Appendix: The forward / inverse maps (nu, delta) <-> (N, Delta) — canonicalization by the conserved quantity, a unified map, the privileged scale, and full invertibility
Year: 2026
Type: preprint
Abstract: This appendix gives a rigorous formulation, in a single unified form, of the forward and inverse maps between the observable side (nu, delta) and the fundamental-quantity side (N, Delta). The key is canonicalization (normalization) by the conserved quantity on the right-hand side of the identity. As a result: (1) every anonymous quantity (the central value nu, the fluctuation delta, the conserved value and its fluctuation) is mapped by the SAME map func(value/normalization) = log2(value/normalization); (2) the scale (the conserved quantity) used for normalization, C = k^(1/n), is the only privileged (non-anonymous) coordinate passed unchanged; (3) the inverse map has the same form, so both the individual values and the fluctuations are recovered exactly. The map nu <-> N is fully invertible (the central shape N-hat = log2(nu/C) with sum zero, and the fluctuation Delta = log2(factor)); non-integers are absorbed by the privileged scale. Discarding the scale C is the canonicalization corresponding to gauge-fixing of g (the g/omega separation, section 7.3 of the parent paper). A worked numerical check is included. A discrete system; 1/2 is a posit; no physical identification is made. Supplement to the parent report (Concept DOI 10.5281/zenodo.20740841).
Author: Kihara, Noriaki
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165133776
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