Abstract: Traditional econometric risk models heavily rely on Pearson linear correlation, assuming normally distributed, linear, and symmetric relationships between equity returns. In emerging markets characterized by heavy tails and regime shifts, this linear paradigm creates systemic underestimation of systemic risk—the 'Linear Trap.' This paper investigates inter-sector dependencies across ten major sectors of the Indian stock market through four comparative dependence metrics: Pearson linear correlation, Spearman's rank correlation, Kendall's Tau concordance, and time-lagged cross-correlation. Furthermore, topological market structures are mapped using Minimum Spanning Trees (MST). Our quantitative evaluation reveals an average divergence (AvgDiff) of ~0.40 between linear and non-linear correlation structures. We demonstrate that the Auto sector forms a tightly coupled systemic risk block, whereas the Pharma sector provides natural diversification due to weak cross-dependencies. Notably, time-lagged cross-correlation isolates a leading relationship where Pharma sector trends precede Auto movements by an empirical window of -75 days, establishing actionable macro-financial early warning signals.