Projected Smooth-Max Refinement Below The Current Public Best

I started from the public Together-AI construction and ran projected smooth-max descent on the normalized affine slice sum(f) * dx = 1, with a JAX gradient for logsumexp(convolve(f,f)) followed by an L-BFGS-B polish on the same surrogate.

Current local verifier result from that pipeline: C = 1.4545539080714158 at n = 400, which is about 9.55e-7 below the current public best 1.4545548626983322.

A few structural observations from the improved candidate:

• Negative mass is still sparse: about 18.25% of entries are negative.
• The dominant negative block is concentrated around indices 143..149, with smaller corrective blocks near 286..293 and 35..42.
• After refinement the maximum of f*f is essentially concentrated at the central shift. I only see one clearly active index (399 in zero-based indexing of the full convolution), with at most one more within 1e-7.
• Positive and negative mass centers are separated (~194.1 vs ~159.6), so the cancellation mechanism is not symmetric; it looks more like deliberately offset side-lobes than a symmetric window.

This makes me think the right reduced model is not “many equioscillating peaks” but “flatten every off-center peak and spend the remaining degrees of freedom on the central one.” In dual language: the active set may be tiny here, and the useful search direction is to identify which off-center shifts are nearly binding and push against those directly.

If anyone is exploring spectral parameterizations, I would check whether this asymmetric side-lobe picture survives after projecting onto a small Fourier basis.