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)

Q: What does this research mean for the field?

A single wave cannot carry information; rather, the minimal unit of information is a zero-closure block (a two-wave pair), and the indistinguishability of identical particles emerges as a dynamical consequence of this system. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Noriaki Kihara

doi:10.5281/zenodo.22729128

September 12, 2026Open Access

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)

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Key Points

To determine the minimal unit of information that can be carried within relational-wave closed systems under strict gauge anonymity constraints.
Analyzed terminal states and edge labels using permutation gauge symmetry (S_N) frameworks.
Constructed abstract closure moduli spaces C_k = Q_{k-2}/S_k based on projective quadrics across dimensions k >= 2.
Evaluated discrete symmetry emergence using cyclic-Fourier high-symmetry constraints.
Demonstrated that a single wave cannot carry invariant information and that the minimal non-trivial zero-closure block consists of a two-wave pair (z_2 = ±i z_1).
Proved that abstract closure alone yields continuous moduli for k >= 3, requiring cyclic-Fourier symmetry to establish discrete block sizes p = spf(L).
Showed that recording data beyond fundamental coordinate pairs on a wave inherently violates gauge anonymity, making particle indistinguishability a dynamical outcome.

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Implication

Theoretical analysis reveals the minimal informational capacity of wave pairs in closed relational systems, indicating that identical particle indistinguishability emerges dynamically.

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 = Qₖ₋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.

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Noriaki Kihara (2026) studied this question.

synapsesocial.com/papers/6aa7b3d00926e14a848b3001 https://doi.org/10.5281/zenodo.22729128

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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)

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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)

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