Appendix: The Logarithmic Map Separates Scale and Quantization — g is Flattened, omega Remains Invariant

This appendix develops the core of the related report, section 7.3: that flattening by the logarithmic map is not a loss of information but a separation of scale (g) and quantization (omega). The two independent real conserved quantities g = sum nu^2 = R^2 (the real part, scale, the Born-norm side) and omega = delta1 delta2 = 1/2 (the imaginary part, area, the quantization side) correspond to the real and imaginary parts of the complex inner product g + i omega. In the fully symmetric limit the logarithmic map acts on g and flattens it to a single value (a gauge), while omega is untouched and remains as the invariant half-bit on the flat surface. Thus even when scale degenerates, quantization survives, because quantization was never in g (the mean) but in omega (the fluctuation). Separation here is operational (the map flattens g and preserves omega); it is not a claim of a canonical transformation strictly preserving the symplectic form. The +/-1/2 is a posit. Holds only in the limited (k=1, fully symmetric) regime.