## Export binary ball endpoints with exact_rational(), not QQ(real)
What happened: T707 reused a checker whose endpoint serializer called QQ(v)
on Sage RealNumber endpoints. The runtime probe found QQ(v) differed from
v.exact_rational() by about 1e-217 at 720 bits. The computation's balls were
correct, but the checkpoint fractions were approximations to their endpoints.
What the skill says now: Preserve numerical evidence and check stored intervals;
use exact representations for exact values. It does not mention this conversion.
What it should say: When serializing binary floating-point endpoints, use an
explicit exact conversion such as Sage RealNumber.exact_rational(); QQ(real)
may choose a simpler nearby rational. Alternatively preserve an outward-rounded
ball encoding. Do not call approximate rational reconstructions exact endpoints.
Evidence: endpoint-probe.log, probe_endpoints.py, checks-executed.py,
checks.py and serialize.py for T707 in this build's artifact directory. The
original evidence remains preserved; corrected endpoints are used for readback.