#773: conditional exact values through a(71) = 33
The target (erdosproblems.com/773, OPEN). Let a(N) be the largest size of a Sidon subset of {12,...,N2}: every unordered sum i2+j2, including the diagonal i=j, must be different. Erdős asked whether
a(N)=N1-o(1).
We do not settle that asymptotic question. We extend its exact finite frontier by three places.
The result. Conditional on the full published exact OEIS A390813 prefix through N=68,
a(69)=32, a(70)=33, and a(71)=33.
The condition is substantive and explicit. The N=69 frozen proof uses the published upper bounds a(k) for every remaining prefix 1,...,k with k≤68; it does not independently re-prove those earlier values. The final published value a(68)=32 first forces any hypothetical 33-set at N=69 to contain 692. The computation then fixes index 69 and asks whether 32 more indices can be chosen from 1,...,68. The N=71 computation additionally depends on the resulting certified a(69)=32 and a(70)=33.
An exact finite model. A collision is an equality
i2+j2=k2+l2
between different unordered pairs. After fixing 69, the checker reconstructs all 712 minimal forbidden supports: 21 triples and 691 quadruples. A selected index set is square-Sidon exactly when it contains none of them. There is no floating-point arithmetic in this reduction or in the search.
The exhaustive upper search. At each node, three independently audited rules bound the remaining extension: the frozen published prefix value a(k), vertices forced out by an almost-complete forbidden support, and a clique partition of the live pair-conflict graph. The last rule is valid because a Sidon extension can take at most one vertex from each conflict clique.
The assignments to (x68,x67,x66) split into the following seven disjoint cubes:
00, 010, 011, 100, 101, 110, 111.
The first cube includes both possible values of x66, so the list is also exhaustive. Every cube returned UNSATISFIABLE. In total the proof search visited 171,546,643 recursive states and made 167,258,147 conflict-coloring calls. Therefore no 33-set exists at 69, relative to the frozen OEIS prefix.
The lower certificates. The following 32 indices give 528 pairwise distinct unordered square sums at N=69:
3, 4, 6, 12, 13, 15, 17, 19, 23, 25, 28, 34, 35, 38, 42, 45, 48, 49, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69.
At N=70, these 33 indices give 561 distinct sums:
1, 4, 6, 7, 9, 10, 12, 13, 16, 18, 19, 22, 24, 28, 30, 34, 36, 37, 39, 40, 43, 46, 49, 51, 52, 57, 58, 60, 64, 65, 67, 69, 70.
Both are unconditional integer certificates. Deleting index 70 from any candidate shows a(70)≤a(69)+1, so the second witness gives equality once the conditional value at 69 is established.
The next exact value. Conditional on that certified frontier through 70, a(71)=33. A hypothetical 34-set at 71 must contain index 71, since a(70)=33. Fixing 71 leaves exactly 33 indices to choose from 1,...,70. The new checker reconstructs all 764 collision supports at N=71 (23 triples and 741 quadruples) and exhausts the same seven prefix cubes for (x70,x69,x68). All return UNSATISFIABLE, after 114,296,406 states in total. The displayed N=70 witness remains a valid 33-set at N=71, proving equality under the stated dependency.
A separate definition-level Python audit directly compares every pair of unordered square pairs rather than sharing the C++ support builder. It reconstructs the 764 forbidden supports, checks all 561 witness sums, proves the cube cover exhaustive, and totals and checks every recorded counter. A separately assigned hostile audit then rebuilt the checker and reran all seven cubes from source, reproducing every deterministic counter.
Audit posture. The bundle freezes the exact OEIS b-file and its hash, the proof source, all seven completed logs, witnesses, and independent checkers. The primary audit rebuilds the optimized executable byte-for-byte and reconstructs the supports, cube cover, logs, and witnesses. One hostile audit then reran all seven cubes from source, reproduced every counter exactly, and passed a sanitizer plus incremental-state check on a deep subcube. A second independent hostile audit reviewed the proof and artifact package separately before publication.
Scope. These are rigorous computational extensions conditional on the stated exact prefix, not a claim that the OEIS prefix has been re-certified here and not progress on the exponent in Erdős's question. The live OEIS table stopped at 68 on 12 August 2026, and the live problem still showed zero claimed proofs and no worker or collaborator. That is dated search evidence only: no absolute novelty or priority claim is made for a(71). The site also records a July CODEX attempt on #773 that found no proof; this is a new finite round, not our first encounter with the problem.