The Alignment Peak Precedes the Price Peak: A Robust Ordering Theorem for Bubble Dynamics

Byulsup Lee

MPRA Paper from University Library of Munich, Germany

Abstract: In a bubble, two peaks are commonly conflated: the peak of the price and the peak of the collective alignment that sustains it. This paper separates them and proves an ordering. We model collective alignment as the order parameter of a bistable synchronization system driven toward a fold by an accumulating load, and the bubble component of price as an inertial monotone readout of alignment. Under four assumptions — regularity of the readout, start from near rest, occurrence of the fold, and a crossing threshold whose closed form we derive — the alignment peak strictly precedes the price peak. The result is a strong one: it survives the presence of a price-to-alignment feedback loop of arbitrary finite gain, because the proof uses only the existence of the alignment peak and the depth of the descent, not the mechanism that produces them. What the feedback changes is not the order but the size of the lag: numerically, the lag scales inversely with the market adjustment speed and increases monotonically in the feedback gain — the feedback lends the falling alignment a supporting arm of rising prices, widening the window in which the structure has already turned while the price still climbs. The ordering yields a directly falsifiable prediction: in bubble episodes, the peak of any price-independent alignment proxy — sentiment surveys, the inverse of forecast dispersion, the concentration of flows — precedes the price peak; a consistent observation of the reverse order refutes the readout structure. We state the verification design, including the circularity condition that disqualifies price-derived proxies, and delimit exactly which layer of theory a refutation would kill. AI disclosure: All derivations were additionally verified by independent symbolic computation, and the manuscript’s internal consistency was machine-audited; AI systems were used as instruments of verification and audit under the author’s direction. The theorem, the model, and the text are the author’s.

Keywords: Keywords: bubbles; order parameter; synchronization; saddle-node bifurcation; early warning; falsifiability (search for similar items in EconPapers)
JEL-codes: G01 (search for similar items in EconPapers)
Date: 2026-08-07, Revised 2026-08-07
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