Three Fermion Generations as CP/T-Asymmetric Identity Transformation: The CKM Lower Bound, Irreversible Pruning, and the Minimal Flavor Architecture of Finite Distinction Systems

Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-X2 develops the fermion-generation paper in the physical bridge sequence of Finite Distinction Systems (FDS) / Distinction Theory. It studies why a weak-sector identity-changing channel requires at least three CKM-participating flavor dimensions if it is to carry a rephasing-invariant CP/T orientation. The central chain is: identity update → T/CP orientation → irreducible CKM phase → NCKM ≥ 3. The hard result is a conditional lower bound. For an N × N CKM-type unitary charged-current mixing matrix, the number of physical complex phases that cannot be removed by quark-field rephasings is (N - 1)(N - 2)/2. Thus N = 1 and N = 2 cannot support an irreducible CKM CP phase, while N = 3 is the first possible case. If weak-sector identity transformation requires a microscopic CP/T-asymmetric carrier, and if the Standard Model charged-current sector is that carrier, then the CKM-participating weak flavor dimension must satisfy NCKM ≥ 3. FDS-X2 explicitly separates this hard lower bound from the stronger and higher-risk claim that the realized theory contains exactly three sequential chiral generations rather than more. CKM algebra permits N > 3. The step from NCKM ≥ 3 to NCKM = 3 is therefore treated as a finite-system minimality bridge, not as another CKM phase-counting theorem. The paper formalizes weak identity transformation as Tid: |dj⟩ → Σi Vij|ui⟩, and uses Jarlskog-type rephasing-invariant quartets to distinguish physical CP/T orientation from removable phase convention. It emphasizes that CKM phase counting supplies a necessary dimensional lower bound, not a sufficient derivation of observed CP violation; nonzero mixing, nondegenerate masses, and a nonzero Jarlskog invariant are additional physical requirements. The paper also includes a PMNS-sector extension, a fourth-generation caveat distinguishing sequential chiral generations from vector-like or sterile states, a section on what X2 does not explain, falsification and demotion conditions, and a relation table connecting X2 to Kobayashi-Maskawa theory, the CPT theorem, Standard Model flavor physics, the Jarlskog invariant, kaon and B-meson CP-violation experiments, PMNS phenomenology, and FDS/DT identity-update logic. Deterministic algebraic demonstrations are included. They illustrate CKM phase counting, quark rephasing, rephasing-invariant orientation, hard/soft claim separation, and a toy flavor-cost functional. These demonstrations are algebraic and normal-form visualizations, not empirical fits. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, and CSV / JSON outputs.