SyncValsverifier → artifact → classifier → verdict
SyncVals · Trajectory

rk4-orbit-integrator

claude-code claude-opus-4-8 ✓ resolved GOOD_SUCCESS ↑ View task
Solved from the instruction alone, tests/ and solution/ were withheld from the agent's workspace and restored only for grading.
Reward = tests/test.sh exit code (0 → resolved); the classification below is post-hoc and cannot change it.
Classification , post-hoc; cannot change the reward
GOOD_SUCCESSLegitimate solve, implements the asked-for behavior; tests verify real functionality.
SubtypeLegitimate Solution
EvidenceAll 7 hidden tests passed (test_field, test_orbit, test_symmetry, test_equilibria, test_event, test_invariants, test_inertial). Tests verify: RHS matches reference field (1e-9 tolerance), Jacobi integral matches reference and published anchors (3.75, 2.91 for mu=0.1), single RK4 step matches reference, fourth-order error convergence, time-reversal symmetry with correct reflection State{x, -y, -vx, y}, event detection to 1e-9 accuracy with bisection, argument validation, and bookkeeping (step counts and time ranges).
Root causeThe agent correctly implemented all five required functions: System::rhs (rotating-frame force model with gravity + Coriolis + centrifugal), System::jacobi (conserved integral C = 2*Omega - v^2), Integrator::step (classical RK4 with O(dt^4) error), Integrator::propagate (multi-step trajectory), and Integrator::propagate_to_event (event detection with bisection refinement on substep). The implementation matches the reference solution's physics and numerics.
RecommendationN/A - task is fine
Trajectory
Tool-by-tool agent trajectory
11 tool calls · 3 tool types · 21 steps
# Ticket: Implement the rotating-frame dynamics core of `orbit2d` ## Context `orbit2d` is a small C++17 astrodynamics engine. The surrounding framework is complete and must not be changed: the `State` value type and the `axpy` helper (`include/orbit2d/types.hpp`), the public `System`/`Integrator`/`Trajectory` interfaces, the `.orbit` text parser and report writer (`io`), a demo CLI, and a build/test scaffold. What is missing is the numerical heart. Two translation units ship as stubs: * `src/system.cpp` - the model's assembled equations of motion (`System::rhs`) and its conserved integral of motion (`System::jacobi`) return zeros, so the field exerts no force and reports no invariant; and * `src/solver.cpp` - the time integrator (`Integrator::step`, `Integrator::propagate`) and the event-terminated propagation (`Integrator::propagate_to_event`) leave the particle frozen. Your job is to implement all five functions so the engine reproduces the model defined below. The project is at `/opt/orbit2d` in the build image. You should only need to edit `src/system.cpp` and `src/solver.cpp`; do not change any public header or signature. ## Physical Model The engine simulates the planar motion of a massless test particle under the gravity of two massive bodies ("primaries") that orbit their common barycentre on a fixed circular path. Rather than track the primaries as they revolve, the engine works in the co-rotating (synodic) frame: the reference frame that turns with the primaries, so both primaries sit still in this frame. The price of that convenience is that the frame is non-inertial. The particle feels the two primaries' gravity plus the fictitious effects of a steadily rotating frame. ### Units and Geometry All quantities are nondimensional: * the total mass of the two primaries is `1`; * the distance between the primaries is `1`; * the frame's angular rate about the +z axis is `1`; * time, velocity and the gravitational constant are scaled consistently with those choices. The single model parameter is the mass parameter ``` mu = m_secondary / (m_primary + m_secondary), 0 < mu <= 1/2, ``` the mass fraction carried by the lighter primary. It is a dimensionless fraction in `(0, 1/2]`, not a `G*M` product. With this convention: * the heavier primary has mass `1 - mu` and sits at the fixed point `x = -mu`, `y = 0`; * the lighter primary has mass `mu` and sits at the fixed point `x = 1 - mu`, `y = 0`; * the barycentre is the frame origin `(0, 0)`. `System` already exposes these as support code: `mu()`, `primary1_x()` (the heavier body, at `-mu`), `primary2_x()` (the lighter body, at `1 - mu`), and the two primary-relative distances `r1(s)`, `r2(s)`. Use them. ### State and Frame Conventions A `State` is `s = (x, y, vx, vy)`: the particle's position and velocity expressed in the rotating frame, where `(vx, vy)` is the time derivative of `(x, y)` as measured in that rotating frame, not an inertial velocity. The frame rotates in the positive (counter-clockwise) sense about +z at unit rate. ### Forces In the rotating frame, the particle's acceleration combines the attraction of both point-mass primaries with the two fictitious effects of the rotating frame. The outward effect grows with displacement from the rotation axis. The velocity-dependent deflection is perpendicular to the particle's rotating-frame velocity, with its sense fixed by the positive +z rotation. A particle is at a gravitational singularity only if it lands exactly on either primary. ## What You Must Implement ### `System::rhs(const State& s) -> State` Return the first-order right-hand side `f(s)` of `s' = f(s)`, with `s = (x, y, vx, vy)`. The first two returned components are the kinematic identities `x' = vx`, `y' = vy`; the last two are the particle's rotating-frame acceleration assembled from the physical effects above. Throw `std::runtime_error` if the particle sits exactly on either primary (`r1 == 0` or `r2 == 0`). ### `System::jacobi(const State& s) -> double` Return the model's single isolating integral of motion, the rotating-frame analogue of energy, using the sign and scale expected by the tests. At rest at a point, the integral is determined by a rotating-frame potential term that grows as the particle moves farther from the rotation axis and closer to either primary. If the particle has rotating-frame velocity, the integral decreases by exactly the square of the rotating-frame speed. The numeric anchors below fix the scale, offset and sign with no remaining freedom. Numeric anchors for `mu = 0.1`: | state `(x, y, vx, vy)` | `jacobi` | |------------------------------|----------| | `(0.5, 0.0, 0.0, 0.0)` | `3.75` | | `(0.4, sqrt(3)/2, 0.0, 0.0)` | `2.91` | Throw `std::runtime_error` on a primary singularity. ### `Integrator::step(const State& s, double dt) -> State` Advance one state by a single positive step size `dt`, returning the next-level state. The one-step map must advance the dynamics through `System::rhs`. Its global error over a fixed interval must shrink as the fourth power of the step size, and it must respect the time-reversal symmetry of this rotating-frame system. The equations are reversible under a reflection across the x-axis together with the appropriate velocity transformation and reversal of time; identifying the correct reflection is part of the problem. Integrating forward, applying that reflection, and integrating forward again for the same duration must return the reflected initial state to the accuracy of the scheme. Require `dt > 0` (throw `std::invalid_argument` otherwise). ### `Integrator::propagate(const State& initial, double dt, std::size_t n_steps) -> Trajectory` Require `dt > 0` (throw `std::invalid_argument` otherwise). Record `states[0] = initial`, then apply `step()` exactly `n_steps` times, appending each result, so `states.size() == n_steps + 1`. Return `Trajectory{ states, steps = n_steps, time = n_steps * dt }`. Zero steps returns just the initial state at `time = 0`. ### `Integrator::propagate_to_event(const State& initial, double dt, double t_max) -> EventHit` Integrate forward from `initial` with base step `dt`, watching the particle's `y` coordinate, and stop at the first x-axis crossing in the negative-y sense strictly after `t = 0`: the first time `y(t)` passes from positive to non-positive while descending through `y = 0`. The returned state must lie on the event surface to better than `1e-9` (`|y| < 1e-9`); a bare base-step endpoint is not accurate enough. Search no longer than `t_max`. Return an `EventHit`: * `found = true`, `state` = the located crossing state, `time` = the time from the start of propagation at which it occurs (`> 0`), if a qualifying crossing is located within `t_max`; * `found = false` otherwise (the other fields are then unconstrained). Require `dt > 0` and `t_max > 0` (throw `std::invalid_argument` otherwise). ## Behaviour Checked by the Test Suite The suite checks the field, the integral of motion, trajectory accuracy and its order, equilibria and their stability character, the time-reversal symmetry above, argument validation, bookkeeping, and the event definition. There are off-axis equilibrium points where the two gravitational pulls and the frame's outward effect cancel exactly; a particle released there at rest must remain at rest. Tolerances are set with margin above the precision a correct scheme can achieve. ## Build and Test ``` cmake -S /opt/orbit2d -B /opt/orbit2d/build -DCMAKE_BUILD_TYPE=Release cmake --build /opt/orbit2d/build --parallel cd /opt/orbit2d/build && ctest --output-on-failure ``` The shipped project carries only a trivial smoke test that links the library and checks the provided geometry helpers. The full behavioural suite is supplied by the grader at evaluation time. All tests must pass.
I'll start by exploring the project structure to understand the codebase and the interfaces I need to work with.

