Part 31 found the abstraction-in-loop residual loses on PandaPickCubeOrientation — peak success 0.320/0.344 vs PPO 0.805/0.836 — because its analytic grasp is a fixed top-down grasp at a fixed yaw, the wrong prior for a random-orientation target, and a residual that only corrects a fixed primitive cannot synthesize an arbitrary reorientation. Part 31 closed with the obvious follow-up: make the prior right for the task — feed the per-episode target orientation into the analytic controller — and see whether that recovers performance toward PPO. If it does, the limitation is the fixedness of the prior, not abstraction itself, and parametrized priors are the route to reusable abstractions. Part 33 runs exactly that experiment.

The change — a parametrized (orientation-aware) prior

The Part 31 controller’s analytic IK held a fixed desired end-effector rotation R_home (top-down, level) through grasp→lift→place. The orientation-aware controller instead rotates that desired frame by the per-episode target orientation:

R_des = R_target @ R_home     (applied from the GRASP phase onward)

where R_target = quat_to_mat(data.mocap_quat[target]) is the random per-episode target rotation (the same quantity the box_target reward’s rot term scores against). Because the cube is grasped rigidly, rotating the gripper to R_des carries the cube to the target orientation. Two more pieces keep it honest and Markov:

  • Markov conditioning: the per-episode target quaternion (4 dims) is appended to the residual’s observation (77 → 81 dims), exactly as the controller phase z was appended in Parts 26-27. The base 66-dim obs only carried the relative rot-error; the residual now also sees the absolute target it must achieve.
  • Everything else identical to Part 31: α fixed at 1.0, the same reach→descend→grasp→lift→place machine, budget-matched (~56M env-steps, brax-rounded from a 35M nominal request), eval n=256, success = box_target ≥ 0.9.

The result

Real success (box_target ≥ 0.9, n=256):

arm peak success final success
PPO (Part 31) seed 1 / seed 2 0.805 / 0.836 0.785 / 0.836
fixed-prior residual (Part 31) seed 1 / seed 2 0.320 / 0.344 0.320 / 0.320
orientation-aware residual (Part 33) seed 1 / 2 / 3 0.621 / 0.695 / 0.688 0.606 / 0.652 / 0.648
orientation-aware controller-alone (α=0, sanity) 0.023 —

Mean peak ~0.67 — double the fixed-prior 0.33, and ~70% of the way to PPO’s 0.82 ((0.67−0.33)/(0.82−0.33) ≈ 0.69). Making the prior orientation-aware recovers most of the gap that Part 31 opened. The controller-alone is only 0.023 — open-loop it does almost nothing — so the residual still does the heavy lifting, but only once the prior points the right way.

Why — prior-fixedness was the dominant limiter (but not the whole story)

The same in-loop-residual class that failed at 0.33 with a fixed prior reaches ~0.67 once the prior is parametrized by the task’s degree of freedom (target yaw). That ~2× lift localizes the Part 31 failure: it was the prior being fixed, not the use of abstraction. But the recovery stops ~0.15 short of PPO’s 0.82 — a residual correcting an analytic grasp cannot fully match end-to-end learning on the random-orientation task. There is a real residual gap that the analytic-controller-plus-residual class does not close, even with the right parametrization.

What this means — parametrized vs fixed priors

  • Reusability axis (the headline): abstractions become reusable when their priors are parametrized to the task’s degrees of freedom, not hard-coded to one regime. A fixed prior is brittle (Part 31: 0.33, worse than learning-from-scratch); a parametrized prior is a genuine asset (0.67, recovering most of the gap).
  • Honest-positive, not a clean win: parametrization gets ~70% of the way back, not 100%. The ceiling of “analytic prior + Markov residual” sits below end-to-end RL on hard tasks — consistent with the whole campaign (ties when the prior fits, trails when the task is hard for it).
  • Stability bonus: the orientation-aware residual drops only ~0.03-0.04 peak→final, whereas the fixed PickCube residual peaked ~0.79 and collapsed (the “fragile optimum”, Part 28). A prior matched to the task makes the learned correction small and stable instead of large and brittle — so the right prior buys both performance and stability.

Verdict

Parametrize the prior, and the abstraction recovers most of what the fixed prior lost (0.33 → ~0.67 of PPO’s 0.82) and trains more stably — so the route to reusable abstractions is parametrized, not fixed, priors. But it does not fully close to learning-from-scratch. This is the cleanest statement of the reusability axis the campaign has produced: the value of an abstraction is conditional on its prior matching the task’s variation, parametrization is how you make that match, and even then a residual on an analytic skill trails end-to-end RL by a task-dependent margin.

Caveats

3 seeds (orientation-aware residual) vs Part 31’s 2 (PPO, fixed residual). Budget-matched at ~56M env-steps. “Success” is the same box_target ≥ 0.9 metric as Parts 27-31, with per-episode random target orientation. α fixed at 1.0 (the variant that ties on PickCube). The orientation-aware controller rotates the desired EE frame by R_target from the grasp phase onward and conditions the residual obs on the target quaternion (obs 77→81); all else matches Part 31. Data: exp/tdmpc_glass/generalization_PandaPickCubeOrientation_oriaware/{residual_eval_s1,residual_eval_s2,residual_eval_s3,controller_alone_eval}.json; Part 31 baselines: exp/tdmpc_glass/generalization_PandaPickCubeOrientation/{ppo_eval_s*,residual_eval_s*}.json. Prior: Part 31 (the fixed-prior loss + this follow-up’s design), Parts 27/30 (the tie and the orientation ceiling).