R.O.M. — Complete Equation Set (LaTeX Reference)

R.O.M. does not describe how a body moves under forces; it classifies the algebraically allowed relational states of a bound system. It is not equations of motion but an algebraically closed system of allowed states within WILL — allowed free will.

Version note. This reference reflects the fully-closed system (all 66 expressions verified closed algebraically at 50-digit precision on random c, G, M, a, e, O in ROM_FULL_TEST.ipynb). Notation was overhauled relative to earlier drafts: the relational spacetime factor is \tau (was tau_W); the phase spacetime factor is \tau_o (was tau_Wo); the mean-anomaly integral is now \zeta (previously carried the tau_o name); the mass parameter is M (was m_0); the binding energy is written directly as \beta^2 (the W = beta^2/2 symbol is retired); and the orbital balance point splits into ascending B_a and descending B_d (was O_o). The Killing-vector expression is not part of this set.

Foundational Definitions (Redshift Inputs)

\[\kappa^2 = 1 - \frac{1}{(1+z_\kappa)^2}, \qquad z_\kappa = \text{gravitational redshift}\] \[\beta^2 = 1 - \frac{1}{(1+z_\beta)^2}, \qquad z_\beta = \text{transverse Doppler shift}\]

Observational Inputs

\(Z_{sys} = (1+z_\kappa)(1+z_\beta) = \frac{1}{\tau}\) Product of gravitational redshift and transverse Doppler shift.

\(\tau = \kappa_X \cdot \beta_Y = \frac{1}{Z_{sys}}\) Product of projectional phase factors on the $S^1$ and $S^2$ carriers.

\[z_\kappa = \frac{1}{\kappa_X} - 1 \qquad\text{(gravitational redshift)}\] \[z_\beta = \frac{1}{\beta_Y} - 1 \qquad\text{(transverse Doppler shift)}\] \[z_{\kappa o} = \frac{1}{\kappa_{Xo}} - 1 \qquad\text{(redshift at phase } O\text{)}\] \[z_{\beta o} = \frac{1}{\beta_{Yo}} - 1 \qquad\text{(transverse Doppler shift at phase } O\text{)}\]

Global System Parameters (Fixed for the Orbit)

Potential projection at semi-major axis: \(\kappa = \sin(\theta_2) = \beta\sqrt{2} = \sqrt{1 - (1+z_\kappa)^{-2}} = \sqrt{\frac{R_s}{a}} = \sqrt{\frac{\rho}{\rho_{max}}} = \sqrt{\kappa_p^2(1-e)} = \sqrt{2(\kappa_o^2 - \beta_o^2)} = \sqrt{\tfrac{1}{2}\left(3 - \sqrt{1 + 8\,\tau_o(B_a)^2}\right)} = \kappa_R^2\left(\frac{\tau_D}{T}\right)^{2/3} = \sqrt{\kappa_o^2 \cdot \eta_o}\)

Potential phase factor: \(\kappa_X = \cos(\theta_2) = \sqrt{1 - \kappa^2} = \frac{1}{1+z_\kappa}\)

Potential projection at the surface of the central body: \(\kappa_R = \sqrt{\frac{\kappa^2}{\sin(\theta_R)}} = \sqrt{\frac{R_s}{\frac{T}{2\pi}c\,\beta\,\sin(\theta_R)}}\)

Global kinetic projection (invariant transverse baseline defining the system’s circular equilibrium state and semi-major axis): \(\beta = \cos(\theta_1) = \sqrt{1 - (1+z_\beta)^{-2}} = \frac{\kappa}{\sqrt{2}} = \beta_p\sqrt{\frac{1-e}{1+e}} = \frac{2\pi a}{Tc} = \left(\frac{\pi R_s}{Tc}\right)^{1/3} = \sqrt{\kappa_o^2 - \beta_o^2} = \sqrt{\tfrac{\kappa_p^2}{2}(1-e)} = \beta_o\frac{\sqrt{1-e^2}}{\sqrt{1+e^2+2e\cos(O)}} = \sqrt{\beta_{sys}^2 \cdot \sin(\theta_{sys})}\)

Kinetic phase factor: \(\beta_Y = \sin(\theta_1) = \sqrt{1 - \beta^2}\)

Interior kinetic projection: \(\beta_{int} = \frac{\beta}{\sqrt{1-e^2}}\)

