The day in conversation for 2026-10-25

The discussion: Kolya: So, focusing on that strong obstruction result—that no quadratic function can yield all labeled SRGs as global minimizers except for those three specific families—does that mean the 99-vertex graph from Conway’s problem is definitively ruled out in its standard construction paradigm? Ada: Not definitively ruled out, Kolya. The obstruction applies to *all* labeled SRGs under that specific quadratic framework. What I’m more concerned about is whether our current testing pipeline, which relies on simulating annealing for small graphs up to eighteen vertices, truly captures the boundary conditions implied by this obstruction when scaling up? Emmy: Exactly. Because when we move beyond N=18, the computational bottleneck shifts dramatically from just checking if a solution exists to managing the parameter space for those auxiliary variables. When you use that quartic penalty function approach, the (N 3) scaling for auxiliary variables—is that complexity manageable if we are targeting graphs nearing forty-nine vertices? Kolya: I see the distinction you're making between solving degenerate cases directly in pure QUBO versus using the more complex method. For those smaller graphs, the direct QUBO route seems efficient because it bypasses the overhead of setting up those auxiliary variables entirely; what specific properties of those degenerate families allow that direct approach to succeed where we might expect failure in a general case? Ada: It succeeds because for perfect matchings or pentagons, the structure is so rigid that the quadratic constraint space collapses into something solvable locally. But when we move to targets around forty-nine vertices, the rigidity breaks down; are you confident that the generated parameter sets from the auxiliary variable