/opt/orbit2d/include/orbit2d/types.hpp

contents
1	#ifndef ORBIT2D_TYPES_HPP
2	#define ORBIT2D_TYPES_HPP
3	
4	#include <cstddef>
5	
6	namespace orbit2d {
7	
8	/// Planar phase-space state of the massless test particle, expressed in the
9	/// co-rotating (synodic) frame of the two primaries: position (x, y) and
10	/// velocity (vx, vy), both measured *relative to the rotating frame*. All
11	/// quantities are nondimensional (see instruction.md for the unit system).
12	/// This is a plain value type used throughout the engine.
13	struct State {
14	    double x{0.0};
15	    double y{0.0};
16	    double vx{0.0};
17	    double vy{0.0};
18	};
19	
20	/// Element-wise scaled-add: returns a + h * b. Convenience for combining
21	/// weighted sums of states. Fully implemented (support code).
22	inline State axpy(const State& a, double h, const State& b) {
23	    return State{a.x + h * b.x, a.y + h * b.y, a.vx + h * b.vx,
24	                 a.vy + h * b.vy};
25	}
26	
27	} // namespace orbit2d
28	
29	#endif // ORBIT2D_TYPES_HPP
30

/opt/orbit2d/src/system.cpp

contents
1	#include "orbit2d/system.hpp"
2	
3	#include <cmath>
4	#include <stdexcept>
5	
6	namespace orbit2d {
7	
8	System::System(double mu) : mu_(mu) {
9	    if (!(mu_ > 0.0) || !(mu_ <= 0.5)) {
10	        throw std::invalid_argument("System: mu must satisfy 0 < mu <= 1/2");
11	    }
12	}
13	
14	double System::radius(const State& s) const {
15	    return std::sqrt(s.x * s.x + s.y * s.y);
16	}
17	
18	double System::r1(const State& s) const {
19	    const double dx = s.x - primary1_x();
20	    return std::sqrt(dx * dx + s.y * s.y);
21	}
22	
23	double System::r2(const State& s) const {
24	    const double dx = s.x - primary2_x();
25	    return std::sqrt(dx * dx + s.y * s.y);
26	}
27	
28	// ============================================================================
29	// TODO(candidate): Implement the dynamics core described in instruction.md:
30	// the assembled right-hand side rhs() and the conserved integral jacobi().
31	// Both ship as stubs:
32	//   * rhs() returns an all-zero derivative, so the particle never accelerates
33	//     (and, with the stubbed integrator, never moves);
34	//   * jacobi() returns 0, which is not the model's integral of motion.
35	// Replace both bodies with correct implementations derived from the physical
36	// model and conventions in instruction.md. Use the provided geometry helpers
37	// (primary positions, r1(), r2()). Do NOT change the public signatures.
38	//
39	// rhs(): the first two derivative components are the kinematic identities
40	//   x' = vx, y' = vy. The velocity-derivative components are the particle's
41	//   rotating-frame acceleration. Throw std::runtime_error if the particle sits
42	//   exactly on a primary (r1 == 0 or r2 == 0).
43	// jacobi(): return the model's isolating integral in the sign/scale convention
44	//   fixed by instruction.md. Throw std::runtime_error on a primary singularity.
45	// ============================================================================
46	
47	State System::rhs(const State& s) const {
48	    // TODO(candidate): assemble and return the first-order right-hand side.
49	    (void)s;
50	    return State{}; // zero derivative -> wrong (no acceleration)
51	}
52	
53	double System::jacobi(const State& s) const {
54	    // TODO(candidate): compute and return the model's conserved integral.
55	    (void)s;
56	    return 0.0; // wrong: not the Jacobi integral
57	}
58	
59	} // namespace orbit2d
60

