The MSJS Finite-State Framework for Resolving the Collatz Conjecture: Master Reference Compendium & Grand Unified Mathematical Theory

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-15
DOI
https://doi.org/10.5281/zenodo.22776310
Primary Topic
Benford’s Law and Fraud Detection
Type
preprint
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preprint

The MSJS Finite-State Framework for Resolving the Collatz Conjecture: Master Reference Compendium & Grand Unified Mathematical Theory

mahmoud sabri jaber (sobeh)
Zenodo (CERN European Organization for Nuclear Research)
Benford’s Law and Fraud Detection
preprint

The MSJS Finite-State Framework for Resolving the Collatz Conjecture: Master Reference Compendium & Grand Unified Mathematical Theory

mahmoud sabri jaber (sobeh)
preprint en

Abstract

This master compendium presents the comprehensive mathematical resolution of the Collatz Conjecture (3n+1) via the Modular Special Jump System (MSJS) framework. The framework rigorously decouples any natural number N into two orthogonal components—Quality (the deterministic modular pattern S_n) and Quantity (the digital magnitude fuel M)—reducing the infinite Collatz dynamics to a closed, localized, finite-state machine (PSM). The global convergence to the unique attractor (1 → 4 → 2 → 1) is algebraically established through three pillars: (1) modular isolation and universal scale invariance; (2) the sovereign contraction inequality 2^D > 3^R; and (3) the inescapable terminal highways—the Backbone (powers of 2) and Decimal Bridge 5. The document also includes: (a) cryptographic applications including the Rippling Carries One-Way Function and the Mahmoud Sabri One-Way Conjecture; (b) a complete Python simulator (v15) for verification; and (c) an exploratory conjecture on a possible homology between the MSJS contraction factor λ = 1/2 and the Riemann Zeta critical line Re(s) = 1/2. The Riemann homology is presented strictly as a conjecture, not a proof. Keywords: Collatz conjecture, finite-state reduction, pattern invariance, dynamical systems, finite automata, cryptographic one-way functions, Riemann Zeta. --- Author's Name Note: The official family name is registered as "Suobeh" (as per ORCID and official identification documents). The alternative spelling "Sobeh" represents the phonetic pronunciation of the same family name in English.

Zenodo (CERN European Organization for Nuclear Research)
Oldham Council (GB)
Benford’s Law and Fraud Detection
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