We present a framework in which the universe consists of two Riemannian 3-manifolds—a spher-ical ball B3 and a hyperbolic ball H3—sharing an interface at |z| = 1. A single sign flip in theconformal factor, (1 + |z|2) →(1 −|z|2), generates the dual structure. Particles are identified withknots embedded at this interface, and the mass of each particle is given by m(K) = m0 ·exp(V (K)),where V (K) is the hyperbolic volume of the knot complement—a topological invariant fixed byMostow rigidity—and m0 = 0.06712 MeV is a normalisation constant determined by identifyingthe electron with the figure-eight knot (41).This formula, derived from the magnitude of the analytically continued SL(2, C) Chern–Simonspartition function, reproduces the masses of 12 Standard Model particles with zero tunable mod-uli, requiring only a single global scale factor to calibrate the topological volumes to physicalenergy scales: 9 fermions (electron 0.00%, up 1.54%, down 0.09%, strange 0.15%, muon 0.42%,charm 0.28%, tau 0.06%, bottom 0.00%, top 0.02%) and 3 bosons (W 0.01%, Z 0.03%, Higgs 0.01%).Mean fermion error: 0.28%. Statistical significance: p < 0.00002 (50,000 Monte Carlo trials usingthe actual knot volume distribution).The formula extends naturally to composite particles. We identify the proton with the 3-component hyperbolic link o10 149348, with volume V = 9.540, predicted mass 933.1 MeV (0.55%error), and a 2 + 1 cusp structure mirroring the uud quark content. The Chern–Simons invariantof the link complement encodes parity, uniquely selecting this manifold from a volume-degeneratetwin and retrodicting the absence of a light parity-odd baryon partner.All computations use SnapPy 3.3.2 and are reproducible. A verification script is provided asancillary material
Bastin et al. (Mon,) studied this question.