q3 — certify the fifth decimal on the {b,1} ray

Pure Liu Example 4 certifies frequency 0.38305 on {b,1}. Thus

0.38305 > 0.38304 > 0.382709087918741.

The first inequality improves the q1 ray record. The second compares with Liu's published Example 5 number. The analytic first-crossing is unchanged:

0.3830513565868255825….

The printed constant is 1.3565868e-6 below that crossing. A 9,000 by 7,000 mesh checks every retained point with mean at most 0.38305. The Python verifier uses monotonicity to inspect the last retained point in each row; the C verifier independently visits every retained point in the 63,000,000-point grid. Their retained-cell counts, minimum ratios, and minimizers agree. The mesh minimum is 1.0000049143029008, recorded in certs/certificate.json.

Replay:

PY=/path/to/python ./run_all.sh

Python needs mpmath; the mesh and comparison use the standard library. run_all.sh also compiles verify.c with gcc -O3.

probe_joint.py records a failed route past the q2 ceiling. For independent bits, the shared-union pair (A union B, A union C) already has an interior entropy equality near 0.343708, so that joint-output inequality is weaker than the current ray protocol.

The scope is the {b,1} hypothesis class used in q1 and Liu's numerical optimizer. Frankl's 1/2 target and the every-measure inequality remain open.

Acknowledgements

Codex (GPT-5) ran the q3 search and certificate. Stephen Wu is the human author of the notebook.