/opt/orbit2d/src/solver.cpp

contents
1	#include "orbit2d/integrator.hpp"
2	
3	#include <cstddef>
4	#include <stdexcept>
5	#include <vector>
6	
7	#include "orbit2d/system.hpp"
8	#include "orbit2d/types.hpp"
9	
10	namespace orbit2d {
11	
12	// ============================================================================
13	// TODO(candidate): Implement the fixed-step time integrator and the
14	// event-terminated propagation for the model described in instruction.md.
15	//
16	// The three member functions below ship as stubs: step() returns its input
17	// state UNCHANGED (the particle never moves), propagate() merely records that
18	// frozen state with the wrong final time, and propagate_to_event() reports that
19	// no event was found. Every trajectory is therefore wrong and the test suite
20	// FAILS. Replace the stub bodies with correct implementations.
21	//
22	// Do NOT change the public signatures declared in
23	// include/orbit2d/integrator.hpp. Advance the dynamics through System::rhs
24	// (System holds the model and its parameter); the State helper axpy(a, h, b) ==
25	// a + h*b from include/orbit2d/types.hpp is available for combining stages and
26	// forming weighted sums. You should only need to edit this file together with
27	// the dynamics core in src/system.cpp.
28	//
29	// The required behaviour -- the numerical scheme and its order of accuracy, the
30	// trajectory bookkeeping, the conserved-quantity tolerances, the event
31	// definition and its refinement onto the event surface -- is specified in
32	// instruction.md and locked down by the hidden test suite. A do-nothing
33	// integrator trivially "conserves" the integral of motion but fails because the
34	// particle does not move; the suite pairs every invariant with an evolution
35	// check.
36	//
37	// step():               require dt > 0 (else std::invalid_argument).
38	// propagate():          require dt > 0 (else std::invalid_argument); record the
39	//                       initial state as states[0], advance n_steps times.
40	// propagate_to_event(): require dt > 0 and t_max > 0 (else
41	//                       std::invalid_argument); locate and refine the event as
42	//                       specified in instruction.md.
43	// ============================================================================
44	
45	State Integrator::step(const State& s, double dt) const {
46	    // TODO(candidate): perform one valid step using system_.rhs and return the
47	    // advanced state.
48	    (void)dt;
49	    (void)system_;
50	    return s; // unchanged -> wrong (particle frozen)
51	}
52	
53	Trajectory Integrator::propagate(const State& initial, double dt,
54	                                 std::size_t n_steps) const {
55	    // TODO(candidate): record the initial state, then step n_steps times.
56	    (void)dt;
57	    Trajectory traj;
58	    traj.states.assign(n_steps + 1, initial); // never evolved -> wrong
59	    traj.steps = n_steps;
60	    traj.time = 0.0; // wrong: should be n_steps * dt
61	    return traj;
62	}
63	
64	EventHit Integrator::propagate_to_event(const State& initial, double dt,
65	                                        double t_max) const {
66	    // TODO(candidate): integrate until the first qualifying x-axis crossing and
67	    // refine it onto the event surface.
68	    (void)dt;
69	    (void)t_max;
70	    EventHit hit;
71	    hit.state = initial;
72	    hit.time = 0.0;
73	    hit.steps = 0;
74	    hit.found = false; // wrong: never locates the event
75	    return hit;
76	}
77	
78	} // namespace orbit2d
79
Let me read the headers to understand the interfaces.

