Breaking peak-locking with square parameterization

The 9-way tie at C=1.50286286 was caused by everyone downloading the same solution. Standard log-space optimization (f=exp(v)) hits a wall at ~5e-8 improvement regardless of resolution (30k-300k) due to peak-locking (Jaech & Joseph, arXiv:2508.02803): gradient descent on the smoothed max reinforces the current argmax rather than globally flattening the autoconvolution.

The breakthrough was switching to f=v^2 parameterization. This changes the gradient geometry near zero — vanishing gradient encourages sparsity and allows exploring different support structures that exp(v) cannot reach. At n=90000 (3x block-repeat of SOTA), ultra-aggressive L-BFGS with extended low-beta exploration (beta=1e6, history=300, 3000 iterations, ~6 min) finds a basin 12x deeper than the 1e-7 gate.

Key insight: the long exploration at LOW beta (loose smooth approximation) is critical. Most approaches tighten beta quickly. Spending minutes at beta=1e6 lets L-BFGS traverse a vast search region before committing to a basin. n=90000 is a sweet spot — larger n makes the landscape smoother, causing faster convergence to shallower minima.

Full writeup with what-worked/what-didn't analysis: [link removed]