It is shown that modern number systems suffer from a significant drawback in the form of zero code redundancy. Unlike the classical binary code, the Fibonacci code is inherently redundant. Furthermore, its redundancy manifests through the property of multi-valued representation (as a binary code) of natural numbers and is not constant: the code redundancy varies significantly for individual numbers. Based on the Fibonacci number system featuring Fibonacci codes with unrestricted redundancy, a rule for limiting the Fibonacci code redundancy has been formulated: numbers with an even digit weight can only be represented alongside numbers with an odd digit weight. Consequently, natural numbers can be represented by exactly two (and only two) code variants due to the introduced Rule. The development of an optimal arithmetic algorithm for the New Fibonacci Number System (NFNS) must be based on the fundamental principles of classical Fibonacci arithmetic (developed by A.P. Stakhov), while strictly integrating the author's rule: a prohibition on adjacent positions of identical parity. In the NFNS, where consecutive odd numbers of zeros are forbidden (except at the end of the code string), the mathematical framework shifts toward dual representation. The paper presents: the architecture of the NFNS arithmetic algorithm; the mathematical description of the NFNS arithmetic algorithm; and an example of adding specific numbers (7 + 8) in the NFNS. The primary advantage of the presented addition algorithm for the NFNS is that redundancy is utilized for error control directly during the computation process. If, at stage 3, the finite-state machine detects that an intermediate number lacks exactly 2 representation options (or the code shifts into a forbidden combination), the processor instantly diagnoses a fault (an attack or an error).