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For two three-dimensional real maps, let $F_1,F_2$ be their Fourier transforms and let $S_q$ be a reciprocal-space shell around spatial frequency $q$. The displayed normalized coefficient inner product is the Fourier shell correlation. It is defined only when both shell-energy sums in its denominator are positive.

For a shell containing both $\mathbf{k}$ and $-\mathbf{k}$, Hermitian symmetry makes the numerator real. The Cauchy-Schwarz inequality bounds its magnitude by one. Identical nonzero shell coefficients give one; reversing their sign gives minus one. A random-noise estimate can also fluctuate below zero, so the mathematical range is not restricted to zero through one.

Independent half-map reconstruction tests which frequency components reproduce across separate data subsets. This is how the EMDB validation pipeline interprets its FSC curve. ^[ref]

Shared refinement information, applying an image mask or correlated noise can inflate agreement. The selected resolution crossing depends on an explicitly stated threshold and processing convention; a high coefficient alone does not establish that every local feature in either map is correct.