# A collision is a boundary, not a missing frame

This worked reference helps a learner or scientific-software developer check
one exact three-body state. It is a classical special family, not a solution of
arbitrary starting positions and velocities.

Choose G=1 and symmetric Euler masses (1,7/4,1), shape (-1,0,1), fixed horizontal
axis, no center drift, collision time t_c=0. Then μ=7/4+1/4=2.
For collapse, t<0 and τ=-t:

\[
\rho=\sqrt[3]{9\tau^2},\qquad
\dot\rho=-\sqrt{4/\rho},\qquad \ddot\rho=-2/\rho^2.
\]

At t=-1/3, ρ=1, ρ'=-2, ρ''=-2. All y/z components are zero.

| Body | Mass | x | v_x | a_x |
|---|---:|---:|---:|---:|
| 1 | 1 | -1 | +2 | +2 |
| 2 | 7/4 | 0 | 0 | 0 |
| 3 | 1 | +1 | -2 | -2 |

For body1, the middle body contributes7/4 to acceleration; body3 contributes
1/4 because its separation is2. The total is+2, matching the equation.
Internal kinetic energy is4, potential energy is−4; E=0 and L=0.

**Try a plausible mistake.** Change both nonzero velocities to their negatives.
Energy and angular momentum still agree. But the bodies now move outward, so
the state fails the specified collapse branch. Checking conservation alone
misses this error. Compare labeled positions, velocities, accelerations and time.

Closer to collision, at t=-1/24, ρ=1/4, ρ'=-4 and ρ''=-32. The end bodies are
at±1/4, their speeds are4, and accelerations point inward with magnitude32.
At t=0 the formula has no finite classical state. A stored position of zero
and placeholder velocity zero must not be mistaken for a valid solution.

**Reproduce the software check.** In the project repository, use its already
qualified Rust toolchain and independent C++ checker:

```sh
cargo build --manifest-path native/three-body-conformance/Cargo.toml --frozen --offline --release
c++ -std=c++20 -O2 -Wall -Wextra -Wpedantic -Werror tests/radial-reference.cpp -o /tmp/ergentics-radial-reference
python3 tests/radial-controls.py native/three-body-conformance/target/release/radial-cases docs/contracts/radial-cases.csv /tmp/ergentics-radial-reference
```

The existing hosted qualification script additionally verifies exact compiler
pins and compares Swift on macOS. The demonstration uses16 accepted states,
8 rejected parameter cases and a suite of injected output faults. Its acceptance
and rejection counts describe that finite workflow, not a measured educational
effect or general guarantee about external software. See the [equation
supplement](equations-019.md) and contract for domain and tolerance details.