# When the shape remains, and the scale falls to zero

## Exact radial reference — equation supplement 019

A triangle can preserve its shape while all its sides shrink together. A line
of three bodies can do the same. The limiting instant is not another ordinary
point on the trajectory: the Newtonian force law becomes singular there.
Writing both the equation and its domain makes that distinction inspectable.

This supplement gives classical zero-energy homothetic solutions for the
Lagrange equilateral and symmetric Euler collinear configurations. It extends
the explicit circular and nonradial parabolic references in supplement018.
The contribution here is a bounded implementation and checking resource in
Ergentics Rust, with independent native formulas; no discovery priority is
claimed over these classical solutions. The previous historical source trace
remains contextual, not an exhaustive novelty or priority review.

## Shape, units and physical state

Let positive masses m_i and gravitational constant G use one consistent unit
system, with G in length³/(mass·time²). The fixed complex shape coordinates c_i
have zero mass-weighted center. For Lagrange, center the unit-side triangle
(0,1,1/2+i√3/2) and take μ=GΣm_i. For symmetric Euler take (-1,0,1), masses
(m,m₀,m), μ=G(m₀+m/4). Adjacent spacing is one. These shapes satisfy

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

Fix phase φ and orthonormal real three-vectors u,v. Define the embedding
P(w)=u Re(w)+v Im(w), and center C(t)=C₀+V₀(t−t₀). The physical bodies are

\[
\boxed{r_i(t)=C(t)+\rho(t)P(e^{i\phi}c_i)}
\]
\[
v_i(t)=V_0+\dot\rho(t)P(e^{i\phi}c_i),\qquad
a_i(t)=\ddot\rho(t)P(e^{i\phi}c_i).
\]

The single length ρ is the triangle side length or the Euler adjacent spacing.
μ has units length³/time²; ρ, its derivatives and the frame retain their physical
units. Body identities and masses are fixed throughout each branch.

## Finite elementary equations and open time branches

Choose a collision time t_c and s=-1 for collapse or s=+1 for expansion. Put
τ=s(t−t_c). Require **τ>0 before evaluating the formula**. Then

\[
\boxed{\rho(t)=\left(\frac{9\mu}{2}\right)^{1/3}\tau^{2/3}},\qquad
\boxed{\dot\rho=s\sqrt{\frac{2\mu}{\rho}}=
s\frac{2\rho}{3\tau}},\qquad
\boxed{\ddot\rho=-\frac{\mu}{\rho^2}=-\frac{2\rho}{9\tau^2}}.
\]

Collapse exists for t<t_c, expansion for t>t_c. Cube roots and arithmetic give
explicit finite elementary expressions in physical time. No time anomaly needs
numerical inversion; no infinite series is used here. These are zero-energy
radial members of the reduced Kepler equation, not every radial trajectory.

Derivation begins with the reduced zero-energy relation
\(\dot\rho^2/2-\mu/\rho=0\). Selecting the sign yields
\(\dot\rho=s\sqrt{2\mu/\rho}\). Integration gives
\(\rho^{3/2}=\tfrac32\sqrt{2\mu}\,\tau\).
Differentiating on either open branch gives the derivatives above. Direct
Newton substitution uses r_j−r_i=ρP(e^{iφ}(c_j−c_i)), hence

\[
G\sum_{j\ne i}m_j\frac{r_j-r_i}{|r_j-r_i|^3}
=-\frac{\mu}{\rho^2}P(e^{i\phi}c_i)=a_i.
\]

At t_c all pair separations tend to zero while nonzero shape components of
velocity and acceleration diverge. The middle Euler body may have zero internal
position/velocity/acceleration, but the pair force terms are still undefined at
the common collision. There is no finite three-body state there. Placing the two
branches side by side does not define a classical continuation through collision.

