Three formulas stand at the foundations of modern physics: the spacetime interval s² = c²t² - x², the Lagrangian L = T - V, and the action S = -mc²τ. Each belongs to a different domain—relativity, analytical mechanics, and dynamical principles—yet all three employ subtraction. This paper demonstrates that this shared structure is not coincidence but consequence: all three formulas express a single physical principle, the partition of energy between internal and external modes. We show that conservation of energy, distributed between complementary modes (internal energy associated with temporal evolution, external energy associated with spatial displacement), necessitates subtraction in any formula isolating one mode from the other. The spacetime interval subtracts spatial from temporal contributions to yield proper time. The Lagrangian subtracts potential from kinetic energy to yield the driver of dynamical change. The action accumulates this driver, producing a quantity directly proportional to proper time. The mathematical chain connecting these formulas—L → S → τ → s—displays the unity explicitly. The Lagrangian's subtraction propagates through integration to the action, which is proportional to proper time, which is determined by the interval with its characteristic minus sign. This interpretation explains why geodesics maximize proper time (energy optimization), why the action principle works (proper time extremization), and why the metric signature distinguishes time from space (energetic complementarity). The minus sign is the mathematical signature of conservation applied to complementary modes. It appears because it must appear.
Francis J Martin (2026) studied this question.