Reciprocal Price Representation and Nonlinear Risk Response in Financial Systems

Reciprocal price representation is intrinsic to foreign exchange markets and has become increasingly explicit in digital asset markets through inverse-quoted and inverse-settled products. While inversion preserves relative returns, it reshapes absolute variations and practical risk perception in a state-dependent manner. This paper studies reciprocal price representation as a system-level reparameterization induced by a fixed-capital constraint, under which attainable asset quantity expands nonlinearly as price declines. We show that reciprocal representation introduces a curvature-driven amplification mechanism: small price changes remain approximately symmetric across representations, whereas large drawdowns generate disproportionately large responses in the reciprocal coordinate. This amplification is geometric rather than leverage-induced and arises purely from representation choice. As a result, identical market events can appear qualitatively different when observed through conventional versus reciprocal coordinates. The framework further clarifies how reciprocal representation reshapes boundary interpretation and sustainability under stress. Conventional short exposure is structurally associated with unbounded adverse directions and liquidation risk, whereas the reciprocal coordinate relocates the unbounded direction and imposes a natural floor, transforming failure modes from debt-driven blow-ups into compression-dominated boundary regimes. We interpret reciprocal representation as a risk-response coordinate that magnifies stress proximity and reveals compensatory structural responses during deep drawdowns. The contribution of this work is conceptual and interpretive. Rather than proposing a trading strategy or performance claims, we provide a formal framework for understanding price representation as a non-neutral design element in digital asset markets, with implications for stress interpretation, boundary awareness, and sustainability-oriented risk analysis.