The Hopfion Universe Part II : A 4D Topological Encoding of Spacetime, Matter, and Quantum Fields via Conserved Hopf Charge

We propose that spacetime’s fundamental structure is a 4-dimensional Hopfion—a topologically
non-trivial soliton with a conserved, non-zero Hopf invariant Q ∈ Z [2], generated by the
recursive normed division algebra Kn = Kn−1 ⊗ (R ⊕ Ren ) (with e2n = −1, en x = −xen for
x ∈ Kn−1 , K0 = R [4]). Matter, forces, and time emerge from nested Hopf fibrations up to
the octonion level (n = 3), with quarks as topological data units encoding Q, unisolatable
due to infinite energy. Gravity arises as a residual electromagnetic energy bleed (scaling
as Q/m2 ), while entropy reflects irreversible topological hashing, validated by simulation
variance drops (e.g., 1.207 × 10−1 to 1.088 × 10−1 on a 323 lattice). Simulations confirm a
stable Q ≈ 1.97, supporting the model. Recent LHCb results (Nature, 2025) report a 2.45%
CP violation asymmetry in baryon decays [3], consistent with a topological Λ ∝ Q/(4π 2 ℓ4P ).
This framework unifies physics, refining the concepts introduced in our prior work [7] by
incorporating the recursive division algebra framework and expanded simulation validations,
predicting testable signatures (e.g., fractal CMB) falsifiable by 2025 experiments (DESI, LHC,
LISA).