Core Theorem Paper - Fiber Framework

We study extremal distributions on the Boolean cube {0,1}^m under low-order marginal constraints. A distribution lies in P_t(m) if its Walsh-Fourier coefficients of orders 1 through t all vanish, making it invisible to every observer of depth t. We introduce the SHOM functional (Hidden Higher-Order Mass) as a measure of spectral energy permanently beyond any depth-k observer. The Boolean cube decomposes into the eigenspaces E_0,...,E_m of the Hamming association scheme H(m,2); fixing the first t marginals eliminates E_1,...,E_t, making the extremal problem one of spectral energy optimization over the remaining eigenspaces. Main results: (1) the Observer-Kernel Theorem establishes the exact mathematical boundary between what any depth-t observer can reconstruct and what remains permanently hidden; (2) the Projective Extremizer Theorem proves that max SHOM_2 over P_2(2^k-1) equals 2^{m-k}-1, achieved if and only if the distribution is uniform on a simplex code coset, proved via a frame identity and equidistant code classification; (3) for depth-3 observers, max purity over P_3(2^k-1) = 2^{-(k+1)}, proved via an explicit Delsarte dual certificate with analytically verified Krawtchouk coefficients. The t=4 phase transition is stated as an open problem with proved cases at k=3,4.