Why we tried it
Seek larger compatible square-sum constructions and turn the search into algebraic-curve questions.
What the evidence shows
The manuscripts retain exact integer/rational certificates and bounded search ranges; finite boxes do not exhaust all rational points.
Where the work stands
The existing construction is not the required 5×5 solution, and curve decompositions and bounded searches leave the global rational-point problem open.
Possible next steps
Replay the certificates and negative controls, then isolate the missing rational-point, infinitude or completeness argument; a search timeout is not an exclusion.
Suggested next steps are untested research questions, not new experimental results. The archive records manuscript provenance, licensing and status.