Pseudogout (Calcium Pyrophosphate Deposition Disease, CPPD): The presence of joint pain and swelling, especially with the significant improvement not seen with Lyme disease treatment, along with the aspiration results showing a high number of red blood cells (RBC) and polymorphonuclear white blood cells (PMN WBC), suggests a crystalline arthropathy. The absence of rheumatoid factor and antinuclear antibody (ANA) further supports this diagnosis, as these are more commonly associated with autoimmune conditions.

Gout: Although the aspiration results do not specifically mention urate crystals, gout remains a possibility, especially given the joint pain and swelling. The lack of response to Lyme disease treatment and the presence of a significant inflammatory response in the joint aspirate could also be consistent with gout.
Septic Arthritis: The high number of PMN WBCs in the joint aspirate raises the concern for a septic joint, especially in an elderly patient. The fact that the patient has been treated for Lyme disease without significant improvement does not rule out a concurrent infection.
Hemarthrosis: The presence of +4 RBC in the joint aspirate could suggest hemarthrosis, possibly due to trauma or anticoagulation therapy, which is common in elderly patients.

Infectious Arthritis (including Lyme Arthritis): Despite the patient being treated for Lyme disease, it's crucial to ensure that the treatment is adequate and that the diagnosis is correct. Other infectious causes, such as bacterial or fungal infections, could present similarly and have severe consequences if not promptly treated.
Malignancy-related Arthropathy: Although less common, certain malignancies can cause joint symptoms. Given the patient's age, it's essential to consider this possibility, especially if there's no clear explanation for the symptoms after thorough investigation.

Amyloid Arthropathy: This condition can cause joint pain and swelling, especially in older adults, but it is relatively rare. The diagnosis would typically require biopsy and specific staining for amyloid.
Sarcoid Arthropathy: Sarcoidosis can cause joint symptoms, but it is less common and would typically be associated with other systemic symptoms or findings suggestive of sarcoidosis.
Charcot Arthropathy (Neuropathic Arthropathy): This condition is associated with sensory loss and can lead to severe joint destruction. It's more commonly seen in patients with diabetes or other conditions causing neuropathy.