Let σ and ω be locally finite positive Borel measures on ℝn. We assume that at least one of the two measures σ and ω is supported on a regular C1,δ curve in ℝn. Let Rα,n be the α-fractional Riesz transform vector on ℝn. We prove the T1 theorem for Rα,n: namely that Rα,n is bounded from L2(σ) to L2(σ) if and only if the A2α$${\cal A}_2^\alpha $$ conditions with holes hold, the punctured A2α$$A_2^\alpha $$ conditions hold, and the cube testing condition for Rα,n and its dual both hold. The special case of the Cauchy transform, n = 2 and α = 1, when the curve is a line or circle, was established by Lacey, Sawyer, Shen, Uriarte-Tuero and Wick in [LaSaShUrWi]. This T1 theorem represents essentially the most general T1 theorem obtainable by methods of energy reversal. More precisely, for the pushforwards of the measures σ and ω, under a change of variable to straighten out the curve to a line, we use reversal of energy to prove that the quasienergy conditions in [SaShUr7] are implied by the A2α$${\cal A}_2^\alpha $$ with holes, punctured A2α$$A_2^\alpha $$, and quasicube testing conditions for Rα,n. Then we apply the main theorem in [SaShUr7] to deduce the T1 theorem above.