Equilibrium Mapping from Gradient Topology

Hypothesis

Gradient topology information can be used to locate multi-objective equilibria more precisely than loss minimisation alone. Two candidate approaches are tested:

1. Cosine-based \(d\mathcal{M}/d\theta\): Descend on the gradient of the interaction matrix magnitude to push toward equilibrium.
2. Balance residual (B-flow): Descend on the balance residual \( B(\theta) = \frac{\|\sum_i g_i\|^2}{\sum_i \|g_i\|^2} \), which is exactly zero when gradients cancel.

At equilibrium the I-ratio should equal \(-0.5\) (theoretical fixed point for balanced 2-objective systems).

Method

Attempt 1 (cosine-based): Compute pairwise cosine matrix \(\alpha_{ij}\), form scalar summary \(\mathcal{M} = \sum_{i < j} |\alpha_{ij}|\), and descend on \(\nabla_\theta \mathcal{M}\). Hypothesis: pushing toward orthogonality approaches equilibrium.

Attempt 2 (B-flow): Directly descend on balance residual \( B(\theta) \). Test on three problems: 1D convex (2 objectives), 2D convex (2 objectives), non-convex (multiple local equilibria). 1000 steps per method. Compare with standard loss-flow \(\nabla \sum_i L_i\).

I-ratio verification: At converged equilibrium, compute the I-ratio and compare against the theoretical prediction of \(-0.5\).

Results

Attempt 1: Cosine-Based \(d\mathcal{M}/d\theta\)

The cosine matrix \(\alpha_{ij}\) is a normalised ratio that discards gradient magnitude. Descending on \(\nabla_\theta \mathcal{M}\) pushes gradients toward orthogonality but does not drive them toward cancellation. Orthogonality (\(\cos = 0\)) is necessary but not sufficient for equilibrium (\(\sum g_i = 0\)).

Attempt 2: Balance Residual (B-Flow)

B-flow achieves exact or near-exact equilibrium in convex settings. In the 2D case, \(B = 4.7 \times 10^{-34}\) vs loss-flow's \(2.3 \times 10^{-4}\) — approximately \(4.9 \times 10^{29}\)× more precise. In the non-convex case, B-flow converges to a saddle point where \(B = 0\) but the solution is suboptimal.

Two-Phase Strategy

The two-phase strategy combines loss-flow's ability to navigate the global landscape with B-flow's surgical precision at equilibrium refinement.

I-Ratio at Equilibrium

The I-ratio converges to exactly \(-0.5\) at the B-flow equilibrium, confirming the theoretical prediction from Theorem 13.

Analysis

• Cosines are insufficient: \(\alpha_{ij}\) strips magnitude information. Equilibrium requires \(\sum g_i = 0\) (vector cancellation), not \(\cos(g_i, g_j) = 0\) (orthogonality). The cosine-based approach cannot distinguish "small opposing gradients" from "large orthogonal gradients."
• B-flow targets the right objective: \(B\) is exactly zero iff gradients sum to zero. Descending on \(B\) directly optimises for the equilibrium condition, yielding 1D exact solutions and 2D precision of \(4.7 \times 10^{-34}\).
• Non-convex limitation: \(B = 0\) can occur at saddle points, not just true equilibria. The two-phase strategy resolves this by using loss-flow to find the correct basin first.
• I-ratio confirmation: \(I = -0.5\) at equilibrium provides independent validation — the equilibrium found by B-flow satisfies the Theorem 13 fixed-point condition exactly.

Conclusion

Pass — Theorem 14 validated. B-flow achieves \(B = 0\) (1D exact) and \(B = 4.7 \times 10^{-34}\) (2D), vastly outperforming loss-flow. Cosine-based \(d\mathcal{M}/d\theta\) fails — too coarse for equilibrium location. The recommended approach is a two-phase strategy: loss-flow to explore and find the correct basin, then B-flow to refine to high-precision equilibrium. I-ratio = \(-0.5\) confirmed at equilibrium.

Reproducibility

# Equilibrium mapping experiment
../simplex/build/sxc exp_equilibrium_mapping.sx -o build/exp_equilibrium_mapping.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_equilibrium_mapping.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_equilibrium_mapping \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_equilibrium_mapping

# Balance residual experiment (B-flow data)
../simplex/build/sxc exp_balance_residual.sx -o build/exp_balance_residual.ll

clang -O2 build/exp_balance_residual.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_balance_residual \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_balance_residual

Related Theorems

• Theorem 14 — B-Flow Convergence
• Theorem 13 — I-Ratio Fixed Point
• exp-balance-residual — B-Flow Convergence (standalone)
• exp-iratio-proof — I-Ratio Theorem Validation
• exp-structure-discovery — Gradient Topology as Structural Probe