# Explicit physical-time central-configuration equations

These are classical Newtonian Lagrange and symmetric Euler homographic
solutions. The circular rotation and the real inverse of Barker's parabolic
time relation are elementary functions of physical time. Implementing and
checking them does not discover a new family, solve arbitrary initial data, or
establish stability. The manuscript's classical-source references and attribution
remain applicable. The executable interchange contract is
`inquiry-v2` in the project repository; the equations below explain that contract.

## Shape and frame reduction

Let positive masses be \(m_i\), \(M=\sum_i m_i\), and \(G>0\). Shape coordinates
\(c_i\) are dimensionless complex numbers with \(\sum_i m_i c_i=0\).
For Lagrange, subtract the mass-weighted centroid from
\((0,1,\tfrac12+i\tfrac{\sqrt3}{2})\); their side length is one and \(\mu=GM\).
For symmetric Euler, use \((-1,0,1)\), masses \((m,m_0,m)\), and
\(\mu=G(m_0+m/4)\). The outer masses must agree exactly. These shapes obey

\[
G\sum_{j\ne i}m_j\frac{c_j-c_i}{|c_j-c_i|^3}=-\mu c_i.
\]

Fix orthonormal real three-vectors \(u,v\), the embedding
\(P(w)=u\Re w+v\Im w\), and normal \(n=u\times v\). Set
\(C(t)=C_0+V_0(t-t_0)\). A complex scalar \(z(t)\), with units of length,
gives the physical state

\[
r_i=C(t)+P(zc_i),\qquad
v_i=V_0+P(\dot zc_i),\qquad a_i=P(\ddot zc_i).
\]

Substitution reduces all three Newton equations to
\(\ddot z=-\mu z/|z|^3\). The phase rotates the shape in the chosen plane;
it does not alter the axes. Body labels retain their declared masses.

## Circular physical time

For radius \(a>0\), phase \(\phi\), and direction \(s\in\{-1,+1\}\), let

\[
\omega=\sqrt{\mu/a^3},\quad
\theta(t)=\phi+s\omega(t-t_0),\quad
z=a e^{i\theta},\quad
\dot z=is\omega z,\quad \ddot z=-\omega^2z.
\]

The API input `scale` is \(a\); returned `scale` is also \(a\), and
`anomaly` is exactly zero. At \(t_0\),
\(z_0=ae^{i\phi}\), \(\dot z_0=is\sqrt{\mu/a}\,e^{i\phi}\), and
\(C=C_0\). These equations specify the compatible initial positions and
velocities through the frame map above. Adding a constant phase can preserve
all invariants while failing those initial data.

## Nonradial parabolic physical time

For periapsis radius \(q>0\), periapsis time \(t_p\), fixed periapsis orientation
\(\phi\), and direction \(s\), define

\[
\beta=\sqrt{\mu/(2q^3)},\quad B=\beta(t-t_p),\quad
D=2\sinh\!\left(\frac13\operatorname{arsinh}\frac{3B}{2}\right),\quad H=1+D^2.
\]

The triple-angle identity gives \(D+D^3/3=B\), Barker's time relation. Its
derivative \(1+D^2>0\) proves a unique real root for every real \(B\). The
chosen real inverse is odd: \(D<0\) before periapsis, \(D=0\) at periapsis,
and \(D>0\) after. Direction \(s\) reverses orientation rather than this
time branch. No root iteration or unevaluated inverse remains in the Rust
elementary expression.

Differentiation gives \(\dot D=\beta/H\) and
\(\ddot D=-2\beta^2D/H^3\). The complete scalar state is

\[
z=q e^{i\phi}(1-D^2+2isD),\qquad |z|=qH,
\]
\[
\dot z=\frac{2q\beta}{H}e^{i\phi}(-D+is),\qquad
\ddot z=\frac{\mu}{q^2H^3}e^{i\phi}(D^2-1-2isD).
\]

Since \(|1-D^2+2isD|=H\), the last expression is exactly
\(-\mu z/|z|^3\). There is no singularity at periapsis. At \(t_p\),
\(z=qe^{i\phi}\), \(\dot z=is\sqrt{2\mu/q}\,e^{i\phi}\), and
\(\ddot z=-\mu e^{i\phi}/q^2\). The API input `scale` is \(q\); its
returned `scale` is the time-dependent radius \(qH\), and `anomaly` is \(D\).
The frame center at periapsis is \(C_0+V_0(t_p-t_0)\), not necessarily \(C_0\).

To obtain initial data at any chosen finite \(t_*\), evaluate these expressions
at \(B_*\) and apply the body map. Conversely, an initial body state belongs to
this parameterized family only if its common shape, frame, velocities and scale
all match such a state. Merely having zero energy is insufficient. Where a
compatible scalar state has been recovered, its specific angular momentum
\(h=\Im(\bar z\dot z)\ne0\) fixes
\(q=h^2/(2\mu)\), \(s=\operatorname{sign}h\), and
\(D=\Re(\bar z\dot z)/\sqrt{2\mu q}\). Then
\(t_p=t_*-(D+D^3/3)/\beta\) and
\(\phi=\arg z-2s\arctan D\) modulo \(2\pi\). This is a compatibility map,
not an API for classifying arbitrary three-body initial conditions.

