Modular Invariants of the Exceptional Series (3,4,5,7), Combinatorial Derangement Defects, and Projective Embeddings of the Fano Plane

This paper establishes a rigorous algebraic linkage within the series of exceptional discrete states n in {3, 4, 5, 7}. We prove that the combinatorial singularity of the first-order discrete difference operator governing partial derangements maps uniquely to the isolated bridge n=3. By evaluating multi-scale projective coverings under the constraints of the Bruck–Ryser–Chowla theorem at the non-existent order q=12, we derive the exact algebraic superorder I = 157 as a modular defect invariant. Finally, we demonstrate that the known non-trivial integer solutions to the Brocard–Ramanujan Diophantine equation (n! + 1 = m^2) are strictly bounded and isomorphically encoded by the geometric capacity limits and representation submodules of the Fano plane PG(2,F_2) inside the symmetric group S_7.