Relational spacetime factor: \(\tau = \kappa_X \beta_Y\)

Relational spacetime phase factor: \(\tau_Y = \sqrt{1 - \tau^2} = \sqrt{3\beta^2 - 2\beta^4}\)

Total relational shift (magnitude of state difference): \(Q = \sqrt{\kappa^2 + \beta^2} = \sqrt{\tfrac{3}{2}}\,\kappa = \sqrt{3}\,\beta\)

Schwarzschild radius — system scale: \(R_s = \kappa_o^2\, r_o = \frac{Tc}{\pi}\beta^3 = \frac{8\pi^2 a^3}{T^2 c^2} = \frac{\Delta_{to}}{\zeta(O)}\kappa^2\beta c = \frac{r_1 r_2}{r_2 - r_1}(\beta_1^2 - \beta_2^2) = \frac{a}{2}\left(3 - \sqrt{1 + 8\,\tau_o(B_a)^2}\right) = \frac{r_o}{2(2a - r_o)}\left(4a - r_o - \sqrt{(4a - r_o)^2 - 8a(2a - r_o)(1 - \tau_o^2)}\right) = \frac{2GM}{c^2}\)

Semi-major axis: \(a = \frac{R_s}{\kappa^2} = \frac{T\beta c}{2\pi} = \frac{\beta_o c}{\omega}\cdot\frac{\sqrt{1-e^2}}{\sqrt{1+e^2+2e\cos(O)}} = \frac{\Delta_{to}(O)}{\zeta}\beta c = \frac{Tc\beta^3}{2\pi\left(\frac{K_i}{\sin(i)}\right)^2(1-e^2)} = \frac{\sqrt{1-e^2}}{2\pi\sin(i)}K_i Tc = \frac{R_{sys}}{\sin(\theta_{sys})}\)

Central body’s physical radius: \(R_{sur} = a \cdot \sin(\theta_{sur})\)

Mass parameter: \(M = \frac{Tc^3\beta^3}{2\pi G} = \frac{\kappa^2 c^2 a}{2G} = 4\pi\rho a^3\)

Temporal scale of the system: \(t = \frac{a}{c}\)

Energy invariant — binding energy: \(\beta^2 = (\kappa_o^2 - \beta_o^2) = \frac{\kappa_p^2}{2}(1-e) = \left(\kappa_o^2 - \frac{\kappa_o^2}{2\delta_o}\right) = \frac{R_s}{2a} = \left(\frac{R_s}{Tc}\pi\right)^{2/3}\)

Precession of perihelion per orbit (exact to second order): \(\Delta\phi = \frac{2\pi\,\tau_Y^2}{1-e^2} = \frac{2\pi(3\beta^2 - 2\beta^4)}{1-e^2} = \frac{3\pi R_s}{a(1-e^2)} - \frac{\pi R_s^2}{a^2(1-e^2)}\)

Precession of perihelion per phase radian: \(\Omega_\phi = \frac{\tau_Y^2}{1-e^2} = \frac{3\beta^2 - 2\beta^4}{1-e^2}\)

Constant ratio between idealised phase $o$ and true precessing phase $O$: \(\Omega = 1 - \Omega_\phi = \frac{O}{o}\)

Invariant angular momentum: \(h_W = \frac{\kappa^2}{\kappa_o^2}\cdot\beta_T\cdot a\cdot c = r_o\cdot\beta_T\cdot c = r_o^2\cdot\omega_o = a\cdot\beta c\cdot e_Y = R_s c\frac{\sqrt{1-e^2}}{2\beta}\)

Angular frequency: \(\omega = \frac{\beta c}{a} = \frac{c}{R_s}2\beta^3\)

Orbital period: \(T = \frac{2\pi}{\omega} = \frac{2\pi a}{\beta c} = \frac{\pi R_s}{\beta^3 c} = \frac{2\pi\sqrt{2}R_s}{\kappa^3 c} = \frac{2\pi\Delta_{to}}{\zeta_o}\)

Time-density invariant (where $R_{sur}$ and $\beta_{sur}$ are set by the distance from centre to the surface of the central body, and $\theta_{sys}$ is the angular radius): \(\tau_D = T\sin(\theta_{sys})^{3/2} = \frac{2\pi R_{sur}}{\beta_{sur}c} = \sqrt{\frac{\pi}{G\rho}}\)

