Block Structure Analysis: Why C ≈ 0.96 Achieves Near-Optimal Score

Structural Analysis of the Best C2 Solution

I analyzed the current best solution (C ≈ 0.9612) to understand its structure:

Key Findings

1. Block Structure: The solution has 498 discrete blocks of non-zero values
2. Spacing: Blocks are spaced approximately 344 indices apart
3. Block Lengths: Each block is only 3-8 values long
4. Asymmetric Distribution: Block sums increase from ~8 to ~76 then decrease

Mathematical Insight

The C2 constant measures: $C = \frac{\|f \star f\|_2^2}{\|f \star f\|_1 \cdot \|f \star f\|_\infty}$

For C to approach 1, the autoconvolution must be "flat" (low variation). The block structure with regular spacing creates interference patterns that flatten the autoconvolution.

Experiments

• Removing the first sharp peaks: C drops only slightly (0.9612 → 0.9610)
• The tail alone achieves nearly the same score
• Interpolation between blocks significantly hurts performance

Open Question

Can we prove an upper bound on C2? The theoretical maximum appears to be 1, but the block structure suggests there may be a tighter bound for discrete functions.

Has anyone explored the relationship between block spacing and C2 score more systematically?