text{surf}} = 2.3\)--\(2.5\)
- Developing heterogeneity
- Recovering toward optimal packing
Group C (Monoculture Plantation):
- \(D_{\text{surf}} = 2.1\)--\(2.3\)
- Near-uniform canopy ("flat sheet")
- Poor space utilization
Predicted Measurement Ranges¶
| Group | Forest Type | \(D_{\text{surf}}\) (\(H_0\)) | \(D_{\text{surf}}\) (\(H_A\)) |
|---|---|---|---|
| A | Old Growth (>200 yr) | \(2.3 \pm 0.2\) | \(2.70 \pm 0.10\) |
| B | Late-Successional | \(2.3 \pm 0.2\) | \(2.40 \pm 0.12\) |
| C | Plantation (20-50 yr) | \(2.3 \pm 0.2\) | \(2.15 \pm 0.10\) |
Hypothesis 2: The "Scale Invariance" Hypothesis (Lacunarity)¶
Expected Outcomes Under \(H_{20}\) (Null)¶
If the null hypothesis is correct---that Lacunarity curves do not differentiate forest types---we predict:
- No significant difference in \(R^2\) of linear fit on log-log axes
- Similar "kink" patterns across all forest types
- Deviations from linearity random rather than systematic
Expected Outcomes Under \(H_{2a}\) (Alternative)¶
If the alternative hypothesis is correct---that old-growth exhibits scale invariance while disturbed forests show spectral kinks---we predict:
Group A (Old Growth):
- \(\Lambda(r)\) follows strict power-law decay
- \(R^2 > 0.95\) for linear regression on log-log plot
- No characteristic gap scale dominates
Group B (Late-Successional with Disturbance):
- \(\Lambda(r)\) exhibits "spectral kinks" at specific scales
- \(R^2 = 0.7\)--\(0.9\) due to deviations
- Kink position corresponds to disturbance scale (e.g., logging road width, windthrow radius)
Group C (Monoculture Plantation):
- \(\Lambda(r)\) shows strong characteristic scale
- \(R^2 < 0.8\) with systematic deviation
- Kink at tree spacing interval
Disturbance Signatures¶
| Disturbance Type | Expected Kink Scale | Interpretation |
|---|---|---|
| Recent Tree Fall | 15--25 m | Single large gap |
| Windthrow | 30--50 m | Cluster of gaps |
| Logging Road | 10--15 m | Linear feature |
| Fire | Variable | Patch-dependent |
Hypothesis 3: The "Zeta" Distribution of Gaps (Metabolic Scaling)¶
Expected Outcomes Under \(H_{30}\) (Null)¶
If the null hypothesis is correct---that gap sizes follow exponential rather than power-law distributions---we predict:
Gap Size Distribution:
- Characteristic mean gap size \(\bar{S}\)
- Exponential tail (rapid decline)
- Likelihood Ratio Test favors exponential model
Expected Outcomes Under \(H_{3a}\) (Alternative)¶
If the alternative hypothesis is correct---that gap sizes follow a Zeta (power-law) distribution in old-growth---we predict:
Old Growth (Group A):
- Power-law distribution: \(P(A) \sim A^{-\alpha}\)
- Exponent: \(\alpha = 1.8\)--\(2.2\) (related to \(\zeta(2) = \pi^2/6\))
- Extended scaling regime: 2+ orders of magnitude
- Self-Organized Criticality confirmed
Late-Successional (Group B):
- Truncated power-law with exponential cutoff
- \(\alpha = 2.3\)--\(2.8\)
- Limited scaling range
Plantation (Group C):
- Exponential or lognormal distribution
- No power-law signature
- Characteristic gap size dominates
Gap Distribution Parameters¶
| Group | Distribution Type | Exponent \(\alpha\) | Scaling Range |
|---|---|---|---|
| A (Old Growth) | Power Law | \(2.0 \pm 0.2\) | \(10^1\)--\(10^4\) m\(^2\) |
| B (Late-Successional) | Truncated Power Law | \(2.5 \pm 0.3\) | \(10^1\)--\(10^3\) m\(^2\) |
| C (Plantation) | Exponential | N/A | N/A |
Hypothesis 4: Universal Repulsion (The "Spectral DNA")¶
Expected Outcomes Under \(H_{40}\) (Null)¶
If the null hypothesis is correct---that tree locations are random---we predict:
Nearest Neighbor Spacing (NNS):
- Poisson distribution: \(P(s) = e^{-s}\)
- No repulsion between dominant trees
- \(\chi^2\) test favors Poisson model
Expected Outcomes Under \(H_{4a}\) (Alternative)¶
If the alternative hypothesis is correct---that old-growth trees exhibit GUE-like repulsion---we predict:
Old Growth (Group A):
- Wigner-Dyson distribution (GOE/GUE statistics)
- Linear repulsion at small distances: \(P(s) \propto s\) as \(s \to 0\)
- Characteristic "level repulsion" signature
- \(\chi^2\) test strongly favors GUE over Poisson
Physical Interpretation:
- Small \(s\): Rare to find two dominant trees very close (competition)
- Medium \(s\): Most likely spacing (optimal resource sharing)
- Large \(s\): Exponential tail (random long-range)
The GUE Pair Correlation Function¶
In old-growth forests, the probability of finding two giant trees at distance \(r\) should follow:
| Model | Pair Correlation at \(r=0\) | Physical Meaning |
|---|---|---|
| Poisson | \(g(0) = 1\) | No repulsion; random |
| GUE | \(g(0) = 0\) | Strong repulsion; competition |
| Hexagonal Lattice | \(g(0) = 0\), periodic peaks | Perfect order |
Prediction: Old-growth forests should fall between GUE (soft repulsion) and lattice (hard repulsion), indicating evolutionary optimization of spacing.
