The Fano Plane at the Intersection of Division Algebras, Calibrated Geometry, and Coding Theory

The Fano plane PG(2, F_2) simultaneously encodes three apparently unrelated mathematical structures: (i) the multiplication table of the octonion algebra O, (ii) the G_2 calibration 3-form on R^7, and (iii) the stabilizers of the Steane [7,1,3] quantum error-correcting code. We prove that this triple role is unique: the Fano plane is the only Steiner triple system (n >= 7) whose incidence data simultaneously defines a normed division algebra and a perfect binary code (Fano Universality Theorem). We show that the same incidence data determines the Harvey-Lawson calibration 3-form with stabilizer G_2 (Fano-Calibration-Code Correspondence), operating over distinct base fields (R vs F_2). We prove that matroid regularity and algebraic associativity are equivalent for Steiner triple systems, with both holding only at n = 3. These results are organized into a convergence catalog of mathematical arguments for the distinguished role of d = 7, each graded on a four-level epistemic scale (R1-R4). The pair (d_c, N_gen) = (7, 3) is the unique intersection of six mathematical constraints and the global minimum of the combined chi^2 = 2.63 across 8 fermion mixing observables (zero free parameters, p = 0.96), a factor ~370 better than the nearest competitor. Independence analysis identifies 2-3 genuinely independent threads.