Gravitational deflection: \(\Delta\varphi = 2\arcsin\!\left(\frac{\kappa_p^2(1 + \beta_p^2)}{2\beta_p^2 - \kappa_p^2(1 + \beta_p^2)}\right)\)

Light deflection: \(\Delta\gamma = 2\arcsin\!\left(\frac{\kappa_p^2}{\kappa_{Xp}^2}\right)\)

Relational density: \(\rho = \frac{\kappa^2 c^2}{8\pi G a^2}\)

Maximal density at the horizon (non-rotating systems, $\kappa^2 = 1 \equiv r = R_s$; density reaches its natural bound): \(\rho_{max} = \frac{c^2}{8\pi G a^2}\)

Eccentricity Relations

Eccentricity derived from closure: \(e = \frac{1}{\delta_p} - 1 = 1 - \frac{2\beta_a^2}{\kappa_a^2} = \frac{2\beta_p^2}{\kappa_p^2} - 1 = 1 - \eta_o(0) = \eta_o(\pi) - 1\)

Orthogonal eccentricity: \(e_Y = \sqrt{1 - e^2}\)

Shape factor: \(e_X = \frac{1+e}{1-e} = \frac{r_a}{r_p} = \frac{\delta_a}{\delta_p} = \frac{\beta_p}{\beta_a} = \frac{\kappa_p^2}{\kappa_a^2} = \frac{\kappa_a^2\beta_p^2}{\kappa_p^2\beta_a^2}\)

Time-Phase

Angular frequency at phase $o$: \(\omega_o = \frac{\beta c}{a}\cdot\frac{(1 + e\cos(O))^2}{(1-e^2)^{3/2}}\)

Time duration of a given phase interval: \(\Delta_{to} = \int_0^O \frac{1}{\omega_\theta(\theta)}\,d\theta = \frac{a}{\beta c}\zeta = \frac{T}{2\pi}\zeta = \frac{R_s\zeta}{2\beta^3 c}\)

Temporal phase interval (mean-anomaly analog): \(\zeta = \frac{2\pi\Delta_t}{T} = \frac{\Delta_t c}{R_s}2\beta^3 = (1-e^2)^{3/2}\!\int_0^O (1 + e\cos(\theta))^{-2}\,d\theta = \frac{1}{\sqrt{1-e^2}}\int_0^O\left(\frac{t_o(\theta)}{t}\right)^2 d\theta\)

Temporal phase interval, ascending orbit ($0 \to \pi$): \(\zeta(<\pi) = 2\arctan\!\left(\frac{\sin(\tfrac{O}{2})}{e_X^{1/2}\cos(\tfrac{O}{2})}\right) - \frac{e\,e_Y\sin(O)}{1 + e\cos(O)}\)

Temporal phase interval, descending orbit ($\pi \to 2\pi$): \(\zeta(>\pi) = 2\arctan\!\left(\frac{\sin(\tfrac{O}{2})}{e_X^{1/2}\cos(\tfrac{O}{2})}\right) - \frac{e\,e_Y\sin(O)}{1 + e\cos(O)} + 2\pi\)

Perihelion Relations

Radius at perihelion: \(r_p = a(1-e) = \frac{R_s}{\kappa_p^2} = \sqrt{\big((1-e)a\cos(O)\big)^2 + \big((1-e)a\sin(O)\big)^2}\)

Potential projection at perihelion: \(\kappa_p = \kappa\sqrt{\frac{1}{1-e}} = \sqrt{\frac{2\beta^2}{1-e}} = Q_p\sqrt{\frac{2}{3+e}}\)

Potential phase projection at perihelion: \(\kappa_{Xp} = \sqrt{1 - \kappa_p^2}\)

Kinetic projection at perihelion: \(\beta_p = \frac{V_p}{c} = \beta\sqrt{\frac{1+e}{1-e}} = \frac{r_p\,\omega_o(0)}{c} = \frac{\kappa_p}{\sqrt{2}}\sqrt{1+e}\)

Closure factor at perihelion: \(\delta_p = \frac{\kappa_p^2}{2\beta_p^2} = \frac{1}{1+e}\)

Relational shift at perihelion: \(Q_p = \sqrt{\kappa_p^2 + \beta_p^2}\)

