Modular Walsh rigidity for Boolean functions of degree at most four

The Walsh coefficients of a bent Boolean function have constant absolute value. We give a sufficient condition based on their residues modulo an odd integer. Let m ≥ 4 and M = 2^m. A Boolean function on 4m variables of algebraic degree at most four is bent if every Walsh coefficient lies in one of the two residue classes ±M² modulo p, where p > 7M/12 is odd. The modulus may be composite. The proof uses the binary digits of the normalized Walsh spectrum, low-weight Reed–Muller codewords, and Fourier inversion on affine subspaces. As an application, we give a sufficient condition for periodic restrictions of rotation-symmetric bent functions of degree at most four to remain bent. Version 2 places the result in the modular Hadamard literature, clarifies the distinction for moduli with distinct prime factors, and proves obstructions to small direct-sum counterexamples modulo 5, 7, and 9. The main bound remains 7M/12.