# Per-case fixed-topology refit interpretation

## Kármán

The frozen joint topology does not acquire one monotonic coefficient trend with Reynolds number. The rear constant remains the most stable element: its magnitude is 3.58, 4.18, 3.41 and 2.55 for Re 50, 100, 200 and 400, respectively. The rear lift coefficient is weak at Re50 (0.22), peaks at Re200 (1.70), and decreases at Re400 (1.05). The front drag-asymmetry coefficient varies non-monotonically (0.58, 0.32, 0.50, 0.29).

This supports a common steady rear-rotation backbone, but not a simple low-Re/high-Re two-group coefficient law. The independent broad-discovery inventory reaches the same conclusion: Re50 favours simple front-force/affine structures, Re400 repeatedly selects `Cl_rear_s-Cl_tot`, and Re100/Re200 are heterogeneous lift-based intermediates. Re400 is distinct chiefly through a smaller rear constant and poorer rear-head offline fit, consistent with the loss of joint-law closed-loop performance. Re50 is also structurally special because its rear lift feedback is nearly absent. The data therefore suggest non-monotonic retuning plus distinct endpoint signatures rather than two clean clusters.

## Illusion

The frozen four-term topology changes signs and relative weights between diameters. At 0.75L the front action is dominated by negative front-lift feedback and the rear refit is poorly identified. At 1L the front coefficients reverse relative to the joint formula and the rear topology has strongly negative R2, indicating that this fixed rear family is not a useful open-loop action surrogate for that single trajectory. At 1.5L the rear coefficients become an order of magnitude larger and the front action is almost purely positive front-lift feedback.

Broad discovery makes the diameter progression clearer than the frozen-topology coefficients alone: 0.75L is rear-drag dominated (`Cd_rear_a` 15 and `Cd_rear_s` 10 occurrences), 1L is mixed/transitional (`Cd_F` 13 and `Cl_rear_s` 12), and 1.5L is overwhelmingly lift-contrast dominated (`Cl_F` 54 and `Cl_rear_s` 46). The endpoint families are therefore clearly distinct, while 1L is a genuine transition rather than a member of either clean group. Coefficient sign changes still show that one joint static formula is an averaging compromise rather than a universal target-tracking law.

## Scope

These per-case refits are diagnostics on PPO-visited trajectories. They are not promoted to controllers without separate closed-loop CFD validation. Negative R2 values are retained because they are useful evidence that the frozen topology is insufficient for a particular head/case.
