A residual factor of 3⁻³⁴ in reconstructing the corner of Nissim's SO(3) deconfinement tube, and computational obstructions to closing it

Nissim's proof that SO(3) lattice Yang-Mills theory fails Wilson's confinement criterion at strong coupling rests on an explicit closed-form evaluation of a "tube" cluster in the cluster expansion, obtained by integrating out edges via a second-moment identity and reducing the remainder to a chain of third-moment identities around the loop. While cross-checking this value by an independent, direct combinatorial reconstruction of the tube — rather than by invoking the closed form — we found a residual discrepancy of exactly 3^-34 between our reconstruction and Nissim's Lemma 3.8, constant across every loop size tested and reproduced identically by four algorithmically independent computational methods. We localize this discrepancy precisely to the corner geometry of the tube: Nissim's construction uses 4 plaquettes per corner, while the natural reconstruction — giving each of the two loop edges meeting at a corner its own complete, independent ring of 4 plaquettes — uses 12. We then report two exhaustive negative results obtained while searching for the correct 4-plaquette corner: first, a proof that no single plaquette can be shared between the two loop edges at a corner without either leaving edges unresolved or reproducing exactly the "pinched corner" configuration that Nissim's own lemma shows vanishes; second, a computational constraint search ruling out, with certainty, every other local 4-plaquette patch drawn from a generous neighborhood of the corner. The discrepancy itself remains open; we record what we found so that it need not be rediscovered.