Step 1: The task is to find the arthritis that does not stimulate the periosteum to form new bone, a key radiographic discriminator between inflammatory arthritides.
Step 2: Three of the choices are classic producers of periostitis: psoriatic arthritis with its shaggy new bone, reactive arthritis from the same seronegative family, and neuropathic (Charcot) joints that show exuberant bone reaction.
Step 3: By contrast, rheumatoid arthritis erodes rather than builds bone. Its radiology shows periarticular osteoporosis, symmetric marginal erosions and uniform joint space loss, with a conspicuous absence of periosteal new bone.
Step 4: Removing the three periostitis-positive options leaves rheumatoid arthritis as the arthritis with no periosteal reaction.
\[\boxed{\text{Rheumatoid arthritis}}\]