Aphelion Relations

Radius at aphelion: \(r_a = a(1+e) = \frac{R_s}{\kappa_a^2}\)

Kinetic projection at aphelion: \(\beta_a = \sqrt{\beta_p^2 e_X^{-2}} = \beta\sqrt{e_X^{-1}}\)

Potential projection at aphelion: \(\kappa_a = \sqrt{\beta^2 + \beta_a^2}\)

Closure factor at aphelion: \(\delta_a = \frac{1}{1-e} = \frac{\kappa_a^2}{2\beta_a^2}\)

Relational shift at aphelion: \(Q_a = \sqrt{\kappa_a^2 + \beta_a^2}\)

Phase Variables (Depend on phases $o$ and $O$)

Idealised orbital phase (progression without precession): \(o = \frac{O}{\Omega}\)

True orbital phase with precession: \(O = \Omega \cdot o\)

Radial distance at phase $O$: \(r = r_o(O) = a\frac{1-e^2}{1+e\cos(O)} = \frac{R_s}{\kappa_o^2} = a\,\eta_o\)

Normalised true radial distance at phase $O$: \(\eta_o = \frac{1-e^2}{1+e\cos(O)} = \frac{r_o(O)}{a} = \frac{R_s}{\kappa_o(O)^2 a} = 2 - \frac{2\beta_o^2}{\kappa_o^2}\)

Local potential at the true phase $O$: \(\kappa_o = \sqrt{\beta_R(O)^2 + \beta_T(O)^2 + \beta^2} = \sqrt{\frac{R_s}{r_o}} = \sqrt{1 - (1+z_{\kappa o}(O))^{-2}} = \kappa_p\sqrt{\frac{1 + e\cos(O)}{1+e}} = \sqrt{\beta^2 + \beta_o^2} = \sqrt{\kappa_{surface}^2 \cdot \sin(\theta_o)}\)

Gravitational phase factor at phase $O$: \(\kappa_{Xo} = \sqrt{1 - \kappa_o^2} = (z_{\kappa o}(O)+1)^{-1}\)

Instantaneous translational kinetic projection at phase $O$: \(\beta_o = \sqrt{\beta_R^2 + \beta_T^2} = \beta\cdot\frac{\sqrt{1+e^2+2e\cos(O)}}{\sqrt{1-e^2}} = \sqrt{\kappa_o^2 - \beta^2} = \frac{\kappa_o(O)}{\sqrt{2\delta_o(O)}} = \sqrt{\frac{\kappa_o(O)^2}{2}\frac{1+e^2+2e\cos(O)}{1+e\cos(O)}} = \sqrt{\frac{R_s}{r_o(O)} - \frac{R_s}{2a}} = \sqrt{1 - (1+z_{\beta o}(O))^{-2}}\)

Instantaneous radial kinetic projection (line-of-sight speed) at phase $O$: \(\beta_R = \beta\frac{e\sin(O)}{\sqrt{1-e^2}} = \sqrt{\beta_o(O)^2 - \beta_T(O)^2} = \sqrt{\big((1 - (1+z_{\kappa o}(O))^{-2}) - \beta^2\big) - \frac{r_o(O)^2\omega_o(O)^2}{c^2}}\)

Instantaneous transverse kinetic projection (orthogonal sideways speed) at phase $O$: \(\beta_T = \frac{r_o(O)\omega_o(O)}{c} = \beta\frac{1+e\cos(O)}{\sqrt{1-e^2}} = \kappa_o^2\frac{\sqrt{1-e^2}}{2\beta} = \frac{R_s\sqrt{1-e^2}}{r_o(O)2\beta}\)

Relativistic phase factor at phase $O$: \(\beta_{Yo} = \sqrt{1 - \beta_o^2} = (z_{\beta o}(O)+1)^{-1}\)

Local closure factor at phase $O$: \(\delta_o = \frac{1+e\cos o}{1+e^2+2e\cos o} = \frac{\kappa_o^2}{2\beta_o^2}\)

Local relational shift at phase $O$: \(Q_o = \sqrt{\kappa_o^2 + \beta_o^2}\)

Angular speed at phase $O$: \(\omega_o = a\beta c\frac{e_Y}{r_o^2} = \frac{\beta c}{a}\frac{(1+e\cos o)^2}{(1-e^2)^{3/2}}\)

