Musculoskeletal Back Pain: Given the history of nerve characterization to the lumbar spine and the centralized back pain radiating to the lower abdominal area, musculoskeletal back pain is a plausible diagnosis. The absence of dysuria or hematuria and the presence of nausea and vomiting could be associated with the pain itself or other non-renal causes.

Kidney Stones: Although the patient denies dysuria or hematuria, kidney stones (nephrolithiasis) could still be a cause, especially with a history of kidney stones. The pain from kidney stones can radiate to the lower abdomen and be associated with nausea and vomiting.
Lumbar Disc Herniation: This condition could cause centralized back pain radiating to the lower abdominal area due to nerve root compression. The history of nerve characterization to the lumbar spine supports this possibility.
Degenerative Disc Disease: Similar to lumbar disc herniation, degenerative disc disease could cause back pain due to the degeneration of the discs in the lumbar spine, potentially radiating pain to the lower abdominal area.

Abdominal Aortic Aneurysm (AAA): Although less likely, AAA is a critical diagnosis that must not be missed due to its high mortality rate if ruptured. Back pain and abdominal pain can be symptoms, especially if the aneurysm is expanding or rupturing.
Pyelonephritis or Kidney Infection: Despite the denial of dysuria or hematuria, an upper urinary tract infection could present with back pain and systemic symptoms like nausea and vomiting. It's crucial to consider, especially if the patient has a history of urinary tract infections or kidney stones.
Ovarian Torsion: This is a gynecological emergency that could present with acute lower abdominal pain, which might radiate to the back. Nausea and vomiting are common associated symptoms.

Spinal Tumors: Tumors in the spine, whether primary or metastatic, could cause back pain radiating to the abdominal area. This would be less likely without other systemic symptoms or a known history of cancer.
Inflammatory Spondyloarthropathies: Conditions like ankylosing spondylitis could cause chronic back pain and stiffness, potentially radiating to the abdominal area, though this would typically be part of a longer-standing symptom complex.
Pheochromocytoma: A rare tumor of the adrenal gland that could cause episodic back pain, abdominal pain, nausea, vomiting, and other systemic symptoms due to catecholamine release.