Limitations and future work

Scope of the model

• The impact law is kinematic Newton restitution plus a tangential retention factor, not a full Coulomb friction cone. Sticking and sliding contact modes are out of scope, and trajectories entering them terminate with status=2 rather than being silently integrated.

• Gradients are exact for the continuous hybrid system at fixed event topology. Across bounce-count boundaries the objective is discontinuous, which is inherent to contact and visible in E1.

• Event detection resolves terrain features wider than |vx|·dt/3, documented on the input schema. That rule is necessary, not sufficient: what has to be resolved is the width of the window where the guard is negative, and near tangency that width scales as the square root of the penetration depth regardless of how wide the bump is. A grazing contact of depth 1e-6 is missed at step sizes more than an order of magnitude below what the rule suggests, and the run reports status = 0 with one fewer impact rather than signalling anything. Optimizing bump amplitudes can steer a design toward tangency, so this is reachable rather than theoretical.

• Grazing impacts carry an inherent δ^(-1/2) sensitivity growth near tangency. The solver raises if the saltation denominator falls below 1e-12, but that denominator scales as δ^(1/2) and δ cannot fall below the guard’s own floating-point resolution, so on metre-scale terrain it floors near 1e-7, five orders above the threshold. Tangency therefore never raises. What you get instead is a large finite gradient, measured growing from 54 to 8e6 as δ falls from 1e-2 to 1e-13; then, for grazes shallow enough that the rebound falls under v_stop, a status = 2 report; and below that a silently missed event at status = 0.

• The Jacobian is discontinuous across the status boundary as well as across bounce-count changes. A run that ends at t_final differentiates the state at a fixed time; a run truncated at the event budget differentiates it at a parameter-dependent event time. Straddling that boundary, qf moves continuously while its Jacobian can flip sign. The two cases are distinguishable only through status, which a client has to check.

• The tangential law removes a fixed fraction μ of tangential velocity at every impact, independent of the normal impulse. That is not merely “less than a full Coulomb cone”: the implied friction coefficient grows without bound as the normal velocity goes to zero, so near-grazing and settling contacts are the regime where it is least physical. The Routh reset in future work is what fixes this.

• Two dimensions, single body. The saltation machinery is dimension-agnostic; the scope is a deliberate trade for verified correctness inside the hackathon period.

Honest caveats on the results

• The separator’s low-restitution particle is stopped by the event budget rather than by coming to rest, so its separation surface is “position after eight impacts”, not “position at rest”. A fixed-length chute imposes an analogous cut, though not the identical one.

• The wall-clock ordering reversed when we made our own gradient cheaper: Adam is ahead on all five seeds under both accountings now. That is evidence about this implementation, not about gradient methods against gradient-free ones. Both accountings are reported in studies.

• The gradient is not demonstrated to be the better search method on this problem. At a matched budget on the E5b ensemble objective, tuned CMA-ES reaches a better design than Adam, and on the E5 held-out purity Nelder-Mead matches Adam with a wider margin. What buys the robustness is optimising the ensemble objective, not the gradient that optimises it. See studies.

• The separator is defined at its own horizon: evaluated at t_final = 3.0 instead of 2.2, the trained e = 0.8 particle lands 0.67 m from its target rather than 0.42 mm.

• Designing environment geometry through contact-driven simulation is not new in itself. What we could not find in the literature is design using exact event-time sensitivities and targeting per-impact routing rather than a bulk flow statistic.

Known rough edges in the implementation

These were found by audit rather than by a failing test, and are recorded because none of them is currently guarded:

• A degenerate crossing raises a Julia exception rather than returning a status code. Tangency does not reach it, for the reason given above, and neither does chatter: the 1e-13 lift applied after each reset so detection re-arms also floors the approach speed at 1.4e-6, which is where a settling sequence’s denominator stops falling. What does reach it is a launch height small enough that the first impact speed sqrt(2 g y0) falls under 1e-12. y0 = 1e-26 is accepted by the schema and raises instead of returning a status, and y0 is checked only for finiteness. v_stop = 0 is separately a legal input that makes the settling branch unreachable, so a run that would have settled reports status = 1 at the event budget.

• A second guard rejects a crossing whose velocity is not approaching. It fires only when the fixed RK4 step amplifies the drag mode instead of damping it, so the located crossing is an artifact of the integrator rather than of the trajectory. Bisecting the threshold gives drag * dt = 2.7898 at dt = 1e-3 and 2.7859 at dt = 1e-4, converging on the RK4 stability limit 2.785293563. The schema rejects drag * dt past that limit, so this one is a backstop for direct Julia use rather than a reachable client failure.

• The trajectory history is accumulated on every step even when n_samples = 0, which is what every gradient call uses, and the step count is unbounded. A request with a very small dt and a long t_final will consume memory proportional to the step count.

• t_end == t_final holds to floating-point accumulation, not exactly, so a client comparing them should use a tolerance. The solver’s own status field is the reliable signal.

Future work

Reverse-mode saltation adjoint. This used to head the list, on the grounds that forward-variational VJP cost scales with parameter count and was measured at 93 microseconds per parameter. That figure turned out to be an artifact of the implementation rather than of the method: the flow is affine in the state, so the RK4 variational update collapses to a fixed tangent map that can be composed across a smooth segment and applied only at events. With that factoring the slope is 15 microseconds per parameter and a VJP at 581 parameters costs 1.8x a forward solve, so the parameter-count argument for an adjoint is largely gone.

It remains the right extension for large state dimension. The tangent map is 4x4 here; in three dimensions with multiple bodies it is not, and a backward costate integration with Sᵀ jumps at events would then be the cheaper side of the trade. That is the justification to keep, and it is a different one from the one we started with.

Coulomb friction cone. The impulse-ratio (Routh) reset is a small change to the reset map, at the cost of re-deriving the closed-form oracle, and would extend the model into sticking and sliding contact.

Three dimensions and multiple bodies. The work is in guard geometry and event bookkeeping, not in the sensitivities.

Upstream. Test the tesseract-jax fix for issue 234 against this solver once it lands.