No binary [[14,3,d≥5]] quantum stabilizer code exists

We prove that no binary qubit stabilizer code with parameters [[14,3,d>=5]] exists, settling the remaining distance gap for stabilizer codes encoding three qubits into fourteen. The proof first applies a signed affine-shadow inequality and an integrality congruence to restrict low-weight stabilizer words. Exact split-shadow infeasibility certificates then exclude weight-two words and all configurations with three weight-four checks, leaving a unique weight-four check. That check yields a ten-coordinate binary-additive code of size 1,024, minimum distance at least five, and symplectic hull dimension four. We reconstruct all 37 relevant equivalence classes by exhaustive lengthening: 25 have the wrong hull dimension, and the other twelve violate a necessary 64-coset capacity bound. Together with the known [[14,3,4]] construction, this proves d_max(14,3)=4 for binary stabilizer codes.