Why Discrete? Topological Quantisation in Field Topology Theory

Abstract. The apparent discreteness of quantum numbers — spin, charge, baryon number — is typically imposed as an axiom rather than derived from first principles. We show that discreteness emerges necessarily from the topology of the space of Lorentzian metrics on a four-dimensional manifold. The fundamental group π₁(ℝP³) = ℤ₂ establishes that the configuration space of Lorentzian metrics admits non-contractible loops, and by the Finkelstein-Rubinstein theorem, topological defects in such a space automatically carry half-integer spin and fermionic exchange statistics. Fermion statistics are therefore not imposed — they are built into Lorentzian geometry itself. We identify the Skyrme model as the established precedent for this programme and specify the Wess-Zumino-Witten term required to enforce fermionic statistics in the Field Topology Theory Lagrangian. The diffeomorphism invariance problem is resolved through asymptotic framing — fixing the metric frame at spatial infinity protects the global topological charge against coordinate transformation. Fleming’s two electromagnetic rules are identified as frame-dependent perspectives of one geometric interaction, connecting FTT directly to Einstein’s 1905 derivation of special relativity; CPT invariance follows as geometric necessity. Planck’s constant ℏ is identified as the minimal symplectic flux of the 4D Lorentzian vacuum itself — preceding and determining the properties of any particle; the electron is the minimum stable defect costing exactly one ℏ unit of vacuum action. The McKay correspondence anchors the first fermion generation to E₆ via binary tetrahedral symmetry, and the Koide mass formula parameter b = −√2 is confirmed to six decimal places. FTT’s derivation of gauge structure from 4D topological winding is strictly distinct from Kaluza-Klein theory, requiring no extra dimensions. Three unsolved problems are documented honestly: the gravitational WZW term, RG running, and the completion of the gauge group derivation.