A Singer set with one point per level is maximal Sidon in Z_q x H

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_proof: Maximality follows from the perfect difference property at every level; the strong Sidon property reduces to three families of conditions on the level map, and a greedy count satisfies them when h <= p/4.

evidence/: Build Singer sets for primes from 13 to 53, run the greedy rule of the proof for odd groups of order at most p/4, and verify maximality and the strong Sidon property by brute force.

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This author-recorded result is due to Plasma AI. The proof is elementary and complete; the finite check runs the proof's greedy rule for twelve pairs $(p,H)$ and verifies the resulting sets by brute force.

Significance. Perfect difference sets of the required kind exist for every prime power $p$ by [[../library/additive_bases/singer_1938_theorem_finite_projective_geometry_some_applications_number_theory/theorem_p380|Singer's theorem, which supplies the set B]]. Choosing a cyclic $H$ of odd order $h\le p/4$ coprime to $q$ with $h\ge p/8$, which is possible for every large $p$, gives cyclic groups $\mathbb Z_{qh}$ of order between about $p^3/8$ and $p^3/4$ with maximal strong Sidon sets of at most $5p/4$ elements, that is $O(|G|^{1/3})$. In a group of odd order the counting bound recorded in [[theory/additive_bases/L208_two_parabola_maximal_sidon/_index|L208, the non-cyclic analogue in F_q^3]] shows that the cube-root order cannot be improved. The construction answers the cyclic-group analogue of Problem 156 for these orders only; it gives nothing for other cyclic orders or for integer intervals.

Mechanism. Maximality is automatic: the point $(c(s),s)$ of level $s\ne0$ and the differences of $B$, which cover every non-zero residue, block every other element of that level, and level $0$ is blocked by $B+B-B$. The map $c$ only has to keep $A$ Sidon, and three families of conditions do that. The worst of them excludes about half of $\mathbb Z_q$ for each value $c(x)$, because $c(x)+c(-x)$ must avoid $B+B$; the others exclude $O(hp)$ residues, and a greedy choice therefore succeeds while $h$ is at most about $0.3p$. The constant is not optimized: the evidence program also checks retained valid maps with $(p,h)=(5,7)$ and $(11,13)$, where $h$ exceeds $p$.

Relation to the integers. The integer analogue of this set is the anchored Singer lift, with $B$ at height $0$ and one point $c(s)+qs$ at each height $s$. There the residue sum of a blocking triple can wrap past $q$, and the corresponding integer is then off by a multiple of $q$. That carry defeats the automatic maximality; the [[research/small_maximal_sidon_attack/integer_transfer|integer transfer analysis]] records the precise failure.

Standing. Tier 0. No independent review is recorded. Verification consists of reading the proof, chiefly the count of excluded residues in the greedy step. The evidence program checks the greedy rule and the conclusion on finite cases and does not replace the proof. There is no remaining obligation inside the stated scope. The attack this result serves is recorded in the [[research/small_maximal_sidon_attack/proof_sketch|proof sketch for the small maximal Sidon problem]].