1. **State the problem:** Find the product of $\left(m-\frac{1}{2}\right)\left(m+\frac{1}{2}\right)\left(m^2+\frac{1}{4}\right)$. 2. **Recall the formula:** The product of two binomials of the form $(a-b)(a+b)$ is $a^2 - b^2$. 3. **Apply the formula to the first two factors:** $$\left(m-\frac{1}{2}\right)\left(m+\frac{1}{2}\right) = m^2 - \left(\frac{1}{2}\right)^2 = m^2 - \frac{1}{4}.$$ 4. **Rewrite the product:** Now the expression becomes $$\left(m^2 - \frac{1}{4}\right)\left(m^2 + \frac{1}{4}\right).$$ 5. **Use the difference of squares formula again:** $$(a-b)(a+b) = a^2 - b^2,$$ where $a = m^2$ and $b = \frac{1}{4}$. 6. **Calculate:** $$\left(m^2 - \frac{1}{4}\right)\left(m^2 + \frac{1}{4}\right) = (m^2)^2 - \left(\frac{1}{4}\right)^2 = m^4 - \frac{1}{16}.$$ 7. **Final answer:** $$\boxed{m^4 - \frac{1}{16}}.$$