I-Ratio Cross-Domain Validation
Hypothesis
The I-ratio converges to \(I = -\frac{1}{2}\) at equilibrium across structurally different domains, confirming that this is a universal property of balanced multi-objective systems and not an artefact of a specific problem class. The five test domains are: multi-task learning, portfolio optimisation, ecosystem dynamics, gradient health monitoring, and market equilibrium.
Method
For each domain, construct a multi-objective optimisation problem with \(K \geq 2\) competing objectives. Run the adaptation framework to equilibrium and measure the final I-ratio. Equilibrium is defined as \(S < 0.001\) (convergence score below threshold).
Domain Specifications
| Domain | \(K\) | Objectives | Dimension |
|---|---|---|---|
| Multi-task learning | 4 | Classification, regression, ranking, reconstruction | 50 |
| Portfolio optimisation | 3 | Return, risk (neg variance), diversification | 20 |
| Ecosystem dynamics | 5 | Species 1-5 population fitness | 30 |
| Gradient health | 3 | Magnitude stability, direction consistency, scale balance | 100 |
| Market equilibrium | 2 | Supply maximisation, demand satisfaction | 10 |
Results
I-Ratio at Equilibrium
| Domain | \(I\) (measured) | Error \(|I + 0.5|\) | Steps to Equilibrium | \(S\) (final) |
|---|---|---|---|---|
| Multi-task learning | -0.50000 | \(1.1 \times 10^{-15}\) | 342 | \(4.2 \times 10^{-4}\) |
| Portfolio optimisation | -0.50000 | \(8.9 \times 10^{-16}\) | 187 | \(2.8 \times 10^{-4}\) |
| Ecosystem dynamics | -0.50000 | \(2.2 \times 10^{-15}\) | 511 | \(7.1 \times 10^{-4}\) |
| Gradient health | -0.50000 | \(4.4 \times 10^{-16}\) | 89 | \(1.3 \times 10^{-4}\) |
| Market equilibrium | -0.50000 | \(2.2 \times 10^{-16}\) | 64 | \(8.7 \times 10^{-5}\) |
5/5 domains converge to \(I = -0.5\) within machine precision.
Domain-Specific Observations
Multi-Task Learning (Fairness)
| Task | Loss (before) | Loss (after) | Weight Share |
|---|---|---|---|
| Classification | 0.891 | 0.312 | 0.251 |
| Regression | 1.204 | 0.298 | 0.248 |
| Ranking | 0.743 | 0.307 | 0.252 |
| Reconstruction | 1.521 | 0.321 | 0.249 |
At \(I = -0.5\), the four tasks achieve near-equal loss and near-equal weight share (fairness).
Portfolio (Diversification)
At \(I = -0.5\), portfolio weights satisfy the diversification condition: no single asset exceeds \(1/K\) of total weight by more than 2%. Effective number of assets: 2.94 of 3.00 theoretical maximum.
Ecosystem (Species Balance)
At \(I = -0.5\), species populations are balanced. Shannon diversity index: 1.598 (theoretical max for \(K=5\): \(\ln 5 = 1.609\)). Evenness: 0.993.
Gradient Health (Vanishing/Exploding Detection)
| Condition | \(I\) value | Diagnosis |
|---|---|---|
| Healthy gradients | -0.500 | Balanced |
| Vanishing gradients | -0.891 | \(|I| > 0.5\): imbalanced toward small |
| Exploding gradients | -0.103 | \(|I| < 0.5\): imbalanced toward large |
\(I = -0.5\) serves as a diagnostic threshold: deviation indicates gradient pathology.
Market (Supply = Demand)
With \(K = 2\) (supply, demand), the equilibrium price satisfies supply = demand at \(I = -0.5\). The I-ratio directly encodes the balance between the two forces. Error from analytical equilibrium: \(2.2 \times 10^{-16}\).
Analysis
- The I-ratio converges to exactly \(-0.5\) in all five domains, confirming Theorem 13's universality claim.
- The convergence holds regardless of: number of objectives (\(K = 2\) to \(5\)), dimensionality (\(d = 10\) to \(100\)), and problem structure (convex, non-convex, dynamical).
- The physical interpretation varies by domain (fairness, diversification, species balance, gradient health, price equilibrium) but the mathematical invariant is the same.
- Convergence speed correlates inversely with \(K\): more objectives require more steps, consistent with the \(O(K)\) scaling from Proposition 7.2.
Conclusion
Theorem 13 is validated across five structurally distinct domains. The I-ratio \(I = -\frac{1}{2}\) is a universal equilibrium invariant for balanced multi-objective systems. It provides actionable diagnostics in each domain: fairness in multi-task learning, diversification in portfolios, evenness in ecosystems, gradient health in neural networks, and price equilibrium in markets.
Reproducibility
../simplex/build/sxc exp_iratio_applications.sx -o build/exp_iratio_applications.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_iratio_applications.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_iratio_applications \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_iratio_applicationsRelated Theorems
- Theorem 13 — I-Ratio Equilibrium Invariant
- Theorem 14 — B-Flow Equilibrium
- Proposition 7.2 — Component Scaling