HOW THE RIT GENERATES THE STANDARD MODEL: DERIVATION OF THE BI-CAYLEY JORDAN TRIPLE AND THE CHAIN OF SYMMETRIES FROM THE RIGHT ISOSCELES TRIANGLE

We present a geometric derivation of the algebraic structures recently obtained by Baez, Bokor, and Boyle (2026) in their paper "Jordan Pair Quantum Theory and the Standard Model," published in the journal Azimuth. Working within the ontology of the Right Isosceles Triangle (RIT) as the universal generator of mathematical reality, we demonstrate that the bi-Cayley triple, the exceptional Jordan algebra, the tripotent formalism, the Peirce decomposition into six fermion types, and the chain of symmetry breakings E₆ ⊃ Spin(10) ⊃ SU(5) ⊃ G_SM all follow rigorously from the axioms of RIT mosaics and their symmetries. We provide explicit formulas linking each algebraic construct to a geometric configuration of RITs. The 16-dimensional spinor representation of Spin(10) is shown to be the space of states of four RITs, each with two possible orientations. The six nonvanishing Peirce spaces are identified with the six ways of attaching an RIT to two fixed axes. This work establishes that the Standard Model gauge group and one generation of fermions are not accidental features of nature but necessary consequences of a single geometric atom: the right isosceles triangle with unit legs.