  A 2015 conjecture of Codesido-Grassi-Mari\~no in topological string theory
relates the enumerative invariants of toric CY 3-folds to the spectra of
operators attached to their mirror curves. We deduce two consequences of this
conjecture for the integral regulators of $K_2$-classes on these curves, and
then prove both of them; the results thus give evidence for the CGM conjecture.
(While the conjecture and the deduction process both entail forms of local
mirror symmetry, the consequences/theorems do not: they only involve the curves
themselves.) Our first theorem relates zeroes of the higher normal function to
the spectra of the operators for curves of genus one, and suggests a new link
between analysis and arithmetic geometry. The second theorem provides
dilogarithm formulas for limits of regulator periods at the maximal conifold
point in moduli of the curves.
