What question did this study set out to answer?

To prove the Collatz conjecture by linking its operations to geometric properties of a hexagonal lattice.

March 7, 2026Open Access

The Collatz Conjecture as Registry Grounding Protocol: Deriving Universal Convergence from Hexagonal Coordination and Bilateral Manifold Structure

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To prove the Collatz conjecture by linking its operations to geometric properties of a hexagonal lattice.
Utilized the Cymatic K-Space Mechanics framework based on a 2D hexagonal lattice model.
Analyzed the operations of n/2 and 3n+1 in the context of lattice coordination and manifold structure.
Established conditions for universal convergence through empirical falsification criteria.
Demonstrated that every odd operation (3n+1) leads to an even number, supporting convergence.
Derived that the sequence always reduces to 1, the unique ground state with zero parity error.
Validated that net cooling dominates over heating, confirming convergence by geometric descent toward the origin.

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Demonstrates universal convergence through a novel framework in discrete hexagonal geometry, suggesting new physics implications.

Geoffrey Howland (2026) studied this question.

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