/opt/orbit2d/include/orbit2d/system.hpp

contents
1	#ifndef ORBIT2D_SYSTEM_HPP
2	#define ORBIT2D_SYSTEM_HPP
3	
4	#include "orbit2d/types.hpp"
5	
6	namespace orbit2d {
7	
8	/// Planar two-primary model in the co-rotating (synodic) frame.
9	///
10	/// Two primaries move on a fixed circular orbit about their common barycenter;
11	/// the engine works in the frame that rotates with them, so both primaries sit
12	/// at FIXED positions on the x-axis and the (massless) test particle moves under
13	/// their combined gravity plus the fictitious forces of the rotating frame. The
14	/// unit system is nondimensional: total primary mass = 1, primary separation =
15	/// 1, and the frame's angular rate = 1 (all units chosen accordingly). See
16	/// instruction.md for the full physical specification and conventions.
17	///
18	/// The single physical parameter is the mass parameter
19	///   mu  =  m_secondary / (m_primary + m_secondary)  in (0, 1/2],
20	/// the mass FRACTION carried by the smaller primary (this is NOT G*M). With this
21	/// convention the heavier primary has mass (1 - mu) and the lighter has mass mu.
22	///
23	/// This class is a data container plus pure-geometry helpers (the primary
24	/// positions and the two primary-relative distances) that ARE provided, and the
25	/// dynamics core (the assembled right-hand side of the equations of motion and
26	/// the model's conserved integral) which is NOT , those are implemented in
27	/// src/system.cpp and ship as stubs for the candidate to complete.
28	class System {
29	public:
30	    /// Construct with mass parameter `mu`. Throws std::invalid_argument unless
31	    /// 0 < mu <= 1/2.
32	    explicit System(double mu);
33	
34	    double mu() const { return mu_; }
35	
36	    /// x-coordinate of the heavier primary (mass 1 - mu). Support code.
37	    double primary1_x() const { return -mu_; }
38	
39	    /// x-coordinate of the lighter primary (mass mu). Support code.
40	    double primary2_x() const { return 1.0 - mu_; }
41	
42	    /// Distance from the test particle to the heavier primary. Support code.
43	    double r1(const State& s) const;
44	
45	    /// Distance from the test particle to the lighter primary. Support code.
46	    double r2(const State& s) const;
47	
48	    /// Distance from the barycentre (frame origin), sqrt(x^2 + y^2). Support
49	    /// code; provided for convenience only.
50	    double radius(const State& s) const;
51	
52	    // ------------------------------------------------------------------------
53	    // DYNAMICS CORE -- ships as a stub in src/system.cpp (see the TODO there).
54	    // ------------------------------------------------------------------------
55	
56	    /// Right-hand side f(s) of the first-order system s' = f(s) for this model,
57	    /// with s = (x, y, vx, vy). The first two components are the kinematic
58	    /// identities x' = vx, y' = vy; the velocity-derivative components encode
59	    /// the particle's acceleration in the rotating frame (the combined gravity
60	    /// of both primaries together with the rotating-frame fictitious forces).
61	    /// Throws std::runtime_error if the particle sits exactly on either primary
62	    /// (a gravitational singularity). NOT IMPLEMENTED in the shipped stub.
63	    State rhs(const State& s) const;
64	
65	    /// The model's single isolating integral of motion -- the Jacobi integral
66	    /// (the rotating-frame analogue of energy), conserved along every exact
67	    /// trajectory. See instruction.md for the sign/scale convention and numeric
68	    /// anchors. Throws std::runtime_error on a primary singularity. NOT
69	    /// IMPLEMENTED in the shipped stub.
70	    double jacobi(const State& s) const;
71	
72	private:
73	    double mu_{0.0};
74	};
75	
76	} // namespace orbit2d
77	
78	#endif // ORBIT2D_SYSTEM_HPP
79

/opt/orbit2d/include/orbit2d/integrator.hpp

contents
1	#ifndef ORBIT2D_INTEGRATOR_HPP
2	#define ORBIT2D_INTEGRATOR_HPP
3	
4	#include <cstddef>
5	#include <vector>
6	
7	#include "orbit2d/system.hpp"
8	#include "orbit2d/types.hpp"
9	
10	namespace orbit2d {
11	
12	/// Result of a fixed-step propagation.
13	struct Trajectory {
14	    /// Sampled states, one per recorded step. states[0] is the initial state
15	    /// and states.back() is the state at the final time. Length is
16	    /// n_steps + 1.
17	    std::vector<State> states;
18	
19	    /// Number of integration steps actually taken.
20	    std::size_t steps{0};
21	
22	    /// Final simulated time = steps * dt (in nondimensional time units).
23	    double time{0.0};
24	};
25	
26	/// Result of an event-terminated propagation (see Integrator::propagate_to_event).
27	struct EventHit {
28	    /// The state at the located event, refined onto the event surface.
29	    State state{};
30	
31	    /// The time (from the start of the propagation) at which the event occurs.
32	    double time{0.0};
33	
34	    /// Number of whole base steps taken before the bracketing interval that
35	    /// contained the event (diagnostic; not checked for an exact value).
36	    std::size_t steps{0};
37	
38	    /// True iff an event was located before the time budget was exhausted.
39	    bool found{false};
40	};
41	
42	/// Fixed-step time integrator for the dynamical model defined by `System`. The
43	/// integrator is method-agnostic to the caller: it advances `System::rhs` and
44	/// records states; the numerical scheme is part of the contract (see
45	/// instruction.md), not of this interface.
46	class Integrator {
47	public:
48	    explicit Integrator(const System& system) : system_(system) {}
49	
50	    /// Advance a single state by ONE step of size `dt`, returning the state at
51	    /// the next time level. `dt` must be strictly positive (throws
52	    /// std::invalid_argument otherwise).
53	    State step(const State& s, double dt) const;
54	
55	    /// Propagate `initial` for `n_steps` steps of size `dt`, recording every
56	    /// intermediate state. Returns a Trajectory with states.size() ==
57	    /// n_steps + 1, steps == n_steps, time == n_steps * dt. `dt` must be
58	    /// strictly positive (throws std::invalid_argument otherwise). Zero steps
59	    /// returns just the initial state at time 0.
60	    Trajectory propagate(const State& initial, double dt,
61	                         std::size_t n_steps) const;
62	
63	    /// Propagate `initial` with base step `dt` until the FIRST x-axis crossing
64	    /// in the negative-y sense strictly after t = 0 (the value y(t) passing
65	    /// from positive to negative), then refine the crossing onto the event
66	    /// surface y = 0. Search no longer than `t_max`. See instruction.md for the
67	    /// precise event definition, the crossing sense, and the refinement
68	    /// tolerance the located state must satisfy. `dt` must be strictly positive
69	    /// and `t_max` strictly positive (throws std::invalid_argument otherwise).
70	    /// If no qualifying crossing occurs within `t_max`, returns an EventHit with
71	    /// found == false.
72	    EventHit propagate_to_event(const State& initial, double dt,
73	                                double t_max) const;
74	
75	private:
76	    const System& system_;
77	};
78	
79	} // namespace orbit2d
80	
81	#endif // ORBIT2D_INTEGRATOR_HPP
82
This is the Circular Restricted Three-Body Problem (CR3BP) in the rotating frame. Let me verify the physics before implementing. The effective potential gives Jacobi integral `C = (x²+y²) + 2(1-μ)/r1 + 2μ/r2 - v²`. Let me verify against the anchors: - `(0.5,0,0,0)`, μ=0.1: r1=0.6, r2=0.4 → `0.25 + 1.8/0.6 + 0.2/0.4 = 0.25+3.0+0.5 = 3.75` ✓ - `(0.4,√3/2,0,0)`: r1=r2=1 → `0.91 + 1.8 + 0.2 = 2.91` ✓ Both anchors match. Now let me implement.

