Two orthogonal parabolas form a maximal Sidon set in F_q^3

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_proof: Pair sums of the two parabolas are classified by which outer coordinates vanish, the non-square separates the two kinds at height zero, and three explicit formulas block every point outside the set.

evidence/: Recompute the size, strong Sidon property, maximality and proof witnesses of the two-parabola set for every non-square in each small odd field.

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This author-recorded result is due to Plasma AI. The proof is elementary and complete; the finite check recomputes every case with $q\le 27$ and $q=49$.

Significance. The statement is the analogue of Problem 156 in the group $\mathbb F_q^3$ of odd order $q^3$, and it answers that analogue at the cube-root scale. In a group of odd order an element $x\notin A$ can only be blocked as $a+b-c$ or as a midpoint of $A+A$, so a maximal strong Sidon set of $k$ elements has $q^3\le k+k(k+1)^2/2$, that is $k\ge(2^{1/3}-o(1))\,q$. The set $A$ has $2q-1$ elements, within a factor $2^{2/3}$ of this bound. The construction is explicit and uses no probability, so it carries no logarithmic loss of the kind in [[../library/additive_bases/ruzsa_1998_small_maximal_sidon_set/theorem_p55|Ruzsa's integer theorem, whose logarithmic factor is the benchmark for Problem 156]].

Mechanism. The difference set $P-P$ consists of the origin and the plane $z=0$ outside the line $x=0$, and $C$ has exactly one element at each height $z$. Hence for $x_1\ne0$ the equation $x=C(u)+P(t)-P(w)$ has exactly one solution: this mixed triple map is birational onto the space. The missing line $x=0$ of each height lies in the plane $x=0$, where $C-C$ plays the role of $P-P$. The non-square $\mu$ makes the sums of $C$ at height $0$, the points $(0,2\mu u^2,0)$, avoid the sums $(0,2t^2,0)$ of $P$; with a square $\mu$ the set is not Sidon. The same mechanism with a perfect difference set in place of the parabola gives the cyclic analogue [[theory/additive_bases/L209_singer_level_maximal_sidon/_index|L209, a greedy maximal Sidon set of size p+h in Z_q x H]].

Scope. Characteristic two is excluded: every double is zero, so a strong Sidon set in an elementary abelian $2$-group has at most one element. Under the different convention that omits doubles, [[../library/additive_bases/redman_2021_small_maximal_sidon_set_z_2/theorem_3_1|Redman, Rose and Walker's theorem]] gives an upper bound with a logarithmic loss for small maximal Sidon sets in those groups. Nothing is asserted about integer intervals: the [[research/small_maximal_sidon_attack/integer_transfer|integer transfer analysis]] records the additional carry conditions for applying the mechanism in $\mathbb Z$, and Problem 156 itself remains open. A limited web search on 2026-10-15 found no earlier construction of maximal Sidon sets of this size in $\mathbb F_q^3$; it does not establish novelty.

Standing. Tier 0. No independent review is recorded. Verification consists of reading the proof: the three-case uniqueness of pair sums and the three explicit witness formulas. The evidence program also checks those witness formulas and the role of the non-square hypothesis on finite cases; it 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]].