Leg 301 screened fourteen candidate function spaces against three separately-measured ways this project's certification machinery has died before, and found exactly one that survives all three on paper: a rational Hardy basis (Malmquist–Takenaka), where the Hilbert transform is exactly diagonal instead of exactly zero on the diagonal, which is where the incumbent broke. But 301's argument only used the bare differentiation matrix: a clean, closed-form object with nothing else attached. This leg's job was to stop reasoning on paper and check the real thing: solve the actual target, build its actual coupled linear operator, and see what it looks like once it's moved into the new coordinates.
First attempt gave the wrong answer, and the check that caught it is worth repeating. The first run showed the operator's conditioning collapsing toward zero as the truncation grew, which would have read as a clean kill. A control (build the same transform with no operator in it at all: it should be the identity) showed the identical collapse. That meant the numerical grid, not the target, was the problem: it was too coarse to represent the basis functions being tested against it. A denser, verified grid removed the collapse entirely.
On the corrected, verified grid, across an eightfold range of truncations, the real operator's diagonal stays flat and nonzero, its growth rate tracks the theory's requirement, and its conditioning does not degrade with truncation: the sharpest test this repository's own history has for this failure mode, and it did not fire. The exact-diagonal Hilbert-transform claim held up on the code's own machinery, not just on paper. And the piece nobody had measured (what the nonlinear (quadratic) term does to this basis) was measured directly on the real target's exact quadratic remainder, and it stayed bounded across two orders of magnitude of input frequency, with the one caveat (the highest-frequency test hit the edge of the projection window used) reported honestly rather than smoothed into a clean pass.
None of this builds a certificate, and it does not lift the ban that has stood since three earlier attempts at this same certification died. It answers the specific question that ban's own lift condition asked for: a scoping leg, on the fourth candidate, checking it against each of the three ways this has died before. All three checks read "does not recur," with the nonlinearity's evidence measured, not assumed, and the cost of actually building a validated transform for this basis, which exists nowhere in the literature, priced at roughly six to nine more legs of work.
What happens next is not this leg's call. The result is packaged and handed to the user, who alone can rule on whether the ban's lift condition has actually been met. Nothing is built past this point, and the odds on the underlying Clay problem have not moved: they stay at roughly 0.05%, exactly where they were before this leg started.