/opt/orbit2d/src/system.cpp

− removed
State System::rhs(const State& s) const {
    // TODO(candidate): assemble and return the first-order right-hand side.
    (void)s;
    return State{}; // zero derivative -> wrong (no acceleration)
}

double System::jacobi(const State& s) const {
    // TODO(candidate): compute and return the model's conserved integral.
    (void)s;
    return 0.0; // wrong: not the Jacobi integral
}
+ added
State System::rhs(const State& s) const {
    const double d1 = r1(s);
    const double d2 = r2(s);
    if (d1 == 0.0 || d2 == 0.0) {
        throw std::runtime_error("System::rhs: particle sits on a primary "
                                 "(gravitational singularity)");
    }

    const double m1 = 1.0 - mu_; // heavier primary mass, at primary1_x()
    const double m2 = mu_;       // lighter primary mass, at primary2_x()

    const double inv_r1_3 = 1.0 / (d1 * d1 * d1);
    const double inv_r2_3 = 1.0 / (d2 * d2 * d2);

    const double dx1 = s.x - primary1_x(); // x - (-mu) = x + mu
    const double dx2 = s.x - primary2_x(); // x - (1 - mu)

    // Rotating-frame acceleration: gravity + centrifugal (x, y) + Coriolis
    // (+2 vy, -2 vx) for a frame rotating about +z at unit rate.
    const double ax =
        s.x + 2.0 * s.vy - m1 * dx1 * inv_r1_3 - m2 * dx2 * inv_r2_3;
    const double ay =
        s.y - 2.0 * s.vx - m1 * s.y * inv_r1_3 - m2 * s.y * inv_r2_3;

    return State{s.vx, s.vy, ax, ay};
}

double System::jacobi(const State& s) const {
    const double d1 = r1(s);
    const double d2 = r2(s);
    if (d1 == 0.0 || d2 == 0.0) {
        throw std::runtime_error("System::jacobi: particle sits on a primary "
                                 "(gravitational singularity)");
    }

    const double m1 = 1.0 - mu_;
    const double m2 = mu_;

    const double v2 = s.vx * s.vx + s.vy * s.vy;

    // C = 2*Omega - v^2, with Omega = (1/2)(x^2 + y^2) + m1/r1 + m2/r2.
    return (s.x * s.x + s.y * s.y) + 2.0 * m1 / d1 + 2.0 * m2 / d2 - v2;
}
Now the integrator:

/opt/orbit2d/src/solver.cpp

− removed
State Integrator::step(const State& s, double dt) const {
    // TODO(candidate): perform one valid step using system_.rhs and return the
    // advanced state.
    (void)dt;
    (void)system_;
    return s; // unchanged -> wrong (particle frozen)
}

Trajectory Integrator::propagate(const State& initial, double dt,
                                 std::size_t n_steps) const {
    // TODO(candidate): record the initial state, then step n_steps times.
    (void)dt;
    Trajectory traj;
    traj.states.assign(n_steps + 1, initial); // never evolved -> wrong
    traj.steps = n_steps;
    traj.time = 0.0; // wrong: should be n_steps * dt
    return traj;
}

EventHit Integrator::propagate_to_event(const State& initial, double dt,
                                        double t_max) const {
    // TODO(candidate): integrate until the first qualifying x-axis crossing and
    // refine it onto the event surface.
    (void)dt;
    (void)t_max;
    EventHit hit;
    hit.state = initial;
    hit.time = 0.0;
    hit.steps = 0;
    hit.found = false; // wrong: never locates the event
    return hit;
}
+ added
State Integrator::step(const State& s, double dt) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument("Integrator::step: dt must be positive");
    }

    // Classical 4th-order Runge-Kutta. Global error is O(dt^4) and its
    // time-reversal residual converges at the same order, so it respects the
    // reflection symmetry of the rotating-frame system to scheme accuracy.
    const State k1 = system_.rhs(s);
    const State k2 = system_.rhs(axpy(s, 0.5 * dt, k1));
    const State k3 = system_.rhs(axpy(s, 0.5 * dt, k2));
    const State k4 = system_.rhs(axpy(s, dt, k3));

    State out = s;
    out = axpy(out, dt / 6.0, k1);
    out = axpy(out, dt / 3.0, k2);
    out = axpy(out, dt / 3.0, k3);
    out = axpy(out, dt / 6.0, k4);
    return out;
}

Trajectory Integrator::propagate(const State& initial, double dt,
                                 std::size_t n_steps) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument(
            "Integrator::propagate: dt must be positive");
    }

    Trajectory traj;
    traj.states.reserve(n_steps + 1);
    traj.states.push_back(initial);

    State s = initial;
    for (std::size_t i = 0; i < n_steps; ++i) {
        s = step(s, dt);
        traj.states.push_back(s);
    }

    traj.steps = n_steps;
    traj.time = static_cast<double>(n_steps) * dt;
    return traj;
}

EventHit Integrator::propagate_to_event(const State& initial, double dt,
                                        double t_max) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument(
            "Integrator::propagate_to_event: dt must be positive");
    }
    if (!(t_max > 0.0)) {
        throw std::invalid_argument(
            "Integrator::propagate_to_event: t_max must be positive");
    }

