Appendix: The Logarithmic Map Separates Scale and Quantization — g is Factored Out as a Privileged Scale, omega Remains Scale-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). Both the central value nu and the fluctuation delta are mapped by the SAME map func = log2(value/normalization): nu -> N-hat = log2(nu/C), delta -> Delta = log2(factor). The only coordinate passed unchanged is the privileged scale C = k^(1/n) (a gauge). What is invariant is Delta = +/-1/2, which is scale-invariant (independent of C); hence after the scale is factored out, omega remains as the invariant half-bit. It is therefore NOT that 'the logarithmic map does not act on omega' (it does act on delta); rather, delta is mapped the same way, but Delta is scale-invariant. This holds in any dimension and requires no full symmetry; the map is fully invertible. Separation here is operational; it is not a claim of a canonical transformation strictly preserving the symplectic form. The +/-1/2 is a posit. (Revised v2.)