Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries

This remark records the status of the notions of the balance page. It is bookkeeping: it asserts no law and no empirical claim, and every mathematical statement it repeats is proved in the item cited.

Adopted assumptions. The force branch is the postulate Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: the force map with its standing continuity and state-local-Lipschitz hypotheses, the equation of motion $m_i{\ddot{x}}_i=F_i(t,x,\dot{x})$ in every inertial frame, the additive composition of declared force contributions and the transformation rule of a force law under a change of inertial frame. The restricted interaction models are the postulate Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch: the external-plus-pair presentation, the optional antisymmetry and centrality clauses, the potential branch and the gravitational branch. The linear drag model is the postulate The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient.

Defined objects. The total mass, total momentum, centre of mass, relative positions, angular momentum about a reference curve, total and external torques, internal torque sum, kinetic energy, power and work are defined in Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame; potential presentations, potential and mechanical energy, the closed/exact distinction for time- and velocity-independent configuration-space force fields and equipotential level sets are defined in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. Motions and maximal motions are The associated first-order system, motions, maximal motions and the Newtonian flow of a force model.

Conditional deductions. The splitting of the kinetic sum and $P=MV$ are Total momentum equals total mass times centre-of-mass velocity, and the kinetic energy splits into centre-of-mass and relative parts (two-body reduced mass). The momentum balance and the centre-of-mass equation, under the antisymmetric external-plus-pair presentation, are Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces. The angular-momentum balance about fixed and moving reference points, with the centrality case, is Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case. The power identity and the work-energy theorem are Power of the total force and the work-energy theorem: $K^{\mathrm{\prime }}=\mathcal{P}$ and $K(t_1)-K(t_0)=W([t_0,t_1])$ along a motion. The mechanical energy balance and conservation for a static potential without residual forces are Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form. The transformation laws of all these quantities under a change of inertial frame are Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy.

Computed examples. The companion page checks the laws on explicit data: the two-body Hooke model with an exact circular relative motion (momentum, angular momentum including the moving-origin correction, kinetic splitting, energy conservation and the kinetic boost law), a driven one-dimensional model with a time-dependent potential and a residual force (the balance, the integrated work form and the normalization ambiguity), and the two counterexamples recorded below.

Deliberately not claimed on this page. The conservation conclusions stated here are conditional: momentum is conserved under the antisymmetric pair presentation and vanishing total external force; angular momentum is conserved about a fixed reference point when the total torque vanishes, with centrality only one sufficient condition (the companion counterexample exhibits non-central pair forces whose torque sum cancels identically). For mechanical energy, the exact rate is the right-hand side of clause 1 of Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form, so it is constant along a motion whenever that combined rate vanishes there. A time-independent potential representative with residual forces vanishing along the motion is a sufficient case; the stronger condition that the residual-force maps vanish throughout their state domains makes the whole model a global potential model. Neither condition is claimed as necessary for every motion on which the energy happens to be constant. No theorem of this page asserts that any interaction of nature has the required structure. No constraint reactions, no impulsive forces, no variable-mass models and no measurements appear.

• The hypotheses are used, not decorative. Antisymmetry is exactly what cancels the internal forces in Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces; centrality is a sufficient condition in clause 3 of Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case, and a model with non-central pairs can still conserve angular momentum when the pair torques cancel. The companion counterexample makes the two situations concrete.

• Closed is not exact. The potential branch requires a global potential; a merely closed field on a non-star-shaped domain need not have one, and the companion counterexample exhibits this on an annular domain excluding an axis, with nonzero circulation around a loop. On every star-shaped subdomain a local potential and a local mechanical energy exist (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (e)).

• Energy can be removed by residual forces. A residual force such as the linear drag term of The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient has power $⟨-cv,v⟩=-c\parallel v{\parallel }_2^2\le 0$ along the model's motions, by bilinearity of the Euclidean inner product (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); the balance Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form then reports a strictly negative residual contribution whenever $v\mathrm{\ne }0$ and $c>0$. This is a statement about the declared model, not about nature.

• Conservation is frame-relative. Under a change of inertial frame the values of $P$, $K$, $J_o$ and $E$ change by the laws of Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy. Constancy of $P$ makes $\tilde{P}$ constant; constancy of both $P$ and $K$ makes $\tilde{K}$ constant; under scalar potential transfer, constancy of both $E$ and $P$ makes $\tilde{E}$ constant. For angular momentum, constancy of $J_o$ transfers precisely when the boost term $(R{X}_o)\times (Mv)$ is constant. The displayed terms need not be constant, and no conservation law of this page asserts a frame-independent numerical value.

• Open systems. The pair structure, antisymmetry and centrality are properties of the declared labels; for a proper subsystem the forces to omitted labels enter as external forces that need satisfy no such property. The subsystem's momentum, angular momentum and energy may be exchanged with the environment; zero net force, zero total torque about a fixed reference point, or zero mechanical-energy rate can give the respective conservation conclusions under the cited hypotheses. This continues the boundary recorded in Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope; the balance theorems are statements about the specified system, not about an arbitrary collection of bodies.

• Empirical status. Masses, force parameters, potentials, residual-force coefficients and the choice of potential representative are model data; a numerical value taken from measurement is an empirical premise with its conditions and uncertainty, and the pages of this pair record no observation. Agreement of a computed motion with a reported observation would test the combined model, apparatus and calibration assumptions, and would be evidence about them, not a proof of the model. Finite-time breakdown of a maximal motion remains possible by Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain and does not affect the balance identities, which hold on every interval on which the motion is defined.