Experiments › Core Theorem

I-Ratio Proof

Source: exp_iratio_proof.sx · Validates: Theorem 13

Hypothesis

The interaction ratio \( I(\theta) \) equals exactly \( -\frac{1}{2} \) if and only if the system is at equilibrium. For \( K \) competing objectives with gradients \( g_1, \ldots, g_K \):

\[ I(\theta) = \frac{\displaystyle\sum_{i < j} g_i \cdot g_j}{\displaystyle\sum_i \|g_i\|^2} = -\frac{1}{2} \quad \iff \quad \sum_i g_i = 0 \]

This holds for all \( K \ge 2 \) and is independent of the dimension of \( \theta \).

Method

Setup: Three test suites: systematic K sweep, off-equilibrium control, and random problem instances.

Parameters:

  • K sweep: \( K = 2, 3, \ldots, 20 \), with 3 equilibrium configurations per K
  • Off-equilibrium: 10 cases where \( \sum g_i \neq 0 \)
  • Random problems: 70 randomly generated multi-objective instances
  • Tolerance: \( |I + 0.5| < 10^{-10} \)

Procedure: For each test, construct gradient vectors satisfying (or not satisfying) the equilibrium condition, compute \( I(\theta) \), and check whether it equals \( -0.5 \) within tolerance.

Results

Test SuitePassTotalRateStatus
K=2..20 equilibrium sweep5757100%Pass
Off-equilibrium (negative control)1010100% correctly \(\neq -0.5\)Pass
Random problems7070100%Pass

Precision

MetricValue
Maximum absolute error \(|I + 0.5|\)\(2.22 \times 10^{-16}\)
Mean absolute error\(< 10^{-16}\)

The maximum error of \( 2.22 \times 10^{-16} \) is exactly one unit of least precision (ULP) for IEEE 754 double-precision arithmetic. The result is exact to machine precision.

Analysis

The result \( I = -\frac{1}{2} \) follows from the algebraic identity:

\[ \left\|\sum_i g_i\right\|^2 = \sum_i \|g_i\|^2 + 2\sum_{iAt equilibrium, \( \sum_i g_i = 0 \), so the left side vanishes, giving \( \sum_{i

Key observations:

  • The result is dimension-free: it holds regardless of parameter space dimension.
  • The result is K-universal: it holds for any number of objectives \( K \ge 2 \).
  • The off-equilibrium tests confirm the biconditional: non-equilibrium states produce \( I \neq -0.5 \), establishing that \( I = -0.5 \) is both necessary and sufficient.
  • Machine-precision accuracy (\( 2.22 \times 10^{-16} \)) confirms the identity is algebraic, not approximate.

Conclusion

Pass — All 137 tests pass (57 equilibrium + 10 negative control + 70 random). Maximum error is 1 ULP. Theorem 13 is validated to machine precision.

Reproducibility

../simplex/build/sxc exp_iratio_proof.sx -o build/exp_iratio_proof.ll
clang -O2 build/exp_iratio_proof.ll ../simplex/runtime/standalone_runtime.c \
  -o build/exp_iratio_proof -lm -lssl -lcrypto -L$(brew --prefix openssl)/lib
./build/exp_iratio_proof

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