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The Riemann cycling-thought Matrix is a central component in the programming of UNSC Artificial Intelligence constructs. It contains a code directory, where all the impulses in the AI's mind exist. It is effectively the AI's brain. The matrix is not a physical object by itself, rather it is a program, allowing the AI to be stored in a Data Crystal Chip for mobility. Because AIs are essentially copied human minds, the Riemann matrix is likely to be a software model of the human brain, similar to real-life proposals to create AI.
Some AI's have personal fail-safes. Dr. Halsey stated that not all AI have such a fail safe, rather, only the "important" ones. In case of Rampancy or other danger to humans caused by the AI, the fail safe can be activated. It works by generating a pulse beam of high frequency ultraviolet light in the Riemann matrix, destroying the higher thought functions of the AI. This effectively erases the AI. It is unknown how common this information is since Araquiel (Ackerson's AI) failed to recognize the fail safe. This indicates that the information is kept secret from the AIs for obvious reasons.
It should be noted that the matrix failsafe was used by Dr. Halsey to shut down both Araquiel and Kalmiya.