Transport Papers T13. Quark and Lepton Families

T13 of the Transport Papers. Version 1.0. 27 pages. Epistemic status: Tier C. A massive unit is two twistors and the maximal composite three (a bound imported at observation level), so there are three faces, with identical fermion content exchanged by S_3 and mixing by shared slots; that the faces are the generations is a conjecture. The mass bracket is fixed by the exact join, 2|B_Λ|² = m² - Λ C/3, with a Λ-linear source; its magnitude cannot reach the weak scale (the thirteenth refutation), and under the indifference principle its direction is democratic, which under the Gram-square reading gives one heavy family and a Cabibbo check passed. The second half confronts the Koide relation. With √(m) as the candidate linear coordinate, Q_√(m) = 2/3 says the family vector is null for the Minkowski member of the S_3-invariant forms; the framework does not force that member, and the 2π/3 phases are Fourier coordinates, not a derivation. The Gaussian computation is proved in full: the pairing sector of the six-mode face algebra is one Bogoliubov event, the mass form is exactly rank one plus an ordering-shift floor with the light pair degenerate at every strength, and Koide survives only at a depth that is its own number transcribed. The displaced-Gaussian class is solved, by theorem for its pairings and by flagged computation for the rest; the split, the angle, and the depth are one open problem. The paper is Tier C; its lepton sector carries Hypothesis (S); every identity, the Wick evaluation included, is verified in exact arithmetic by a deposited script. Position in the series: Within the series it relies on T01, T08, T09, T11, T12. The series is surveyed in T17, The Transport Programme, whose status table records every paper's tier and check count. Verification: verify_T13.py checks every computational identity in the paper in exact rational arithmetic (73 checks, all passing at deposit). It imports GA.py, the series' shared exact-arithmetic library, deposited alongside so the record is self-contained. Run with a bare Python 3: python verify_T13.py. Files: T13_quark_lepton_families.pdf; T13_quark_lepton_families.tex; verify_T13.py; GA.py; LICENSE-CODE.txt. Licence: the paper (PDF and LaTeX source) is CC BY 4.0. The code files in this record are additionally released under the MIT licence; see LICENSE-CODE.txt. Authorship and disclosure. The founding idea of the series is the author's: that division by zero in a Clifford algebra should be read not as a prohibition but as a transport into a higher-dimensional algebra, and that the transported division need not be accompanied by an inverse operation. The direction of investigation and the critical review of the results are likewise the author's. The mathematical development and the drafting of the text were carried out in extended collaboration with Claude Fable 5 (Anthropic), an AI system.