The BPS Triple-Kink and Strong CP: Why the Kink-Axion Complement is Required

Paper XVI of the Multi-Kink Generation Framework (MKGF) identified four
resolution paths for the tension between strong CP suppression and the fermion
mass hierarchy on the orbifold S
1/Z2. Path D proposed that a single BPS soliton
of winding number three, with an internal Z3 symmetry, might enforce exact phase
factorization arg det(Mq) = P
a αa across all three generations. We compute the
mass matrix for zero modes of the triple-kink background and find that Path D
does not resolve the tension. The factorization fails with O(1) residuals because the
quasi-zero modes are not strictly localized at individual sub-kinks: their tails extend
into adjacent regions and pick up inconsistent phases. This is the same localization–
delocalization mechanism that breaks KK parity. For a single global phase (all
sub-kinks carrying the same α), the factorization is exact to machine precision
(< 10−15 rad), confirming Paper XV’s Theorem 1. For independent kink phases
(required for CKM CP violation), the raw residual is O(1), reduced to ∼ 10−5–10−7
by the quark mass hierarchy—still above the 10−10 bound. We conclude that the
MKGF requires the kink-axion complement (Path B): the geometric Nelson–Barr
mechanism suppresses ¯θ from O(1) to ∼ 10−7
, and a light axion from the kink
modulus relaxes the residual to zero. The geometric pre-suppression reduces the
axion quality requirement by seven orders of magnitude relative to a standalone
Peccei–Quinn axion.