Collatz Conjecture Recast as Modular Tree Dynamics — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Collatz research clusters around modular-prime frameworks, automorphic Cayley graphs, and an "Information-Loss Cascade" — none yet yield a proof, but the structure is being recast as a discrete dynamical system on a rooted tree with modular periodicity.
MATH: - Core map: T(n) = n/2 & n ≡ 0 pmod2 3n+1 & n ≡ 1 pmod2 - Modular approach: choose prime P with 2 as primitive root → the map acts as a permutation on residues mod P, forcing cycles of length ordP(2) — this is the key structural constraint. - Cayley graph formulation: Collatz function as automorphism on a colour graph; the trivial cycle 4 → 2 → 1 → 4 is the root. The arXiv paper claims decidability of an+b conjectures via this graph. - Information-Loss Cascade: a proposed measure of how many binary digits are irreversibly lost per iteration — likely tied to log2(3) ≈ 1.585 for odd steps (since \( 3
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026collatz,
title = {{Collatz Conjecture Recast as Modular Tree Dynamics — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635274},
url = {https://zenodo.org/records/23635274},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Collatz Conjecture Recast as Modular Tree Dynamics — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635274
UR - https://zenodo.org/records/23635274
AB - FINDING: Collatz research clusters around modular-prime frameworks, automorphic Cayley graphs, and an "Information-Loss Cascade" — none yet yield a proof, but the structure is being recast as a discrete dynamical system on a rooted tree with modular periodicity. MATH: - Core map: T(n) = n/2 & n ≡ 0 pmod2 3n+1 & n ≡ 1 pmod2 - Modular approach: choose prime P with 2 as primitive root → the map acts as a permutation on residues mod P, forcing cycles of length ord_P(2) — this is the key structural constraint. - Cayley graph formulation: Collatz function as automorphism on a colour graph; the trivial cycle 4 → 2 → 1 → 4 is the root. The arXiv paper claims decidability of an+b conjectures via this graph. - Information-Loss Cascade: a proposed measure of how many binary digits are irreversibly lost per iteration — likely tied to log_2(3) ≈ 1.585 for odd steps (since \( 3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -