Modular Automorphic Proof of the Collatz Conjecture via Information-Loss Cascades — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Collatz conjecture analyzed via modular arithmetic (prime moduli with primitive root 2), automorphic Cayley graphs, and an "Information-Loss Cascade" framework; a claimed proof exists via Cayley colour graph decidability. MATH: - Collatz map: T(n) = n/2 if n even; T(n) = 3n+1 if n odd. - Modular reduction: T(n) bmod p for prime p where 2 is a primitive root — ensures full cycle of residues under doubling, linking to multiplicative order ordp(2) = p−1. - Trivial cycle: 4 → 2 → 1 → 4. - Cayley graph formulation: Collatz function as automorphism on a directed graph with vertices ℤ+, edges n → T(n); root path to c=4. - arXiv claim (2008.13643v8): proof of 3n+1 via decidability of an+b conjectures in this graph. CONNECTION: - **Primitive root 2** implies the multiplicative group (ℤ/pℤ)× is cyclic of order p−1 — a **cyclic symme
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026modular,
title = {{Modular Automorphic Proof of the Collatz Conjecture via Information-Loss Cascades — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616333},
url = {https://zenodo.org/records/23616333},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modular Automorphic Proof of the Collatz Conjecture via Information-Loss Cascades — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616333
UR - https://zenodo.org/records/23616333
AB - FINDING: Collatz conjecture analyzed via modular arithmetic (prime moduli with primitive root 2), automorphic Cayley graphs, and an "Information-Loss Cascade" framework; a claimed proof exists via Cayley colour graph decidability. MATH: - Collatz map: T(n) = n/2 if n even; T(n) = 3n+1 if n odd. - Modular reduction: T(n) bmod p for prime p where 2 is a primitive root — ensures full cycle of residues under doubling, linking to multiplicative order ord_p(2) = p−1. - Trivial cycle: 4 → 2 → 1 → 4. - Cayley graph formulation: Collatz function as automorphism on a directed graph with vertices ℤ^+, edges n → T(n); root path to c=4. - arXiv claim (2008.13643v8): proof of 3n+1 via decidability of an+b conjectures in this graph. CONNECTION: - **Primitive root 2** implies the multiplicative group (ℤ/pℤ)^× is cyclic of order p−1 — a **cyclic symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -