Appendix: From the Sum-Zero of Fluctuations to Uncertainty — The Uncertainty Relation as the Allocation of a Half-Bit

This appendix develops independently a single point compressed in section 6.3 of the related report (On the Logarithmic Map of Scale and the Separation of Scale and Quantization): that the uncertainty relation in the observable space (constant product) is derived from the sum-zero of fluctuations in the fundamental-quantity N-space. The emphasis is on the direction of the derivation: sum-zero (Delta_a + Delta_b = 0) is placed as the definition, and the constant product is the consequence, the two being connected by the logarithmic map N=log2(nu). Under sum-zero, the half-bit +/-1/2 remains irreducibly on each fundamental quantity with anti-correlated signs, and the product of the observables a*b = 2^K is fixed (whereas the product of the fluctuations delta_a*delta_b is not constant). The continuous squeeze (a hyperbola a*b = 2^K) is the trade-off where tightening one widens the other, with the fixed half-bit as its minimal unit. The +/-1/2 is treated as a posit, not a derivation; no physical identification is made; uncertainty here is not in the sense of Bell's theorem.