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Steering by the Shadow of Chaos
The three-body problem is famously intractable. Sundog asks a smaller question: when instability casts a local shadow, can a controller use that signal before reconstructing the full state?
When enabled, signatures are computed from local measurements only (test particle perspective).
This browser-based visualization demonstrates the restricted three-body problem in real time. Two massive bodies (gold and blue) orbit their common center of mass, while a third test particle (white) moves in their gravitational field.
The general spatial three-body state is 18-dimensional: three position vectors, three velocity vectors. This browser prototype uses a planar 12D visualization, but the question is the same: many practical questions about three-body dynamics don't require tracking the full state at every instant. They require detecting signatures of dynamically important events: near-collisions, ejections, resonance capture, stability transitions.
Sundog does not solve the three-body problem. It asks whether the shadow of that chaos is enough to steer by. The current result is bounded and now deflationary: the high-velocity near-escape survival pocket is real, but Phase 18 shows that at matched duty it reduces to a radius-gated inward thrust reflex, not sophisticated tidal steering. Low-velocity and equal-mass boundary cells still show harms.
Phase 2 Status: The sensor-limited view separates privileged diagnostics from local proxy signals. In the browser, tidal gradients are computed as simulated local probe samples; this models what an accelerometer array or small maneuver experiment would estimate, but it is not yet a hardware-valid sensor demonstration.
Later-trials status: The Phase 13-18 chain tests the earlier pocket through a 16-second horizon, repairs the warning channel with radius, rejects per-step energy and hazard-margin mechanisms, and then shows the envelope is reproduced by radius-gated inward thrust. Lower velocities and equal-mass boundary cells remain the first regions where the controller should not be used.
Adjust the initial conditions using the controls on the right and watch how the indirect signatures respond. The goal is to identify signatures that forecast useful events: escape, close approach, hierarchy break, or transition into chaotic scattering.
The ratio of kinetic to potential energy. For a bound system this oscillates around 1. When it persistently exceeds 1, the system is unbound or about to eject. This single scalar compresses all 18 dimensions into a stability diagnostic.
A 3×3 matrix (or its scalar trace) compressing the 9D positional configuration. When a system undergoes a close encounter or ejection, the eigenvalue spectrum changes character before the event resolves.
Total energy (kinetic + potential) of the three-body system. In the absence of external forces, energy is conserved. Tracking energy helps verify integration accuracy and detect numerical drift.
Planned diagnostic: for each of the three pairs, compute the two-body energy in their center-of-mass frame. In a stable hierarchical triple, one pair's energy is deeply negative (inner binary) and one is weakly negative (outer orbit). When these approach each other, the hierarchy is breaking.
Local proxy signal: The test particle estimates the gravitational field gradient by comparing its acceleration with nearby probe samples. In this prototype those samples are simulated; in a physical sensor model they would require an accelerometer array, small probe maneuvers, or a learned local field approximation. High tidal magnitude indicates proximity to massive bodies or strong field gradients.
A separate strand of the three-body work asks a literature-count question, not a controller question. Li & Liao (2025) report 10,059 three-dimensional periodic orbits of the general three-body problem; in the equal-mass slice they identify 21 "choreographies" — orbits where all three bodies follow a single closed trajectory, the same wire. Our gauge-invariant σ₃ detector, run blind against their supplementary table, accepts 25. The difference is one SO(3) rotation angle: 4 of the 25 are rotating relatives, where σ₃·X equals X up to a global 2π/3 turn per period. The slider below shows the separator collapsing the 25 to the catalog's 21.
That reconciliation became the entry point for a seven-chapter K_facet audit chain. v0.3h remains the first pillar: 20 structural-zero receipts plus the named O_617 quarantine. The later program paused at v0.9 with a substantive v0.7 result: a Floquet velocity-fraction direction shadow stratifies stability on 250 analyzable supplementary-B piano-trio rows (chi^2 = 16.43, alignment 0.698). The full caveats live on the isotrophy page.
1.0 × 10⁻⁶ rad
Select one row from the rail to enlarge its wire. Gold rows are strict single-curve returns; purple rows are relative returns that come back after a global 2π/3 turn.
The central wire will animate after the annotated SVG loads.
Put the selected row beside a reference row to see strict single-wire returns against relative 2π/3 returns.
The contrast will update after the annotated SVG loads.
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Isotrophy bead-maze inspector
Download the selected panel as SVG, or render a short WebM loop with the orbiting bodies and colored comet tails. The copied link opens this inspector directly on the selected orbit.
The σ₃ workbench recovered the 21 strict choreographies from the catalog and cleanly split off 4 rotating relatives. That is a detector and literature-count reconciliation: 25 gauge-invariant = 21 strict + 4 relative, separated by 8 orders of magnitude in the rotation-angle metric the gate already computes. The four relatives sit outside the catalog's "single closed inertial-frame trajectory" convention — they are not literature errors.
It is not theorem evidence for Sundog. The v0.2 daughter-count test that would have used this catalog was retired at its cheap precheck (K1), when the prediction reduced to the equivariance-only null (Kfacet = 0). Kfacet = 0 does not mean "no piano-trios exist"; it means the proposed static operator could not distinguish the Sundog claim from generic equivariant bifurcation. The v0.3 derivation has since shipped as the v0.3h K_facet verdict: 20 structural-zero receipts plus one named quarantine (O_617), produced by a pre-registered three-stage audit chain. Audit chain intact; theorem-facing result is not closed.
The audit chain later extended through v0.9. Its public positive is still narrow: on the 250-row analyzable supplementary-B subset, a Floquet velocity-fraction direction shadow stratifies S/U at chi^2 = 16.43, with branch alignment 0.698. It does not predict stability, and it is not catalog-wide.
results/isotrophy/m3eq1-sigma3-precondition-fixed-inverse-orientation-25/residuals.csv (25 rows, with the rotation-angle column read into data-rotation-rad on each panel).scripts/isotrophy_bead_maze.py (integrates orbits via isotrophy_workbench.py); annotator: scripts/isotrophy_bead_maze_annotate.py.