Paper CVI: Chapter R Uniqueness of the Quark-to-Node Assignment and the Six- Fold Identity of e7

The quark-to-node assignment (u = e1, d = e3, c = e2, s = e6, t = e4, b = e5) used throughout Papers CICV is proven to be unique a theorem that was already implicit in the framework but not previously stated. The uniqueness chain has six steps. The rst three (xing e7 from cosmology, xing generation lines from the Fano structure, and ordering generations by H+/H−mass hierarchy) are unambiguous. Step four (up- quark collinearity, Paper CI) reduces 8 possible within-generation assignments to 4. The remaining 4-fold freedom is shown to be exactly the unbroken SU(2)L gauge symmetry: the 4 assignments are not 4 dierent physical situations they are 1 physical situation expressed in 4 gauge choices. The Higgs mechanism (Paper XXIX: Fano condensation in the e7 direction, ⟨ϕ⟩= vˆe7) breaks SU(2)L and xes the gauge, selecting exactly 1 assignment. The mechanism is: the G2 generator Le7 (= W −boson, Paper CIII) acts on the generation lines as the lowering operator T−= T1 −iT2, mapping e7 × e1 = +e3, e7 × e2 = +e6, e7 × e4 = +e5 uniquely identifying (e1, e3), (e2, e6), (e4, e5) as the (u, d), (c, s), (t, b) doublets respectively. Combined with Q = T3 + Y/2 (Gell-Mann Nishijima) and Y = +1/3 for quarks, this gives Qu,c,t = +2/3 and Qd,s,b = −1/3 the observed quark charges, without free parameters. The node e7 is identied as having a six-fold physical identity: (1) sump/time direction, (2) W-boson rotation axis, (3) 0- hop CKM mediator, (4) preferred entropy microstate, (5) black-hole entropy direction, and now (6) Higgs condensate axis. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.