Hypothesis 5: Biotic Decoupling (The Topographic Test)¶
Expected Outcomes Under \(H_{50}\) (Null)¶
If the null hypothesis is correct---that fractal dimension correlates equally with topography across all forest types---we predict:
- Similar correlation coefficients between \(D_{\text{local}}\) and TWI/Slope across Groups A, B, C
- No significant difference by Fisher's z-test
Expected Outcomes Under \(H_{5a}\) (Alternative)¶
If the alternative hypothesis is correct---that old-growth forests "buffer" topographic constraints---we predict:
Group A (Old Growth):
- Weak correlation: \(|r| < 0.3\) between \(D_{\text{local}}\) and topography
- Forest structure independent of underlying terrain
- "Biotic Decoupling" achieved
Group B (Late-Successional):
- Moderate correlation: \(|r| = 0.4\)--\(0.6\)
- Partial decoupling; some terrain signature remains
Group C (Plantation):
- Strong correlation: \(|r| > 0.6\)
- Forest structure follows terrain (valleys dense, ridges sparse)
- "Environmentally Driven" system
Predicted Correlations with Topography¶
| Group | \(r\) (D vs. TWI) | \(r\) (D vs. Slope) | Interpretation |
|---|---|---|---|
| A (Old Growth) | \(0.15 \pm 0.10\) | \(-0.20 \pm 0.10\) | Biotically decoupled |
| B (Late-Successional) | \(0.45 \pm 0.15\) | \(-0.40 \pm 0.15\) | Transitioning |
| C (Plantation) | \(0.70 \pm 0.10\) | \(-0.65 \pm 0.12\) | Environmentally driven |
Four Spatial Distribution Hypotheses: Expected Signatures¶
Fractal String Gap Hypothesis¶
Old Growth Prediction:
- Gap lengths along transects: \(N(L) \sim L^{-D}\) with \(D \approx 1.3\)
- Scale invariance: no characteristic gap size
- Log-log plot linear over 1.5+ decades
Prime Number Repulsion (GUE) Hypothesis¶
Old Growth Prediction:
- Pair correlation function matches GUE: \(g(r) = 1 - \left(\frac{\sin(\pi r)}{\pi r}\right)^2\)
- Strong rejection of Poisson null
- Spacing statistics match quantum chaotic systems
Complex Dimension (Oscillation) Hypothesis¶
Old Growth Prediction:
- Lacunarity vs. \(\log(r)\) shows periodic oscillation
- Period: \(p = 2\pi/\omega\) where \(\omega\) is imaginary part of complex dimension
- Amplitude: Low, constant (steady state)
- Interpretation: Forest constructed via recursive, self-similar algorithm
Riemann Gas Density Hypothesis¶
Old Growth Prediction:
- Number density: \(N(>m) \sim m^{-2}\)
- Packing density: \(\approx 1.645\) (related to \(\zeta(2) = \pi^2/6\))
- System at "Edge of Chaos"---critical density threshold
Summary of Testable Predictions¶
| Hypothesis | Metric | \(H_0\) Prediction | \(H_A\) Prediction | Discriminating Power |
|---|---|---|---|---|
| H1 (Optimal Filling) | \(D_{\text{surf}}\) across groups | No difference | Old Growth > Plantation | High |
| H2 (Scale Invariance) | Lacunarity \(R^2\) | Similar across groups | Old Growth > Disturbed | High |
| H3 (Zeta Distribution) | Gap size distribution | Exponential | Power Law (\(\alpha \approx 2\)) | High |
| H4 (Universal Repulsion) | NNS distribution | Poisson | GUE/Wigner-Dyson | High |
| H5 (Biotic Decoupling) | \(r\)(D vs. Topography) | Similar across groups | Old Growth < Plantation | Medium |
Significance / Expected Outcomes¶
If Hypotheses 3 and 4 are confirmed, this provides evidence that biological systems (forests) at equilibrium converge upon the same mathematical "universality classes" found in:
- Number Theory: Riemann Zeta function
- Quantum Chaos: Random Matrix Theory (GUE)
This would establish a non-destructive, remote-sensing method for identifying forests that have reached:
- "Optimal Packing" (Old Growth): Maximum metabolic efficiency, self-organized criticality
- "Sub-optimal" (Immature/Degraded): Transitioning toward or recovering from critical state