A Finite Obstruction to Parameter-Stable Extension in Strongly Regular Graphs

This short note exhibits a concrete finite obstruction to parameter-stable extension in strongly regular graphs. Using the Paley graph P(13)P(13)P(13) as an explicit algebraic witness with parameters (13,6,2,3)(13,6,2,3)(13,6,2,3), we verify that the standard feasibility identity holds at n=13n=13n=13. However, attempting to extend the same local parameters to n=14n=14n=14 produces an immediate arithmetic contradiction in the feasibility equation (n−d−1)μ=d(d−λ−1). Consequently, no strongly regular graph with parameters (14,6,2,3)(14,6,2,3)(14,6,2,3) can exist. A spectral verification is also provided. The eigenvalue discriminant associated with the parameter set equals 131313, which is not a perfect square, implying that the required eigenvalue and multiplicity conditions for a strongly regular graph cannot be satisfied. This independently rules out the existence of such a graph. The argument is elementary, finite, and fully checkable. The example illustrates that even highly symmetric algebraic graph constructions need not admit parameter-preserving growth when attempting to extend from size nnn to n+1n+1n+1. Version 2.2 adds a spectral cross-check confirming the obstruction via eigenvalue constraints.