A Geometric Origin of Flavor II: Quark and Mixing Theorems, a Supergravity Vacuum, and a Falsifiable Neutrino Spectrum from the Degree-Three Monopole Bundle

Abstract

A companion paper derived the charged-lepton sector — Koide’s relation and the mass hierarchy 1 : 206.770 : 3477.5 — from a single geometric object: a degree-three monopole bundle over an internal two-sphere, probed at unit charge by one spinorial doublet. This paper retains the same degree-three family geometry unchanged and extends it to the rest of flavor; the extension to quarks requires explicitly declared mediator and isospin structure, every piece of which is listed in a consolidated ledger. The construction is first placed on dynamical ground, at two explicitly separated levels: the compactification background — Minkowski space times the flux-one sphere — is an exact half-BPS solution of two-derivative six-dimensional gauged supergravity with its Green–Schwarz sector, while the coherent matter sector is a supersymmetric effective construction on that background whose full six-dimensional lift is declared open. On that foundation we prove a structural decomposition of the family algebra — the nine parameters of any family-space Hermitian structure split into three circulant (masses), three reality-even transverse (mixing angles), and three reality-odd (CP) directions — from which VCKM = ⊮ follows as a zeroth-order theorem: quark mixing measures the relative tilt of family frames, and vanishes exactly when the frames coincide. Deformations transmit through the heavy portal with an automatic square-root dressing, θij ∝ p mi/mj — a mechanism-level account of the Gatto–Sarma–Tomozawa relation — while the familiar texture-zero mechanism (M11 = 0, M12 ̸= 0) is proven unrealizable within the positive portal-induced mass operators of this framework. The deformation channels available to the frozen field content are classified exhaustively (all seat-field-only, projectively even polynomial operators on the extremal vacuum, to all orders, plus the unique seat–carrier bridge): precision of the lepton sector itself excludes the seat-field route, forcing a two-spurion architecture whose orientation is set, with no adjustable angle, by the lepton orientation −2/9 derived in the companion paper; fed through the fixed Clebsch algebra, that angle suppresses exactly the 1–3 link that nature suppresses. a triangle-closure theorem makes the 1–3 link necessary for any rephasing-invariant CP phase (which the standard convention then places in that corner), carried by a reality-odd channel that a holomorphy argument pins near maximal phase, subject to three declared conditions; the magnitude and phase channels are graded by the reality structure and provably do not interfere. The capacity of the architecture is then measured rather than asserted: with all derived structure and three scale parameters, the framework predicts one exact relation among (|Vus|, |Vcb|, |Vub|,sin δ), from which the measured values deviate by a uniform ∼ 40% — a floor computed exactly, with the missing ingredient identified and priced. Neutrinos are Dirac by construction; their exemption from the shared frame explains the coexistence of large leptonic and small quark mixing, and a frozen spectrum (0.36, 8.7, 50) meV, normal ordering, stands falsifiable by oscillation, cosmological, and future endpoint measurements. A sectorcleanliness law — deviation from Koide ordered by portal dressing: charged leptons 10−5 , down quarks +12%, up quarks +33% — and a spectral appendix classifying the companion paper’s one declared unit (the Schur commutant of the irreducible family space, protected by a family of no-go theorems) complete the accounting. Throughout, derived, conditional, and open statements are separated explicitly; every negative result is stated with the same care as the positive ones.