## Compatible initial data and invariants

At any allowed t*, choose the common shape, phase, frame and a positive scale a.
Compatible data require

\[
r_i(t_*)=C(t_*)+aP(e^{i\phi}c_i),\qquad
v_i(t_*)=V_0+s\sqrt{2\mu/a}\,P(e^{i\phi}c_i),
\]
\[
t_c=t_*-s\sqrt{\frac{2a^3}{9\mu}}.
\]

This reconstructs the event time for compatible data. It is not a classifier
for arbitrary three-body initial states. The internal total energy and angular
momentum are E=0 and L=0. With I=Σm_i|c_i|², the potential is −Iμ/ρ and kinetic
energy relative to V₀ is I·ρ'²/2; they cancel. All internal position and velocity
pairs are parallel, so each angular-momentum contribution is zero. Inertial
energy additionally contains M|V₀|²/2 and total momentum is MV₀.

These zero invariants do not determine phase, event time or the velocity sign.
In particular, reversing every internal velocity preserves E and L but changes
collapse to expansion. A comparison must retain the requested branch and initial
state. The golden negative uses precisely this misleading agreement.

At fixed masses and G, changing an event-relative interval by λ^(3/2) multiplies
ρ by λ, speed by λ^(−1/2) and acceleration by λ^(−2). Equal positive gaps on
opposite branches have equal center-relative positions and accelerations, with
opposite internal velocities when phase and plane agree. With center drift,
the inertial positions at these two times generally differ.

## Numerical scope and implementation

The dedicated `radial` module extends the existing canonical Ergentics Rust
crate0.3.0. It reuses the shape/body types, preserves the v2 API, and uses the
explicit collision time rather than overloading a periapsis or anomaly field.
inquiry-v3 in the project repository specifies all parameters and wire fields.
Its frozen corpus has16 accepted states and8 rejected inputs, giving776 rows.

The exact mathematical domain τ>0 is larger than the binary64 operating range
τ∈[10⁻⁴,10⁴]. G and masses lie in[10⁻³,10³]; absolute t,t_c,t₀≤10⁶ and
|φ|≤10⁴. Origin/drift components are bounded by10³, with evaluated center
components additionally bounded by10⁴ρ and drift components by10⁴|ρ'|. Axes
have squared norm/dot tolerance2·10⁻¹². Results must be finite, radius/speed
positive and pair separations resolved. Unsupported inputs return errors.
These bounds do not prove a uniform floating-point error theorem. Input event
uncertainty and lost relative precision after a large translation remain
distinct from the exact singularity at collision.

Swift derives ρ from cbrt((9/2)μτ²), with differentiated radius ratios. C++ uses
the integrated zero-energy relation and direct pair forces. Different arithmetic
paths, physical controls and the same declared parameters provide useful checks;
they do not constitute independent mathematical certification. Actual precision
and compiler/target identities belong to each build receipt. C++ long double is
not assumed wider than double. Self-generated native output is labeled separately
from actual Rust output.

## Practical benefit and its limit

The accompanying [worked collision-boundary card](radial-reference-card.md)
offers a modest, concrete use: compare a correct state with a plausible wrong
one, and see why a singular event needs an explicit domain. The automated
workflow accepts the reference and rejects reversed velocity, fabricated
collision states and other declared faults. This measures detection on those
injected examples. It does not establish learning gains, improved deployed
software, accessibility qualification, resource savings or safety-critical use.
No participant information is collected. Classical knowledge is credited as
shared scientific inheritance; this work's value depends on making it useful
and honestly inspectable.

## Research-method boundary

Epoch019 starts fresh Astra, Sol and Terra roles at requested high reasoning,
preserves original proposals before comparison, and records subsequent shared
implementation/review. The shared filesystem does not enforce blinding; serving
builds and training independence are not established. This collaboration is
separate from the original exploratory model study and supports no model ranking.
The original paper's general local series and circular instability result remain
unchanged. No new radial stability theorem, general finite solution, general
impossibility result, protected examination or collision continuation is added.