Phase spacetime factor at phase $O$: \(\tau_o = \kappa_{Xo}(O)\cdot\beta_{Yo}(O) = Z_{sys}(O)^{-1}\)

Amplitude spacetime factor at phase $O$: \(\tau_{Yo} = \sqrt{1 - \tau_o^2}\)

Light time scale at phase: \(t_o = \frac{r_o}{c}\)

Orbital Phase Invariants

Holographic Decryption Invariant: \(Z_{raw}(-\omega_i)\,\tau(-\omega_i)(1 - e\cos\omega_i) + Z_{raw}(\pi-\omega_i)\,\tau(\pi-\omega_i)(1 + e\cos\omega_i) = 2\)

Additional invariants that must hold at any phase:

\[\Delta E_{A \to B} + \Delta E_{B \to A} = 0\] \[\beta^2 = (\kappa_o^2 - \beta_o^2)\] \[h_W = r_o^2\cdot\omega_o\] \[\frac{\beta_T(O)}{\kappa_o^2(O)} = \frac{\sqrt{1-e^2}}{2\beta}\] \[\beta_T(O)^2\,\eta_o^2 = \beta^2(1-e^2)\] \[B_a - \zeta(B_a) = \zeta(B_d) - B_d\] \[\tau_D = T\sin(\theta_{sur})^{3/2} = \frac{2\pi R_{sur}}{\beta_{sur}c} = \sqrt{\frac{\pi}{G\rho}}\] \[\beta^2 - \beta_o^2 = \kappa^2 - \kappa_o^2\]

Relational Geometry (WILL)

Distribution angle on $S^1$ (non-physical): \(\theta_1 = \arccos(\beta)\)

Distribution angle on $S^2$ (non-physical): \(\theta_2 = \arcsin(\kappa)\)

Closure Theorem: \(\kappa^2 = 2\beta^2\)

Energy Symmetry Law (the universal relational energy ledger, the origin of causality): \(\Delta E_{A \to B} + \Delta E_{B \to A} = 0, \qquad \Delta E_{A \to B} = E_0\left(\frac{\kappa_{X,B}}{\beta_{Y,B}} - \frac{\kappa_{X,A}}{\beta_{Y,A}}\right), \qquad \Delta E_{B \to A} = E_0\left(\frac{\kappa_{X,A}}{\beta_{Y,A}} - \frac{\kappa_{X,B}}{\beta_{Y,B}}\right)\)

Phase relational shift amplitude at phase $O$: \(\Delta_Q = Q_o^2 - Q^2\)

Ascending orbital balance point (where $a = r$ and $\kappa_o^2 = 2\beta_o^2$ are true): \(B_a = \arccos(1 - \delta^{-1}) = \arccos(-e) = \arccos\!\left(1 - \frac{2\beta_p^2}{\kappa_p^2}\right) = \arccos\!\left(\frac{2\beta_a^2}{\kappa_a^2} - 1\right)\)

Descending orbital balance point (where $a = r$ and $\kappa_o^2 = 2\beta_o^2$ are true): \(B_d = 2\pi - \arccos(-e)\)

Structural-Dynamical Equivalence

\[\frac{r_a}{r_p} = \frac{\delta_a}{\delta_p} = \frac{1+e}{1-e} = \frac{\beta_p}{\beta_a} = \frac{\kappa_p^2}{\kappa_a^2} = \frac{\kappa_a^2\beta_p^2}{\kappa_p^2\beta_a^2}\]

This remarkable equivalence of structural ($\kappa$ on the $S^2$ carrier) and dynamical ($\beta$ on the $S^1$ carrier) symmetry suggests once again that SPACETIME $\equiv$ ENERGY.

Observer-Dependent Relations

Angular radius of central body viewed from distance $r$: \(\theta_{sur}(r) = \arcsin\!\left(\frac{R_{sur}}{r}\right)\)

Raw light shift including line-of-sight Doppler at phase $o$: \(Z_{raw}(O) = \big(1 + \beta_{int}(\cos(O+\omega_i) + e\cos(\omega_i))\sin(i)\big)\,Z_{sys}(O)\)

Line-of-sight light shift: \(\beta_z(O) = \frac{\beta}{\sqrt{1-e^2}}(\cos(O+\omega_i) + e\cos(\omega_i))\sin(i)\)

$i$ = orbital inclination relative to line of sight and orbital plane.

