A search algorithm can stop improving without finding the best possible answer. This candidate identifies a matrix problem where that cannot happen, and then shows how fragile the guarantee can be. The quantity being minimised is the Perron eigenvalue: the dominant growth factor of a positive matrix. Think of the off-diagonal entries as paired interactions. Each pair has a fixed total, while a direction of bias redistributes that total between its two entries. The key assumption is factorisation. Every pair strength comes from one positive weight attached to each of its two members. Under this assumption, every local minimum has a simple form. The members can be placed in a complete order, and each pair is biased as far as allowed in the direction specified by that order. Every such order gives the same global minimum. The proof compresses all these possibilities into one polynomial. The same ordering also remains optimal when the weights, diagonal entries and bias vary within the permitted model. For a rectangular uncertainty range, the worst optimal growth factor occurs at one specified corner. This gives a test for the existence of a fixed robustly stable orientation. It is not a guarantee for arbitrary switching between matrices. There is a sharp warning. In a four-member example, changing just one pair strength by one percent separates twenty-four strict local minima into two spectral levels. Sixteen are worse than the other eight. More generally, arbitrarily small departures from factorisation can create such traps. The archive provides written proofs, exact checks, numerical experiments and replay code. Publication testing found tiny numerical differences despite matching random seeds; these are now checked with explicit tolerances, while exact checks retain exact equality. This remains an unrefereed candidate, not an independently verified theorem or an empirically validated model. This is Evidence Press, Perron minima, released on the twenty-eighth of September twenty twenty-six, version one point zero point one candidate. The paper and evidence are linked on this page. This synthetic AI voice is an explanation, not additional scientific evidence.