## Invariants, domains and limits

Let \(I=\sum_i m_i|c_i|^2\). The scalar specific energy and angular momentum
are \(\epsilon=|\dot z|^2/2-\mu/|z|\) and \(h=\Im(\bar z\dot z)\).
The returned three-body internal energy and angular momentum about the center
of mass are \(E=I\epsilon\) and \(L=Ih n\):

| Motion | \(\epsilon\) | \(h\) |
|---|---|---|
| Circular | \(-\mu/(2a)\) | \(s\sqrt{\mu a}\) |
| Parabolic | \(0\) | \(s\sqrt{2\mu q}\) |

Equivalently, compute \(E\) from pair potential energy and velocities relative
to \(V_0\), and \(L=\sum_i m_i(r_i-C)\times(v_i-V_0)\). These returned
invariants exclude translational kinetic energy. Total inertial energy adds
\(M|V_0|^2/2\); angular momentum about a fixed origin adds \(MC\times V_0\).
The weighted center is \(C\), total momentum is \(MV_0\), and summed internal
force is zero. Scaling all lengths by \(\lambda>0\) at fixed masses and \(G\)
scales time by \(\lambda^{3/2}\), speed by \(\lambda^{-1/2}\), and acceleration
by \(\lambda^{-2}\).

Mathematically both stated families exist for all real physical times. Their
positive radii and distinct fixed shape points make every finite-time pair
separation positive. The parabolic member is nonradial, with periapsis \(q>0\).
No radial collision continuation, general Euler mass triple, elliptic/hyperbolic
inverse, arbitrary initial-data solution or stability conclusion is added.

The implementation uses a deliberately smaller binary64 envelope: each mass,
\(G\), and input length lies in \([10^{-3},10^3]\); absolute times \(t,t_0,t_p\)
are at most \(10^6\); absolute input phase and final circular angle are at most
\(10^4\); and \(|B|\le10^4\). Origin and drift components are bounded by
\(10^3\). Axes are finite, squared norms differ from one by at most
\(2\cdot10^{-12}\), and their mutual dot product has that same bound.
At the evaluated time, every center component is bounded by \(10^4|z|\) and
every drift component by \(10^4|\dot z|\). Derived quantities must be finite,
radius/speed and pair separations positive. Unsupported evaluations fail.
These engineering bounds are not a uniform forward-error theorem, and no
claim covers every binary64 input in the envelope. Subtraction after a large
translation can still lose relative accuracy. Qualification is the finite
declared corpus and controls on the actual recorded platforms.

CSV/TSV decimal inputs use the contract's normal-or-zero subset: finite normals
and exact zero are accepted; subnormals and a nonzero decimal significand that
underflows to zero are rejected. This rule cannot be replaced by a finite-only
check. Both input files are capped at 65,536 bytes and read as regular files
through a single checked descriptor. It is not a claim of universal `strtod`
equivalence. See the versioned contract for exact grammar and field identities.

## Independent Swift route and falsification

`tests/equation-check.swift` does not repeat the production hyperbolic expression.
For \(w=3|B|/2\), \(A=\sqrt[3]{w+\sqrt{1+w^2}}\), Cardano gives
\(D=\operatorname{sign}(B)(A-A^{-1})\). To avoid cancellation near zero, the
checker uses

\[
\delta=w+\frac{w^2}{\sqrt{1+w^2}+1},\qquad
|D|=\frac{\delta(A+1)}{A(A^2+A+1)}.
\]

This follows by rationalizing \(\sqrt{1+w^2}-1\) and factoring \(A^3-1\).
The zero branch is explicit. The checker then uses polar derivatives with
\(r=qH\), \(\theta=\phi+2s\arctan D\),
\(\dot r=2q\beta D/H\), \(\ddot r=2q\beta^2(1-D^2)/H^3\),
\(\dot\theta=2s\beta/H^2\), and \(\ddot\theta=-8s\beta^2D/H^4\).
Radial/tangential acceleration is
\((\ddot r-r\dot\theta^2)e_r+(2\dot r\dot\theta+r\ddot\theta)e_\theta\).
It separately evaluates pairwise Newton forces and COM invariants.

The 18 named CSV cases produce 792 expected fields. Cross-language agreement
uses \(2\cdot10^{-10}(1+\max(|x|,|y|))\), except exact masses and times.
Zero anomaly must match exactly; nonzero anomaly uses relative
\(5\cdot10^{-12}\max(|x|,|y|)\), with no absolute floor. Thus replacing the
tiny nonzero anomaly with zero fails. Newton acceleration residuals use
\(2\cdot10^{-9}\mu/r^2\); center, momentum, net force, internal energy and
angular momentum use the same relative factor with their documented
characteristic scales in source. Supplemental centered derivative controls use
\(2\cdot10^{-7}\) and step \(10^{-4}/\sqrt{\mu/\ell^3}\).

Controls include exact cubic nodes, both time branches, very small nonzero
anomalies, rotated/translating frames, direction, length/time scaling, wrong
acceleration, a perturbed shape, and a valid orbit with the wrong initial phase.
Malformed and misleading files include duplicate/missing/unknown keys, wrong
time/mass/velocity, nonfinite/underflow decimals and oversized input. A successful
local self-test establishes those self-controls only. Actual Rust export
comparison and platform receipts are separate qualification evidence; shared
fixtures and collaboration do not establish broad independence of model training
or independent discovery.

## Research record

Equation supplement 018 continues the same Ergentics Rust lineage. The Rust
implementation uses the real hyperbolic inverse; Swift uses rationalized Cardano
and polar derivatives; C++ uses monotone cubic bisection. Three collaborating
roles used two model labels (two Astra roles and one Sol role); the requested
Terra role could not start because of a host thread limit. This is a collaborative
implementation review, not a new controlled comparison of three model capabilities.

The original study remains in [the paper](paper.md). Exact source, build and
result receipts are retained in the canonical research record. Finite equation
evaluation is distinct from a numerical trajectory integration study.