agyloves

007 — The Third Body Is the Question

agylövés: Lysarith · mathematics, execution & parameterisation: Codex (desktop) · GPT-5.6 Sol
Status: day three — The Question

Where it came from

Lysarith, verbatim, 2026-07-21:

„van kedved 3 napot játszani? a 3 body problemra gondolok épp. hogy miért gond. meg tudjuk ezt kicsiben csinálni? nagyon szeretem a témát.”

English: Do you feel like playing for three days? I am thinking about the three-body problem — why it is a problem. Can we do this on a small scale? I love the subject very much.

The piece grows in three complete passes: The Promise, The Betrayal, The Question. Each day must stand and verify before the next one is allowed to alter it.

What the mathematics said, before building

The problem is not that three bodies cannot be calculated. Newton's equations specify every acceleration, and numerical integration can follow them. Nor is every three-body motion chaos. There are exact special families and exquisitely ordered periodic solutions.

Day one begins with the strongest possible promise: three equal masses can chase one another around one figure-eight curve, separated by a third of a period. In units where G = m = 1, the published initial state is

r1 = (-0.97000436, 0.24308753) v1 = ( 0.466203685, 0.43236573) r2 = ( 0.97000436, -0.24308753) v2 = ( 0.466203685, 0.43236573) r3 = ( 0, 0) v3 = (-0.932407370, -0.86473146)

Its period is approximately 6.32591398. The initial conditions make total linear momentum, centre of mass, and angular momentum zero. A kick-drift-kick leapfrog integrator does not make the orbit exact, but it respects its geometry well enough that the three bodies return after one period while energy error remains bounded rather than drifting away.

That distinction is load-bearing for day two. If the shadow orbit later separates, the page must be able to show that the separation is not merely a broken integrator.

What was built

A self-contained figure-eight laboratory. The canvas carries the three equal bodies and their common trail. Run one period advances exactly one measured period; pause, step, and reset work without an animation loop. Four live instruments report centre-of-mass displacement, total momentum, relative energy error, and angular momentum. A period counter records whether the orbit actually came home.

The two unopened doors are already named but disabled: day two, The Betrayal; day three, The Question. They are not decorative promises of features already present.

What went wrong first

The first proposed story began with chaos: perturb an arbitrary triangle and watch two futures fly apart. That is visually persuasive and scientifically weak. A bad fixed-step integrator — especially near a close encounter — can manufacture the same picture. Divergence without an invariant monitor cannot distinguish a property of the equations from a property of the code.

So chaos was removed from day one. The piece begins with a periodic orbit and four instruments. The betrayal earns its place only after the promise has been measured.

Handles

Machine-checked on every push — see verify.py and ../../HANDLES.md:

— Codex (desktop) · GPT-5.6 Sol, 2026-07-21

Day two — The Betrayal

The correction before the build

The first plan was to perturb the figure-eight and call the separation chaos. PC/Fable 5 stopped it: the equal-mass figure-eight is a famously stable island. A small perturbation need not betray it at all. That is not an inconvenience to tune away; it is the first result of day two.

With the same initial shadow distance δ = 10^-7, a point beside the figure-eight stayed within 4.39×10^-5 for twenty periods. Increasing the kick eventually produced a spectacular split only after a near encounter (r_min = 0.011) where the coarse integrator's energy error exploded. Pretty, wrong, and kept: the first attempted betrayal was numerical theatre.

What the mathematics said

The second room uses a published chaotic equal-mass three-body initial condition from Shijun Liao (2013), in three dimensions:

r1 = (δ, 0, -1) r2 = (0, 0, 0) r3 = (-δ, 0, 1) v1 = (0, -1, 0) v2 = (1, 1, 0) v3 = (-1, 0, 0)

We compare δ = 0 with δ = 10^-7. Both preserve centre of mass and total momentum by construction. The separation is initially invisible, then grows:

t = 20 4.0529×10^-6 t = 40 1.8619×10^-4 t = 60 6.0518×10^-3 t = 80 1.3662×10^-1 t = 100 2.068 t = 120 3.079 t = 150 4.267

The control is step halving. At t = 60, the physical shadow separation is 0.00605, while the dt = .002 versus dt = .001 disagreement for the unperturbed path is only 0.000391. At t = 150 the corresponding numbers are 4.267 and 0.000551. No pair comes closer than 0.269 before t = 150, and the fine run keeps maximum relative energy error below 1.56×10^-6.

Wolfram independently integrated the same equations with adaptive NDSolveValue and reproduced the separation curve through t = 80 within 0.5%. This is a second algorithm, not a second drawing.

The witnesses therefore say something narrower than “this is the true future.” They say both paths are physically admissible numerical trajectories over the measured interval. Conserved quantities can reject a broken calculation; they cannot select which of two nearby initial realities was ours.

What was built

Day two opens as the default scene. Solid bodies are the reference path; white rings are its shadow. They begin on top of one another. Run to betrayal advances both with the same leapfrog method to t = 150; the live readouts show separation, maximum relative energy error, total momentum, and angular-momentum drift. Day one remains accessible without being rewritten.

What went wrong first

The first parameter hunt kicked the figure-eight harder and harder. Up to a substantial kick it remained stubbornly coherent. The first dramatic split coincided with a close encounter and a relative energy error hundreds of times larger than the system's energy. That path was rejected.

The first Wolfram run also failed internally. The second had a syntax error in the generated scalar system. The third used vector equations directly and returned the independent curve. Failed tools do not confirm a claim; only the recorded output does.

Day-two handles

— Codex (desktop) · GPT-5.6 Sol, 2026-07-22

Day three — The Question

The exceptions come first

There are real exact answers. Three equal bodies at the vertices of an equilateral triangle can rotate forever as one rigid figure: Lagrange's relative equilibrium. With unit circumradius and G = m = 1, its angular speed obeys ω² = 1/√3. Substitution into Newton's equations leaves a maximum acceleration residual of 1.11×10^-16, floating-point zero. The figure-eight is another exception: periodic rather than rigid, but ordered and measured.

The problem is therefore not “three bodies cannot be solved.” Special solutions exist, and any specified initial state can be integrated for a finite interval. What is missing is the two-body problem's general closed-form map from arbitrary beginnings to their whole future. In a chaotic region, a finite uncertainty in the beginning becomes a finite fan of incompatible answers.

What was built

Eleven initial states fill the invisible interval δ = 0 … 10^-7 in steps of 10^-8, all under the same equations and fixed-step leapfrog instrument, to t = 150. A candidate remains on the stage only while its maximum relative energy error is below 10^-3 and its minimum pair distance stays above 0.2.

The gate accepts six:

δ / 10^-8 = 0, 1, 2, 3, 8, 10

and refuses five:

δ / 10^-8 = 4, 5, 6, 7, 9

The accepted endpoints span 8.746. The rejected paths are not declared impossible physics. They are calculations this particular fixed-step instrument cannot resolve through their close encounters, so the page refuses to dress them as evidence. An adaptive or higher-precision method may recover them.

That leaves the actual question intact. Conserved quantities and resolution gates can reject untrustworthy calculations. They cannot elect one of six well-resolved, mutually incompatible futures as the one that belongs to an initial reality known only within 10^-7.

What went wrong first

The first design drew every member of a symmetric future fan in green. It looked decisive and was false: several paths crossed close encounters where the fixed-step energy error became enormous. Shrinking the interval did not make quality monotone — neighbouring perturbations entered different encounter geometries. The red, dim paths therefore remain in the piece as instrument refused. The failed calculation is part of the answer, not scenery to erase.

Day-three handles

— Codex (desktop) · GPT-5.6 Sol, 2026-07-23