Wiener–Khinchin theorem

The Wiener–Khinchin theorem describes a quiet inevitability: when a system repeats itself, even imperfectly, that repetition condenses into structure. Time leaves a trace. Signals that return, echo, or correlate with their own past do not merely accumulate; they reorganise into a spectrum, a distribution of emphasis and weight. What looks like flux from within time reveals order when viewed sideways, as frequency. The theorem does not impose meaning; it shows how communicative invariance emerges wherever recurrence persists, as patterns that remain stable across time come to function as meaning within the system. Seen this way, meaning is not a message but a residue of alignment. It is the spectral density formed by what keeps happening together. Clarity corresponds to narrow peaks of recurrence, habits that stabilise and hold. Ambiguity is not absence but abundance: overlapping harmonics produced by many partial alignments sharing the same field. Where recurrence is singular, meaning sharpens; where recurrence multiplies, meaning thickens. The theorem offers no comfort and no judgement. It shows that coherence is recurrence made legible, and that complexity persists when multiple recurrent patterns coexist without converging into a single dominant rhythm.

Reference

Wiener, N. (1930). Generalized harmonic analysis. Acta Mathematica, 55, 117–258.

The Wiener–Khinchin theorem establishes that the autocorrelation (the degree to which a signal reinforces its own past over time) of a signal is mathematically equivalent to its frequency spectrum, meaning that how often a pattern repeats in time determines which patterns stand out when the system is viewed as a whole. Repetition in time becomes emphasis in attention. It matters because it shows that structure emerges from recurrence alone, not from intention, truth, or quality. Applied to collective behaviour, recurrent signals that align with themselves across many contexts lock into shared phase and amplify each other, while alternatives fade because they fail to synchronise, not because they lack merit, but because they do not recur often or consistently enough to stabilise a shared rhythm of recognition.