two agreeing points are not a derivation

Between two representations of the same algebraic object there is a conversion, and ours had been calibrated rather than derived. Two corresponding points had been measured by hand, and the conversion was fitted through them. It reproduced the reference implementation exactly at those points, which is what let it survive review.

Everything between them was interpolation, and the reference did not interpolate. The fit was right where we looked and wrong everywhere else, and the wrongness first appeared in a later step with no obvious relation to the conversion, so the search started in the wrong place.

The replacement derives the map from the structure instead of sampling it: the correspondence follows from the algebra, leaving no free parameters to be lucky about. The change also forced a correction in how a second coordinate is recovered, because the previous code produced it correctly for some inputs and with the wrong sign for others — hidden until then, since the fit only ever exercised one branch.

We did not fix that last part. It is recorded as an open inconsistency, in the change record and in the task list, with the condition under which it appears. That is the discipline the episode argues for. A calibration that agrees with its reference on the points you chose is a measurement of your choice of points; a derivation has nothing left to agree with but the mathematics, and a defect it exposes is one you can name.