Carry-Free Representation in Pascal Rows and Mersenne Bases: The N(p) Bound

This study examines carry-free representations of Pascal rows in Mersenne bases and the function N(p), the index of the last carry-free row for a fixed base. Part I establishes the representation of selected powers of two by row p + 2 and gives central-coefficient relations. Part II develops the N(p) extension through worked examples, threshold sequences, bounds and proofs. Part III describes the accompanying software and its recorded validation scope. The equality proposed for all threshold indices remains an open question. Large parameters do not imply that every digit has been computed or stored. The English manuscript is accompanied by an archive of the public repository, including source code, tests, documentation and Turkish translations. The manuscript and documentation are licensed under CC BY 4.0. Original source code is licensed under MIT. Third-party dependencies retain their own licenses. Source repository: https://github.com/JamesAugustus/pascal-mersenne-rows