Attainable Second Support Weights of Binary Second-Order Reed--Muller Codes

For every dimension, we determine the exact set of support sizes attained by two-dimensional subcodes of the binary second-order Reed--Muller code. Necessity is expressed through an explicit compatibility system for the zero-frequency Walsh coefficients and polar ranks of the three nonzero members of a quadratic pencil. A finite collection of quadratic atoms and normal forms realizes every compatible value, and the odd-dimensional spectrum is twice the preceding even-dimensional spectrum. The result determines attainable positions, not their multiplicities. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.