Topological String Theory of Light and the Photon: MultiLevel String Summation, Boundary Wilson–Maxwell Condensation, and Photon–Gluon Holonomy Duality

In the main treatise, we demonstrated that massive fermions (leptons and quarks) are hetero-chiral open strings bridging distinct chiral boundaries across the excised interstitial Hopf sphere of the four-dimensional tripartite cobordism, whereas gravitons are boundaryless closed string loops whose coherent condensation weaves the curved spacetime metric itself. This Supplementary Material establishes the rigorous mathematical formulation of the Topological String Theory of Light and the Photon. We prove that the physical photon and macroscopic electromagnetic radiation are governed by a three-tier summation of open homochiral boundary strings: (i) an internal Fourier–Virasoro harmonic summation along a single open string with Neumann endpoints sliding at the speed of light on the three-dimensional membrane, whose Fokker–Planck–Kolmogorov difusivity variance converges to the Weil–Petersson modular volume; (ii) an algebraic trace summation across the symmetric metric-gauge sector of Eddington's double-frame Cliford spinor algebra; and (iii) an external Glauber–Polyakov coherent summation of open boundary strings that generates the macroscopic Maxwell field tensor as the boundary dual of closed-string gravitational condensation. Finally, we prove the exact Photon–Gluon Holonomy Duality Theorem, demonstrating why wrapping the inverse electromagnetic coupling around a complete closed holonomy cycle on three color D3-braness transforms the electron conversion key directly into the chromodynamic string tension of the eight gluon flux tubes.