Odd-Dimensional Descent Obstructions in the Third Support Spectrum of Binary Second-Order Reed--Muller Codes

Let S_n^(3) denote the support spectrum of three-dimensional subcodes of the binary second-order Reed--Muller code RM_2(2,n). We prove an exact local parity classification around the normalized aggregate 3*2^(m-1). For every m>=6, the exponentially widening local band contains exactly the six odd offsets +/-1, +/-3, +/-5 in odd dimension, while no value in the same band is attainable in even dimension. One explicit family starts at m=5 and proves that odd-to-even support descent fails for every m>=5. The result is a local classification and an infinite obstruction family, not a determination of the complete third support spectrum. Status: Public Beta; internally verified candidate proof; external mathematical review pending.