Appendix: On the Emergence of Multidimensional Structure from the Logarithmic Fundamental Quantity — A Formal Observation on the Dimension of the Hyperplane Defined by the Additive Identity

This appendix gives a concrete, formal example of the structure that the related report (On the Logarithmic Map of Scale and the Separation of Scale and Quantization) said arises under strong constraints but did not exhibit. From the minimal fundamental quantity N=log2 (nu) and the additive identity of the conjugate relation alone, taking n copies of the fundamental quantity and holding their sum constant, the identity Sum Nᵢ = K defines an (n-1) -dimensional hyperplane in n-dimensional coordinate space. The cases n=2 (a line), n=3 (a plane, a triangle in the positive region), and n=4 (a 3-dimensional hyperplane, a tetrahedron / 3-simplex) are shown with figures. The multidimensionality does not spring from a single axis but comes from the number n of variables placed; placing n>=3 distinct variables presupposes the dimensional generation of section 3 of the related paper (the genesis of an integer mode number via boundedness, i. e. , the breaking of anonymity). No physical connection, no special role of four dimensions, and no necessity is claimed; this is a formal illustration only. Contraction under full symmetry is left to a separate appendix.