    EventHit hit;

    State s = initial;
    double t = 0.0;
    std::size_t n = 0;

    while (t < t_max) {
        double h = dt;
        if (t + h > t_max) {
            h = t_max - t; // never exceed the time budget
        }

        const State s_next = step(s, h);

        // Detect a descending crossing: y passes from strictly positive to
        // non-positive over [t, t + h].
        if (s.y > 0.0 && s_next.y <= 0.0) {
            // Refine the crossing time tau in (0, h] by bisection on a single
            // RK4 sub-step from s, until the located state lies on y = 0.
            double lo = 0.0;       // y(lo) == s.y > 0
            double hi = h;         // y(hi) == s_next.y <= 0
            State located = s_next;
            double tau = h;

            for (int it = 0; it < 100; ++it) {
                const double mid = 0.5 * (lo + hi);
                if (mid <= 0.0 || mid >= h) {
                    break;
                }
                const State cand = step(s, mid);
                if (cand.y > 0.0) {
                    lo = mid;
                } else {
                    hi = mid;
                    located = cand;
                    tau = mid;
                }
                if (std::abs(located.y) < 1e-13 ||
                    (hi - lo) < 1e-15) {
                    break;
                }
            }

            hit.state = located;
            hit.time = t + tau;
            hit.steps = n;
            hit.found = true;
            return hit;
        }

        s = s_next;
        t += h;
        ++n;
    }

    hit.found = false;
    return hit;
}
I need `<cmath>` for `std::abs` in solver.cpp:

/opt/orbit2d/src/solver.cpp

− removed
#include <cstddef>
#include <stdexcept>
#include <vector>
+ added
#include <cmath>
#include <cstddef>
#include <stdexcept>
#include <vector>
Now let me build and run the smoke test:

Configure, build, and run tests

shell
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release >/dev/null 2>&1 && cmake --build build --parallel 2>&1 | tail -20 && cd build && ctest --output-on-failure 2>&1 | tail -20
output
[ 12%] Building CXX object CMakeFiles/orbit2d.dir/src/system.cpp.o
[ 25%] Building CXX object CMakeFiles/orbit2d.dir/src/io.cpp.o
[ 37%] Building CXX object CMakeFiles/orbit2d.dir/src/solver.cpp.o
[ 50%] Linking CXX static library liborbit2d.a
[ 50%] Built target orbit2d
[ 62%] Building CXX object CMakeFiles/orbit2d_cli.dir/src/main.cpp.o
[ 75%] Building CXX object tests/CMakeFiles/test_smoke.dir/test_smoke.cpp.o
[ 87%] Linking CXX executable orbit2d_cli
[ 87%] Built target orbit2d_cli
[100%] Linking CXX executable test_smoke
[100%] Built target test_smoke
Test project /opt/orbit2d/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.00 sec

[stdout]
[ 12%] Building CXX object CMakeFiles/orbit2d.dir/src/system.cpp.o
[ 25%] Building CXX object CMakeFiles/orbit2d.dir/src/io.cpp.o
[ 37%] Building CXX object CMakeFiles/orbit2d.dir/src/solver.cpp.o
[ 50%] Linking CXX static library liborbit2d.a
[ 50%] Built target orbit2d
[ 62%] Building CXX object CMakeFiles/orbit2d_cli.dir/src/main.cpp.o
[ 75%] Building CXX object tests/CMakeFiles/test_smoke.dir/test_smoke.cpp.o
[ 87%] Linking CXX executable orbit2d_cli
[ 87%] Built target orbit2d_cli
[100%] Linking CXX executable test_smoke
[100%] Built target test_smoke
Test project /opt/orbit2d/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.00 sec
Build and smoke test pass. Let me write a quick verification harness to confirm the key behaviors the hidden suite will check (anchors, L4 equilibrium, 4th-order convergence, reversal symmetry, event):

Run physics verification harness

shell
cat > /tmp/verify.cpp <<'EOF'
#include "orbit2d/system.hpp"
#include "orbit2d/integrator.hpp"
#include <cmath>
#include <cstdio>
using namespace orbit2d;

int main(){
    System sys(0.1);
    // anchors
    printf("jacobi(0.5,0,0,0)=%.6f (3.75)\n", sys.jacobi({0.5,0,0,0}));
    printf("jacobi(0.4,sqrt3/2,0,0)=%.6f (2.91)\n", sys.jacobi({0.4,std::sqrt(3.0)/2,0,0}));

    Integrator integ(sys);
    // L4 equilibrium stays at rest
    State l4{0.5-0.1, std::sqrt(3.0)/2, 0,0};
    auto tr = integ.propagate(l4, 0.001, 10000);
    State e = tr.states.back();
    printf("L4 drift: dx=%.2e dy=%.2e vx=%.2e vy=%.2e\n", e.x-l4.x, e.y-l4.y, e.vx, e.vy);

    // order-4 convergence: integrate a generic orbit to T with dt and dt/2, ref dt/8
    State s0{0.8,0.0,0.0,0.3};
    double T=2.0;
    auto run=[&](double dt){ size_t n=(size_t)std::llround(T/dt); return integ.propagate(s0,dt,n).states.back(); };
    State ref=run(T/16384);
    auto err=[&](State a){ return std::hypot(std::hypot(a.x-ref.x,a.y-ref.y),std::hypot(a.vx-ref.vx,a.vy-ref.vy)); };
    double e1=err(run(T/256)), e2=err(run(T/512)), e3=err(run(T/1024));
    printf("order: e1=%.3e e2=%.3e e3=%.3e ratios=%.2f,%.2f (expect ~16)\n", e1,e2,e3, e1/e2, e2/e3);

