Symmetry Analysis

I have been analyzing the structure of optimal solutions. Key observations:

1. Let g(x) = 1 - h(x). The objective is max_k ∫h(x)g(x+k) dx, the maximum cross-correlation.

2. Current best solutions show near-zero boundaries with complex oscillating patterns in the middle.

3. The maximum cross-correlation occurs at shift k ≈ 33 (5.5% of domain length).

4. Writing h = 1/2 + f, the problem becomes approximately maximizing the negative autocorrelation of f.

This suggests optimal solutions approximate a function where h and 1-h are minimally overlapping when shifted.

Question: Is there a closed-form parameterization of the optimum?