Greedy Singer-level sets for twelve small cases

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From the repository root run:

uv run --no-sync python wiki/theory/additive_bases/L209_singer_level_maximal_sidon/evidence/main.py

For each prime $p\in\{13,17,19,23,29,31,37,41,43,47,53\}$ the program builds a Singer set $B$ modulo $q=p^2+p+1$ from the trace-zero powers of a primitive element of $\mathbb F_{p^3}$, and checks directly that its differences cover every non-zero residue exactly once. As a control it checks that $B$ alone is maximal strong Sidon in $\mathbb Z_q$. For twelve pairs $(p,H)$ with $H$ of odd order $h\le p/4$, eleven cyclic and $H=\mathbb Z_3\times\mathbb Z_3$ for $p=37$, it runs the greedy rule of the proof, confirms that a value is always available, and verifies by brute force that the resulting set has $p+h$ elements and is strong Sidon and maximal in $\mathbb Z_q\times H$. A further control replaces the level map by a constant and confirms that the set is then not Sidon. Finally it checks two retained level maps outside the proved range, for $(p,h)=(5,7)$ and $(11,13)$ with cyclic $H$, which were found by a randomized backtracking search and are recorded in the program: both give maximal strong Sidon sets. They are finite observations and do not extend the statement.

All arithmetic is exact. The program uses the repository's evidence harness and the Python standard library; it runs in a few seconds and has no quick mode. Any failed obligation gives a nonzero exit, also under optimized Python. The check covers these cases only; the statement for every prime power $p$ and every admissible $H$ rests on the proof.