What question did this study set out to answer?

This study aims to develop a new cryptographic protocol based on mathematical structures related to the Collatz conjecture.

May 17, 2026Open Access

Collatz-Anchored Inversive Manifold Encryption (CAIME): A Cryptographic Primitive via Tension Symmetry

Key Points

This study aims to develop a new cryptographic protocol based on mathematical structures related to the Collatz conjecture.
Defined a smooth seed pair (a, b) based on Collatz tension distribution with Sk2 = 0
Demonstrated the rarity of smooth pairs at 1.4% empirically
Established computational hardness for the inverse of the encryption transformation.
Proven Feedbackloop Lemma as a structural invariant of the Collatz sequence
Empirical data shows 1.4% rarity of smooth pairs, enhancing encryption security
Achieved O(1) encryption efficiency compared to O(m²·r) for brute-force inversion.

Abstract

We present the Collatz-Anchored Inversive Manifold Encryption (CAIME) protocol, a cryptographic primitive grounded in three independent mathematical structures: the Collatz conjecture's convergence dynamics, complex circle inversion, and the Pearson skewness coefficient as a smoothness gate. The protocol defines a smooth seed pair (a, b) as one whose Collatz tension distribution satisfies Sk2 = 0. This anchor deterministically generates the parameters of a circle inversion transformation IM, whose inverse is computationally hard without knowledge of the anchor. Contributions:- Feedbackloop Lemma: structural invariant of the Collatz sequence (proven)- Smoothness condition: 1.4% rarity of smooth pairs (empirically demonstrated)- Forward-inverse asymmetry: O(1) encryption vs O(m²·r) brute-force inversion- Tamper detection via Euler-Feuerbach relation Related publications:NALP Paper: https://doi.org/10.5281/zenodo.20065953Canonical Audit: https://doi.org/10.5281/zenodo.20066792

Collatz-Anchored Inversive Manifold Encryption (CAIME): A Cryptographic Primitive via Tension Symmetry

Key Points

Abstract

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