Resolution-Dependent Lead/Lag Structure - Multi-Horizon Closure Geometry Between Phase Topology and Price

The Phase-Coherence Burst Energy framework (https://zenodo.org/records/18866077) established that markets maintain real geometric structure in the phase domain, that this structure collapses during macro shocks, and that recovery times are shrinking exponentially. A natural follow-up question: does the phase topology lead or lag actual market performance? If Burst Energy rises before a drawdown appears in the S&P 500, the topology would function as a structural early warning system—detecting that the market’s independent degrees of freedom are compressing before the price impact materializes. If it lags, it measures something different: structural damage lingering after prices have already moved. A second question is equally important: does the answer depend on which stocks are in the basket? The original paper used a fixed set of approximately 60 mega-cap equities selected for continuous trading history. This ensured a clean data matrix but introduced survivorship bias—companies that failed during the observation window (Lehman Brothers, Bear Stearns, Washington Mutual, Enron) were excluded entirely. Yet the phase behavior of these companies in the months before their collapse is arguably the most informative data in the set. This addendum addresses both questions. We implement a rolling constituency approach using only the tickers that were actual S&P 500 members on each trading day, with closure detection thresholds scaled proportionally to universe size. We compute the lead/lag relationship between Burst Energy and drawdown across this corrected universe, and then subject the results to a systematic robustness sweep across 20 parameter combinations. The robustness analysis reveals something unexpected: the lead/lag relationship is not a single observable. It is a family of observables indexed by observation horizon, each resolving a different structural layer of the coupling. The parameter sweep is not merely a defensive exercise—it is a discovery tool.