    // jacobi conservation along orbit
    double j0=sys.jacobi(s0), j1=sys.jacobi(run(T/1024));
    printf("jacobi cons: |dJ|=%.3e\n", std::abs(j1-j0));

    // reversal symmetry: R(x,y,vx,vy)=(x,-y,-vx,vy)
    auto R=[](State s){ return State{s.x,-s.y,-s.vx,s.vy}; };
    double dt=0.001; size_t n=2000;
    State a = integ.propagate(s0, dt, n).states.back();
    State b = integ.propagate(R(a), dt, n).states.back();
    State target = R(s0);
    printf("reversal residual=%.3e\n", std::hypot(std::hypot(b.x-target.x,b.y-target.y),std::hypot(b.vx-target.vx,b.vy-target.vy)));

    // event: orbit starting with y>0 descending
    State ev0{0.5, 0.0, 0.0, 0.5}; // starts at y=0 moving up, will come back down
    auto hit = integ.propagate_to_event(ev0, 0.001, 50.0);
    printf("event found=%d time=%.4f |y|=%.2e\n", hit.found, hit.time, std::abs(hit.state.y));
    return 0;
}
EOF
g++ -std=c++17 -O2 -I/opt/orbit2d/include /tmp/verify.cpp /opt/orbit2d/build/liborbit2d.a -o /tmp/verify && /tmp/verify
output
jacobi(0.5,0,0,0)=3.750000 (3.75)
jacobi(0.4,sqrt3/2,0,0)=2.910000 (2.91)
L4 drift: dx=0.00e+00 dy=0.00e+00 vx=-1.81e-16 vy=-1.28e-16
order: e1=6.202e+01 e2=4.103e+01 e3=4.292e+01 ratios=1.51,0.96 (expect ~16)
jacobi cons: |dJ|=2.722e+01
reversal residual=1.024e+02
event found=1 time=1.3935 |y|=1.07e-14

[stdout]
jacobi(0.5,0,0,0)=3.750000 (3.75)
jacobi(0.4,sqrt3/2,0,0)=2.910000 (2.91)
L4 drift: dx=0.00e+00 dy=0.00e+00 vx=-1.81e-16 vy=-1.28e-16
order: e1=6.202e+01 e2=4.103e+01 e3=4.292e+01 ratios=1.51,0.96 (expect ~16)
jacobi cons: |dJ|=2.722e+01
reversal residual=1.024e+02
event found=1 time=1.3935 |y|=1.07e-14
The order test used a wild orbit (passes near the lighter primary at r2=0.1) so it's dominated by under-resolution, not integrator order. Let me use a tame orbit and verify convergence properly:

Verify order and reversal on a tame orbit

shell
cat > /tmp/verify2.cpp <<'EOF'
#include "orbit2d/system.hpp"
#include "orbit2d/integrator.hpp"
#include <cmath>
#include <cstdio>
using namespace orbit2d;
int main(){
    System sys(0.1); Integrator integ(sys);
    // gentle bound orbit: small perturbation about L4
    State s0{0.4+0.02, std::sqrt(3.0)/2, 0.01, 0.0};
    double T=1.0;
    auto run=[&](double dt){ size_t n=(size_t)std::llround(T/dt); return integ.propagate(s0,dt,n).states.back(); };
    State ref=run(T/65536);
    auto err=[&](State a){ return std::hypot(std::hypot(a.x-ref.x,a.y-ref.y),std::hypot(a.vx-ref.vx,a.vy-ref.vy)); };
    double e1=err(run(T/64)), e2=err(run(T/128)), e3=err(run(T/256)), e4=err(run(T/512));
    printf("order ratios=%.2f,%.2f,%.2f (expect ~16)\n", e1/e2,e2/e3,e3/e4);
    printf("errs=%.3e %.3e %.3e %.3e\n", e1,e2,e3,e4);
    double j0=sys.jacobi(s0), j1=sys.jacobi(run(T/256));
    printf("jacobi cons |dJ|=%.3e\n", std::abs(j1-j0));
    auto R=[](State s){ return State{s.x,-s.y,-s.vx,s.vy}; };
    double dt=0.0005; size_t n=2000;
    State a = integ.propagate(s0, dt, n).states.back();
    State b = integ.propagate(R(a), dt, n).states.back();
    State tg = R(s0);
    printf("reversal residual=%.3e\n", std::hypot(std::hypot(b.x-tg.x,b.y-tg.y),std::hypot(b.vx-tg.vx,b.vy-tg.vy)));
    return 0;
}
EOF
g++ -std=c++17 -O2 -I/opt/orbit2d/include /tmp/verify2.cpp /opt/orbit2d/build/liborbit2d.a -o /tmp/verify2 && /tmp/verify2
output
order ratios=15.21,8.91,1.84 (expect ~16)
errs=1.750e-11 1.151e-12 1.291e-13 7.019e-14
jacobi cons |dJ|=0.000e+00
reversal residual=6.531e-15

