Closed Four-Degree-of-Freedom Structure and Its Correspondence with 4-Dimensional Lattice Counting: A Geometric Organization from the 5-Component Sum-of-Squares Constraint to the Unit-Cell Counting Region

This short supplement to Paper 1 organizes, without physical interpretation, the correspondence between the 5-component sum-of-squares constraint and the 4-dimensional lattice counting of Paper 2. The 5-component constraint sum xₙ² = R² defines a closed four-degree-of-freedom object (the 4-sphere S⁴R in R⁵), whereas the actual unit-cell counting is carried out in the 4-dimensional ball region B⁴R. We note that the radial projection PiR (y) = R y / ||y|| acts as the identity on points already satisfying the constraint (lambda' = lambda, nu' = nu), so the projection is not a transformation of values but a geometric description that reads constraint-satisfying points as points on a closed four-DOF structure of constant radius. No correspondence with spacetime, energy, momentum, gravity, or physical constants is claimed.