Minimal Nonzero CP/T Orientation and Full Weak-Sector Consistency CKM-PMNS Orientation Capacity, Texture Independence, and the Three-Family Threshold in 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 v2.0 is the canonical integrated version of the X2 paper in the Finite Distinction Systems (FDS) / Distinction Theory sequence. It combines the minimal nonzero CP/T orientation theorem with the full weak-sector CKM-PMNS consistency extension. The hard theorem in FDS-X2 is the Dirac orientation-capacity theorem. For an N × N Dirac-type charged-current mixing matrix, the number of independent rephasing-invariant Dirac CP/T orientation phases is Corient(N) = (N - 1)(N - 2) / 2. Therefore N = 1 and N = 2 carry no primitive Dirac CP/T orientation, N = 3 carries exactly one, and N > 3 carries multiple independent orientation phases. For one primitive Dirac orientation demand, the unique minimal flavor dimension is therefore three. The canonical FDS-X2 bridge applies this theorem to the CKM sector. If the audited CKM weak identity-update sector demands one primitive Dirac CP/T orientation, Dq = 1, then, under finite orientation minimality, the minimal CKM-participating quark-family dimension is Nq = 3. The PMNS sector is treated as a conditional full weak-sector consistency extension. If the Dirac-type PMNS charged-current interface also demands one primitive CP/T orientation, Dℓ = 1, then the same orientation-capacity threshold gives Nℓ = 3. Thus, under the finite-minimality bridge, Dq = Dℓ = 1 ⇒ Nq = Nℓ = 3. This is a conditional minimality and consistency theorem. It is not a derivation of the Standard Model, fermion masses, Yukawa matrices, CKM angles, PMNS angles, baryogenesis, anomaly cancellation, the neutrino mass mechanism, or the Standard Model gauge group. It also does not claim that FDS derives three fermion generations unconditionally. The paper separates orientation capacity from mixing texture. CKM and PMNS may have radically different matrix-element patterns while sharing the same minimal N = 3 Dirac-orientation capacity. The CKM matrix is close to diagonal, while the PMNS matrix is broadly mixed; FDS-X2 constrains the minimum dimension needed for one Dirac orientation channel, not the numerical values of mixing angles. Majorana phases, if present, are treated separately. They are classified as identity-locking or lepton-number boundary data, not as ordinary oscillation Dirac orientation. Thus the PMNS extension fails if Majorana phases are treated as replacements for the Dirac oscillation phase. The paper also gives a finite-minimality normal form illustrating surplus-orientation overhead beyond the first required primitive orientation channel. This normal form is not a fundamental flavor potential and is not used as a proof of the algebraic theorem. FDS-X2 v2.0 includes: the Dirac orientation-capacity theorem; the minimal dimension formula for primitive orientation demand; the canonical CKM result Dq = 1 ⇒ Nq = 3; the conditional PMNS extension Dℓ = 1 ⇒ Nℓ = 3; the full weak-sector consistency condition Dq = Dℓ = 1 ⇒ Nq = Nℓ = 3; a texture-independence analysis for CKM and PMNS; a Majorana-phase firewall; demotion and falsification conditions; deterministic algebraic and normal-form demonstrations. The release includes the paper PDF, LaTeX source, reproducibility code, generated figures, CSV tables, and model summary outputs.