Anonymity and the Minimal Unit of Information: A Single Wave Cannot Carry Information, and the Minimal Zero-Closure Block (Two-Wave +-i Pair) Carries the S_N-Invariant Type (Exploratory Paper)

Abstract

Paper 9 of the series "Foundations of Self-Consistent Relational-Wave Closed Systems" (10 papers). An exploratory/study paper. Edge labels are an S_N gauge; terminal states are exactly equimodular and indistinguishable by value, so the indistinguishability of identical particles is a dynamical consequence. The nontrivial solutions of z^2=0 are minimally two waves (z_2=+-i z_1, a two-valued +-), but +- is not an S_N-invariant anonymous bit (S_2 flips it). The moduli of the abstract closure space are C_k = Q_{k-2}/S_k (a projective quadric, dim_C=k-2): k=2 collapses to one orbit, k>=3 is continuous and connected, so closure alone does not generate discrete species. Discreteness = the block size p=spf(L) fixed by the discrete symmetry of the cyclic-Fourier high-symmetry floor. Recording anything beyond (a,b) on a wave violates anonymity. Author: Noriaki Kihara (WF System Co., Ltd.), ORCID 0009-0004-6753-4020.