Energetic Time Theory: Energy Partition and the Unified Minus Sign

Energetic Time Theory series: Paper 2 of 11. Readers new to the series should start with Paper 1. The series in order: 1. A Mechanistic Framework for Proper Time Emergence: https://doi.org/10.5281/zenodo.182839072. Energy Partition and the Unified Minus Sign: https://doi.org/10.5281/zenodo.18478370 (this paper)3. A Resolution of the Problem of Time: https://doi.org/10.5281/zenodo.184811284. Calibrated Measurement and the Observer-Invariance of Light Speed: https://doi.org/10.5281/zenodo.185415345. The Relativity of Internal and External: https://doi.org/10.5281/zenodo.187310136. The Quantum Instant: https://doi.org/10.5281/zenodo.228014857. The Universal Clock: Energy Dispersal and the Arrow of Time: https://doi.org/10.5281/zenodo.228021058. The Mirror of Now: Self-Reference and the Limits of Physical Measurement: https://doi.org/10.5281/zenodo.228020379. Rate and the Spacetime Interval: https://doi.org/10.5281/zenodo.2280111910. The Moment of Measurement: The Mirror Principle and the Quantum Measurement Problem: https://doi.org/10.5281/zenodo.2280214211. The Four-Momentum Foundation: https://doi.org/10.5281/zenodo.22801152 Abstract 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 shows that the shared structure is not coincidence but consequence: all three express one physical fact, the partition of a body's energy between internal and external modes under one fixed magnitude. The fixed quantity is the magnitude of the body's four-momentum, its rest energy mc², which every observer agrees on. In energy terms, (internal energy)² = (total energy)² − (momentum × c)². Read physically, as Energetic Time Theory (ETT) reads it, this is one speed of energy, c, shared between returning within the body and carrying it along. The spacetime interval is this relation written in coordinates. The action of a free body is its rest energy multiplied by its proper time. And the Lagrangian of a slow body in a weak gravitational field is its rest energy multiplied by its clock rate: L = −mc²(dτ/dt) ≈ −mc² + T − V. The familiar T − V is exactly the rest energy times the amount by which the body's clock rate falls short of one, through its motion and through its depth in the field. The chain L → S → τ → s therefore carries one minus sign throughout. This explains why the action principle selects paths of greatest proper time, why the metric distinguishes time from space, and why free energy, such as light, has no clock. ETT changes no number of established physics; it supplies the physical reading of these formulas.