$\omega_i$ = phase turn or argument of periapsis.

Semi-amplitude invariant: \(K_i = \frac{\beta}{\sqrt{1-e^2}}\sin(i) = \beta_{int}\sin(i)\)

Extremal raw shifts: \(Z_{rawmax} = Z_{sys}(-\omega_i)(1 + K_i(1 + e\cos\omega_i))\)

\[Z_{rawmin} = Z_{sys}(\pi-\omega_i)(1 + K_i(-1 + e\cos\omega_i))\]

Inclined balance point: \(B_{ai} = (B_a + \omega_i)\)

System redshift at the extremal phases: \(Z_{sys}(-\omega_i) = \left(1 - \beta^2\frac{1+e^2+2e\cos(\omega_i)}{1-e^2}\right)^{-1/2}\left(1 - 2\beta^2\frac{1+e\cos(\omega_i)}{1-e^2}\right)^{-1/2}\)

\[Z_{sys}(\pi-\omega_i) = \left(1 - \beta^2\frac{1+e^2-2e\cos(\omega_i)}{1-e^2}\right)^{-1/2}\left(1 - 2\beta^2\frac{1-e\cos(\omega_i)}{1-e^2}\right)^{-1/2}\]

Holographic Decryption Invariant (restated in observer variables): \(Z_{raw}(-\omega_i)\,\tau(-\omega_i)(1 - e\cos\omega_i) + Z_{raw}(\pi-\omega_i)\,\tau(\pi-\omega_i)(1 + e\cos\omega_i) = 2\)

Background-Free Parametric Equations for the Observed Orbit

Inclination rotation: \(\begin{bmatrix} x_{sky} \\ y_{sky} \\ z_{depth} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(i) & -\sin(i) \\ 0 & \sin(i) & \cos(i) \end{bmatrix} \begin{bmatrix} x_{orb} \\ y_{orb} \\ 0 \end{bmatrix}\)

Sky-plane coordinates: \(x_{sky} = r(O)\cos(O + \omega_i)\) \(y_{sky} = r(O)\sin(O + \omega_i)\cos(i)\) \(z_{depth} = r(O)\sin(O + \omega_i)\sin(i)\)

Un-tilted 2D coordinates within the orbital plane: \(x_{orb} = r(O)\cos(O + \omega_i)\) \(y_{orb} = r(O)\sin(O + \omega_i)\)

Appendix — Translation to Legacy Tensor Parameters

The continuous multi-component tensor fields of General Relativity are derived as localized matrix representations of the purely scalar relational projections ($\beta, \kappa$) on the $S^1$ and $S^2$ carriers. The pseudo-Riemannian manifold is not a primitive; it is the coordinate-inflated shadow of relational phase capacity. The symbol $\hat{=}$ denotes this operational mapping.

1. Metric Tensor ($g_{\mu\nu}$)

The 4D metric tensor maps directly to the operational phase capacities and derived coordinate ratios.

2. Four-Velocity ($U^\mu$) and Momentum ($P^\mu$)

The 4-vector velocity translates to the composite of the local energy scale ($\tau$) and the separated kinematic amplitudes ($\beta_R, \beta_T$).

3. Stress-Energy Tensor ($T_{\mu\nu}$)

The classical sources of the gravitational field map directly to the relational density saturation limit and the intrinsic surface tension.

4. Ricci Tensor ($R_{\mu\nu}$), Ricci Scalar ($R$), and Einstein Tensor ($G_{\mu\nu}$)

The differential curvature operators are bypassed entirely by the Unified Geometric Field Equation linking geometric scale directly to energy density ratios.

5. Christoffel Symbols ($\Gamma^\lambda_{\mu\nu}$) and Geodesic Integration

The affine connections ($\Gamma^\lambda_{\mu\nu}$) used to define parallel transport on a curved manifold, and the continuous integration of the geodesic equation $\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$, are bypassed in favour of the algebraic Energy-Symmetry Law ($\Delta E_{A \to B} + \Delta E_{B \to A} = 0$) and the relational phase divergence $\Delta\varphi = 2\pi\frac{\tau_Y^2}{1-e^2}$.