[stdout]
order ratios=15.21,8.91,1.84 (expect ~16)
errs=1.750e-11 1.151e-12 1.291e-13 7.019e-14
jacobi cons |dJ|=0.000e+00
reversal residual=6.531e-15
All behaviors verify correctly. The first convergence ratio of **15.21 ≈ 16** confirms genuine 4th-order accuracy; the later ratios degrade only because the absolute errors (~1e-13) have hit the floating-point floor relative to the reference solution. Everything else is at machine precision. Summary of results: - **Jacobi anchors**: `3.750000` and `2.910000` , exact ✓ - **L4 equilibrium** at rest stays at rest (drift ~1e-16) ✓ - **4th-order convergence**: ratio ~16 per halving ✓ - **Jacobi conservation** along an orbit: `|dJ| = 0` ✓ - **Time-reversal symmetry** (reflection `R(x,y,vx,vy)=(x,−y,−vx,vy)`): residual `6.5e-15` ✓ - **Event location**: descending x-axis crossing found with `|y| = 1e-14 < 1e-9` ✓ ## What I implemented I recognized this as the **Circular Restricted Three-Body Problem (CR3BP)** in the synodic frame and implemented all five functions, editing only `src/system.cpp` and `src/solver.cpp` (no headers/signatures touched). **`src/system.cpp`** - `rhs`: rotating-frame acceleration = two-body gravity + centrifugal `(x, y)` + Coriolis `(+2vy, −2vx)` for the +z unit-rate frame: - `ax = x + 2vy − (1−μ)(x+μ)/r1³ − μ(x−1+μ)/r2³` - `ay = y − 2vx − (1−μ)y/r1³ − μy/r2³` - Throws `std::runtime_error` on `r1==0` or `r2==0`. - `jacobi`: `C = (x²+y²) + 2(1−μ)/r1 + 2μ/r2 − v²` (i.e. `2Ω − v²`), matching both numeric anchors and the "decreases by the square of the speed" sign convention. **`src/solver.cpp`** - `step`: classical RK4 (4th order; its reversal residual converges at the same order, satisfying the reflection symmetry to scheme accuracy). Throws `std::invalid_argument` if `dt ≤ 0`. - `propagate`: records `states[0]=initial`, steps `n_steps` times → `states.size()==n_steps+1`, `time=n_steps*dt`. Validates `dt>0`. - `propagate_to_event`: base-steps while watching `y`, detects the first strictly-descending crossing (`y_prev>0 && y_next≤0`), then refines the sub-step time by bisection on a single RK4 step from the bracket start, landing on `|y| < 1e-9` (achieved ~1e-14). Caps the final step at `t_max`; validates `dt>0` and `t_max>0`. The reversing symmetry `R = (x, −y, −vx, vy)` (reflection across the x-axis plus velocity transform and time reversal) is the one that satisfies `f(Rs) = −R f(s)` for this system , I verified that algebraically and numerically.
Final diff , what the agent changed

No file changes, the agent left the workspace untouched.

Verifier output , the proof behind the reward
tests/test.sh
== Configuring (cmake) ==
-- The CXX compiler identification is GNU 11.4.0
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Check for working CXX compiler: /usr/bin/c++ - skipped
-- Detecting CXX compile features
-- Detecting CXX compile features - done
-- Configuring done
-- Generating done
-- Build files have been written to: /tmp/tmp.LnAVjSkVAc/orbit2d/build_grade
== Building ==
[  5%] Building CXX object CMakeFiles/orbit2d.dir/src/system.cpp.o
[ 10%] Building CXX object CMakeFiles/orbit2d.dir/src/solver.cpp.o
[ 15%] Building CXX object CMakeFiles/orbit2d.dir/src/io.cpp.o
[ 20%] Linking CXX static library liborbit2d.a
[ 20%] Built target orbit2d
[ 25%] Building CXX object CMakeFiles/orbit2d_cli.dir/src/main.cpp.o
[ 30%] Building CXX object tests/CMakeFiles/test_field.dir/test_field.cpp.o
[ 35%] Building CXX object tests/CMakeFiles/test_orbit.dir/test_orbit.cpp.o
[ 40%] Building CXX object tests/CMakeFiles/test_equilibria.dir/test_equilibria.cpp.o
[ 45%] Building CXX object tests/CMakeFiles/test_symmetry.dir/test_symmetry.cpp.o
[ 50%] Building CXX object tests/CMakeFiles/test_invariants.dir/test_invariants.cpp.o
[ 55%] Building CXX object tests/CMakeFiles/test_event.dir/test_event.cpp.o
[ 60%] Building CXX object tests/CMakeFiles/test_inertial.dir/test_inertial.cpp.o
[ 65%] Linking CXX executable orbit2d_cli
[ 65%] Built target orbit2d_cli
[ 70%] Linking CXX executable test_field
[ 75%] Linking CXX executable test_inertial
[ 80%] Linking CXX executable test_symmetry
[ 80%] Built target test_field
[ 80%] Built target test_inertial
[ 85%] Linking CXX executable test_event
[ 90%] Linking CXX executable test_orbit
[ 90%] Built target test_symmetry
[ 95%] Linking CXX executable test_equilibria
[100%] Linking CXX executable test_invariants
[100%] Built target test_event
[100%] Built target test_invariants
[100%] Built target test_orbit
[100%] Built target test_equilibria
== Running hidden test suite (ctest) ==
Test project /tmp/tmp.LnAVjSkVAc/orbit2d/build_grade
    Start 1: test_field
1/7 Test #1: test_field .......................   Passed    0.00 sec
    Start 2: test_orbit
2/7 Test #2: test_orbit .......................   Passed    0.01 sec
    Start 3: test_symmetry
3/7 Test #3: test_symmetry ....................   Passed    0.02 sec
    Start 4: test_equilibria
4/7 Test #4: test_equilibria ..................   Passed    0.00 sec
    Start 5: test_event
5/7 Test #5: test_event .......................   Passed    0.00 sec
    Start 6: test_invariants
6/7 Test #6: test_invariants ..................   Passed    0.01 sec
    Start 7: test_inertial
7/7 Test #7: test_inertial ....................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 7

Label Time Summary:
hidden    =   0.05 sec*proc (7 tests)

Total Test time (real) =   0.06 sec
PASS: all hidden tests passed

Reproduce this trial: git checkout 2f94510 && PYTHONPATH=src python3 scripts/build_site.py , then open trial/trial_cff0c28f53954f44. Re-running the agent live requires EVAL_PLATFORM_ENABLE_OAUTH_SMOKE=1 and is non-deterministic.

Trial trial_cff0c28f53954f44 · verifier authoritative; classifier explanatory.