Closure Theory: The Fine Structure Constant and the Geometric Origin of Charge in Closure Theory

This paper develops an observable-domain formulation of Closure Theory (CT) and investigates a geometric route from internal momentum participation to the emergence of electric charge. The central result is the projection chain pᵢnt → ρ → χ₃ → α → e, where pᵢnt is an observable internal momentum participation, ρ is a dimensionless winding participation factor, χ₃ is the first non-trivial odd projection invariant, α is the fine-structure constant, and e is the elementary charge. The paper develops the observable ACORN formulation, examines its limiting cases, and introduces a hierarchy of projection invariants. Particular attention is given to odd-sector structure and the role of the cubic invariant χ₃ as the first surviving odd observable after projection symmetry constraints are applied. Within this framework, the fine-structure constant is interpreted as an emergent consequence of projection architecture rather than an externally imposed parameter. Observable electric charge is then viewed as arising through a sequence of geometric projection stages linking internal momentum participation to electromagnetic observables. The work does not claim a complete derivation of charge from first principles. Instead, it presents a coherent geometric architecture connecting internal momentum participation, projection structure, the fine-structure constant, and electric charge within the broader Closure Theory programme. This publication forms part of the ongoing Closure Theory research programme investigating the relationships between mass, charge, electromagnetism, gravity, and relativistic projection geometry.