Rigidity of Three-Input Majority Compositions on Odd Cyclic Networks

This unrefereed research preprint studies exact density classification in an architecture-constrained cyclic Boolean network. For odd n = 2m + 1, the outer operator A applies the majority of n−2 consecutive cyclic inputs at every site, while the middle operator B applies a uniform majority of three taps with offsets 0, a and b. We characterize when A ∘ B ∘ A outputs the strict global majority consensus for every Boolean input. For every odd n ≥ 25, this identity holds if and only if the unordered circular gap multiset of the three taps is (1, m, m). The successful configuration works for all odd n ≥ 5. The n ≥ 25 threshold is sharp; additional successful gap configurations exist for n = 21 and n = 23. The preprint includes combinatorial proofs, explicit counterexample families and finite verification evidence. The result is an exact classification for this prescribed translation-equivariant architecture, not a claim of improved general majority-circuit depth or fan-in. The research involved substantial generative AI assistance and independent AI cross-checking. No human peer review has occurred. Scholarly priority and novelty remain INCONCLUSIVE. Some relevant prior-publication metadata was checked, but full-text equivalence was not comprehensively established. The PDF and a ZIP archive containing the complete version 1.2 paper, source, claim/audit matrix, prior-art notes and reproducibility evidence accompany this record. Manuscript version 1.2 is dated 20 October 2026; the first public GitHub posting was 21 October 2026. GitHub: https://github.com/yonezaemon1-hub/no30-three-input-majority-rigiditySource GitHub commit: 3d1484ba41d92318536572dc40d2daf3caba7ffc Research Inquiries — Yonezu Research LabContact: yonezaemon1+research@gmail.com © 2026 Ryutaro Yonezu. All rights reserved.