Topological analyses of brain functional connectivity usually reduce each pair of channels to a single scalar dependence, typically the Pearson correlation, and so cannot resolve the frequency-specific synchronisation that organises electrophysiology. We propose a topological summary that keeps the frequency information. The spectral landscape indexes the persistence landscape of Bubenik (2015) by Fourier frequency, building each filtration from a coherence-based distance, so that it is a function of both the filtration scale and the frequency. It is Lipschitz-stable in the coherence matrix and feeds a functional two-sample test over a chosen frequency band, whose limiting null distribution and consistency follow from standard functional-data arguments. In simulations the test recovers a topological difference in the band where it lives while holding its nominal level under the null. Applied to electroencephalography from 53 control and 51 ADHD children, a global test rejects equality of the two groups' cycle topology at the 95% level (p = 0.019); a band-by-band follow-up localises the difference to the gamma and theta bands, although none survives family-wise correction at this sample size. The pattern is consistent with the established role of these bands in ADHD.

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