Exact Regions and Low-Polar-Rank Recursion in the Third Support Spectrum of Binary Second-Order Reed--Muller Codes

This manuscript studies support sizes of three-dimensional subcodes of binary second-order Reed--Muller codes. It proves two sharp all-dimensional spectrum gaps, determines the exact spectra through nine variables, classifies the complete near-full interval, gives a finite sound-and-complete constructive recursion in the nonzero-Walsh polar-rank-two-or-four layer, and proves an infinite forbidden support ray. Status: Public Beta; internally verified candidate proof; external mathematical review pending.